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Volume I - First Edition: December 2019

e: Rs. 1,000/-

CA Karan Chandwani

[email protected]

VSmart Academy

+91 88883 88886

Every effort has been made to present this publication or book in the most authentic form without any errors

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PREFACE

Dear Students,

I am pleased to present before you the study material on Business Mathematics (Volume I of Business

Mathematics, Statistics and Logical Reasoning) for CA Foundation. This book is strictly based upon the latest

syllabus by Board of Studies, ICAI and therefore comprehensively covers everything. I have tried to incorporate

all my practical experiences and learnings in such a way that it helps you all to understand and remember in a

better way.

I always focus on understanding the concepts first but also equally on remembering it so that you can recollect

it when required during the exams. I am sure that this book will surely help you score better in your exams!

I have been blessed to have such an extraordinary support from the colleagues, friends and family who have

helped me in every sphere of my life. These people deserve much more than a deep thanks and love! I express

gratitude to each and every-one for assisting me in all my endeavours.

I am thankful to IDT Guru CA Vishal Bhattad Sir for his continuous support and mentoring.

Thanks to the student community. I am grateful for the opportunity to be of service to you. It has always

been a pleasure spending time with you all, teaching you and learning so much from you all.

I express my respect, love and gratitude to my Parents for not only giving me life but giving your entire life to

me. I am indebted to you a lot.

Let me remind you two important things:

1. This book is not a substitute for study material issued by the BOS, ICAI; it’s only an

aid.

2. There is no shortcut to success; it is resolute hard work that pays.

Best Wishes and Happy Learning!

Constantly yours,

CA Karan Chandwani

+91 8625 00 1263

[email protected]

V’Smart Academy, Pune

Let’s stay connected…

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INDEX

BUSINESS MATHEMATICS (40 Marks)

HEADINGS

SUB HEADINGS

Information and Text

Important words to be noted

Formulae

SR.

NO.

PARTICULARS WEIGHTAGE AS PER ICAI

PAGE NO.

01 Ratio, Proportion, Indices and Logarithm

02 A Equations

02 B Matrices

03 Linear Inequalities

04 Time Value of Money

05 Basic Concepts of Permutations and Combinations

06 Sequence and Series (AP & GP)

07 Sets, Functions and Relations

08 A Basic Concepts of Differential Calculus

08 B Basic Concepts of Integration Calculus

CHAPTER 01: RATIO, PROPORTION, INDICES & LOGARITHM

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RATIO:

A ratio is a comparison of the sizes of two or more quantities of the same kind by division. If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a:b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent. Both terms of a ratio can be multiplied or divided by the same (non–zero) number. Usually a ratio is expressed in lowest terms (or simplest form). Example 1: 12 : 16 = 12/16 = (3 × 4)/(4 × 4) = 3/4 = 3 : 4 The order of the terms in a ratio is important. Example 2: 3 : 4 is not same as 4 : 3. Ratio exists only between quantities of the same kind. Example 3: (i) There is no ratio between number of students in a class and the salary of a teacher. (ii) There is no ratio between the weight of one child and the age of another child. Quantities to be compared (by division) must be in the same units. Example 4: (i) Ratio between 150 gm and 2 kg = Ratio between 150 gm and 2000 gm = 150/2000 = 3/40 = 3 : 40 (ii) Ratio between 25 mins & 45 secs =Ratio between (25×60) sec. & 45 sec. = 1500/45 = 100/3 = 100 : 3 Example 5: (i) Ratio between 3 kg & 5 kg = 3/5 To compare two ratios, convert them into equivalent like fractions. Example 6: To find which ratio is greater: 21/3, 31/3, 3.6, 4.8 Solution: If a quantity increases or decreases in the ratio a:b then new quantity = b of the original quantity/a. The fraction by which the original quantity is multiplied to get a new quantity is called the factor multiplying ratio. Example 7: Ronak weighs 56.7 kg. If he reduces his weight in the ratio 7 : 6, find his new weight. Solution: APPLICATIONS: Example 8: Simplify the ratio 1/3 : 1/8 : 1/6 Solution:


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Example 9: The ratio of the number of boys to the number of girls in a school of 720 students is 3 : 5. If 18 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the number of boys to the number of girls may change to 2 : 3. Solution:

INVERSE RATIO:

One ratio is the inverse of another if their product is 1. Thus a : b is the inverse of b : a and vice- versa. 1. A ratio a : b is said to be of greater inequality if a>b and of less inequality if a<b. 2. The ratio compounded of the two ratios a : b and c : d is ac : bd. For example: compound ratio of 3 : 4 and 5 : 7 is 15 : 28. Compound ratio of 2 : 3, 5 : 7 and 4 : 9 is 40 : 189. 3. A ratio compounded of itself is called its duplicate ratio. Thus a2 : b2 is the duplicate ratio of a : b. Similarly, the triplicate ratio of a : b is a3 : b3. For example, duplicate ratio of 2 : 3 is 4 : 9. Triplicate ratio of 2 : 3 is 8 : 27. 4. The sub-duplicate ratio of a : b is √a : √b and the sub-triplicate ratio of a : b is 3√a : 3√b. 5. If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. 6. Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c. APPLICATIONS: Example 1: The monthly incomes of two persons are in the ratio 4:5 and their monthly expenditures are in the ratio 7:9. If each saves Rs. 50 per month, find their monthly incomes. Solution: Example 2: The ratio of the prices of two houses was 16:23. Two years later when the price of the first has increased by 10% and that of the second by Rs. 477, the ratio of the prices becomes 11:20. Find the original prices of the two houses. Solution:



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PROPORTIONS

If the income of a man is increased in the given ratio and if the increase in his income is given then to find out his new income, Proportion problem is used. Again, if the ages of two men are in the given ratio and if the age of one man is given, we can find out the age of another man by Proportion. An equality of two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion if a:b = c:d (also written as a:b :: c:d) i.e. if a/b = c/d i.e. if ad = bc. The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).

When three or more numbers are so related that the ratio of the first to the second, the ratio of the second to the third, third to the fourth etc. are all equal, the numbers are said to be in continued proportion. We write it as: x/y = y/z = z/w = w/p = p/q = when: x, y, z, w, p and q are in continued proportion.

APPLICATIONS: Example 1: Find the value of x if 10/3 : x : : 5/2 : 5/4. Solution: Example 2: Find the fourth proportional to 2/3, 3/7, 4. Solution: Example 3: Find the third proportion to 2.4 kg, 9.6 kg. Solution: Example 4: Find the mean proportion between 1.25 and 1.8. Solution:



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PROPERTIES OF PROPORTION:



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Example 1: If a : b = c : d = 2.5 : 1.5, what are the values of ad : bc and a + c : b + d? Solution: Example 2: Solution: Example 3: A dealer mixes tea costing Rs. 6.92 per kg. with tea costing Rs. 7.77 per kg and sells the mixture at Rs. 8.80 per kg and earns a profit of 17 ½ % on his sale price. In what proportion does he mix them? Solution:

INDICES

We know that the result of a repeated addition can be held by multiplication. e.g.: 4 + 4 + 4 + 4 + 4 = 5(4) = 20

a + a + a + a + a = 5(a) = 5a Now, 4 × 4 × 4 × 4 × 4 = 45;

a × a × a × a × a = a5. It may be noticed that in the first case 4 is multiplied 5 times and in the second case ‘a’ is multiplied 5 times. In all such cases a factor which multiplies is called the “base” and the number of times it is multiplied is called the “power” or the “index”. Therefore, “4” and “a” are the bases and “5” is the index for both. Any base raised to the power zero is defined to be 1; i.e. ao = 1. We also define:

LAW 1 Now, we extend this logic to negative integers & fractions. 1st let’s consider this for negative integer, i.e. m will be replaced by –n. By am × an = am+n; we get a–n × an=a–n+n = a0 = 1



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Example 1: Simplify 2x 1/2 3x -1 if x = 4 Solution: Example 2: Simplify 6ab2c3 × 4b–2c–3d. Solution:

LAW 2 Example 3: Find the value of: Solution: Example 4: Simplify Solution:



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LAW 3:

LAW 4:

Example 5: Simplify (xa.y–b)3 . (x3 y2)–a Solution: Example 6: Solution:



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Example 7: Find x, if Solution: Example 8: Find the value of k from Solution:

LOGARITHM The logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number, i.e. to make it equal to the given number. If there are three quantities indicated by say a, x and n, they are related as follows: If ax = n, where n > 0, a > 0 and a not equal to 1 then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed as follows: logan = x i.e. the logarithm of n to the base ‘a’ is x 1. The two equations ax = n and x = logan are only transformations of each other and

should be remembered to change one form of the relation into the other. 2. The logarithm of 1 to any base is zero. This is because any number raised to the power

zero is one. Since a0 = 1, loga1 = 0. 3. The logarithm of any quantity to the same base is unity. This is because any quantity

raised to the power 1 is that quantity only. Since a1 = a, logaa = 1.

FUNDAMENTAL LAWS OF LOGARITHM 1. Logarithm of the product of two numbers is equal to the sum of the logarithms

of the numbers to the same base, i.e.

2. The logarithm of quotient of two numbers is equal to the difference of their logarithms to the same base, i.e. 3. Logarithm of the number raised to the power is equal to the index of the power multiplied by the logarithm of the number to the same base i.e.

CHANGE OF BASE If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation:



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Example 1: Change the base of log531 into the common logarithmic base. Solution: Example 2: Solution:

LOGARITHM TABLES: The logarithm of a number consists of two parts, the whole part or the integral part is called the characteristic and the decimal part is called the mantissa where the former can be known by mere inspection, the latter has to be obtained from the logarithm tables.

CHARACTERISTIC: The characteristic of the logarithm of any number greater than 1 is positive and is one less than the number of digits to the left of the decimal point in the given number. The characteristic of the logarithm of any number less than one (1) is negative and numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be –1. The following table will illustrate it.

All numbers lying between 1 and 10 i.e. numbers with 1 digit in the integral part have their logarithms lying between 0 and 1. Therefore, their integral parts are zero only. All numbers lying between 10 and 100 have two digits in their integral parts. Their logarithms lie between 1 and 2.

Therefore, numbers with two digits have integral parts with 1 as characteristic. In general, the logarithm of a number containing n digits only in its integral parts is (n – 1) + a decimal. For example, the characteristics of log 75, log 79326, log 1.76 are 1, 4 and 0 respectively.



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NEGATIVE CHARACTERISTICS All numbers lying between 1 and 0.1 have logarithms lying between 0 and –1, i.e. greater than –1 and less than 0. Since the decimal part is always written positive, the characteristic is –1. All numbers lying between 0.1 and 0.01 have their logarithms lying between –1 and –2 as characteristic of their logarithms. In general, the logarithm of a number having n zeros just after the decimal point is – (n + 1) + a decimal. Hence, we deduce that the characteristic of the logarithm of a number less than unity is one more than the number of zeros just after the decimal point and is negative.

MANTISSA The mantissa is the fractional part of the logarithm of a given number. Thus, with the same figures there will be difference in the characteristic only. It should be remembered, that the mantissa is always a positive quantity. The other way to indicate this is: Log .004594 = – 3 + .6623 = – 3.6623.

Negative mantissa must be converted into a positive mantissa before reference to a logarithm table. For example:

ANTILOGARITHMS If x is the logarithm of a given number n with a given base then n is called the antilogarithm (antilog) of x to that base. This can be expressed as follows: If logan = x then n = antilog x For example, if log 61720 = 4.7904 then 61720 = antilog 4.7904 Example 1: Find the value of log 5 if log 2 is equal to .3010. Solution: Example 2: Find the number whose logarithm is 2.4678. Solution:



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Example 3: Find the number whose logarithm is –2.4678. Solution: Example 1: Solution: Example 2: If logabc = x, logbca = y, logcab = z, prove that Solution: Example 3: If a=log2412, b=log3624, and c=log4836 then prove that 1+abc = 2bc Solution:



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SPACE FOR SUMMARY, FORMULAE, TRICKS AND NOTES



Page13

1. The inverse ratio of 11 : 15 is (a) 15 : 11 (b) √11 : √15 (c) 121 : 225 (d) none of these 2. The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is

(a) 16 (b) 60 (c) 22 (d) 20

3. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the

antecedent is

(a) 5 (b) √5 (c) 7 (d) none of these

4. The ratio compounded of 2 : 3, 9 : 4, 5 : 6 and 8 : 10 is

(a) 1 : 1 (b) 1 : 5 (c) 3 : 8 (d) none of these 5. The duplicate ratio of 3 : 4 is

(a) √3 : 2 (b) 4 : 3 (c) 9 : 16 (d) none of these

6. The sub-duplicate ratio of 25 : 36 is

(a) 6 : 5 (b) 36 : 25 (c) 50 : 72 (d) 5 : 6

7. The triplicate ratio of 2 : 3 is

(a) 8 : 27 (b) 6 : 9 (c) 3 : 2 (d) none of these

8. The sub-triplicate ratio of 8 : 27 is


9. The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4 is


10. The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3

and 9 : 7 is


11. The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate

ratio of 81 : 256 and sub-triplicate ratio of 125 : 512 is


12. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is

(a) 54 : 25 (b) 8 : 25 (c) 17 : 24 (d) 18 : 25

13. Two numbers are in ratio 2 : 3. If 4 be subtracted from each, they are in ratio 3 : 5. The

numbers are

(a) (16,24) (b) (4,6) (c) (2,3) (d) none of these

14. The angles of a triangle are in ratio 2 : 7 : 11. The angles are

(a) (20, 70, 90) (b) (30, 70, 80) (c) (18, 63, 99) (d) none of these

15. Division of Rs. 324 between X and Y is in the ratio 11: 7. X & Y would get Rupees

(a) (204, 120) (b) (200, 124) (c) (180, 144) (d) none of these

QUESTIONS FOR SELF PRACTICE



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16. Anand earns Rs. 80 in 7 hours and Pramod Rs. 90 in 12 hours. The ratio of their

earnings is


17. The ratio of two numbers is 7 : 10 and their difference is 105. The numbers are


18. P, Q and R are three cities. The ratio of average temperature between P and Q is 11 :

12 and that between P and R is 9 : 8. The ratio between the average temperature of Q

and R is


19. If x : y = 3 : 4, the value of x2y + xy2 : x3 + y3 is


20. If 2s : 3t is the duplicate ratio of 2s – p : 3t – p then

(a) p2 = 6st (b) p = 6st (c) 2p = 3st (d) none of these

21. If p : q = 2 : 3 and x : y = 4 : 5, then the value of 5px + 3qy : 10px + 4qy is


22. The number which when subtracted from each of the terms of the ratio 19 : 31

reducing it to 1 : 4 is

(a) 15 (b) 5 (c) 1 (d) none of these

23. Daily earnings of two persons are in the ratio 4:5 and their daily expenses are in the

ratio 7 : 9. If each saves Rs. 50 per day, their daily earnings in Rs. Are


24. The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 kms. in

5 hours, the speed of the first train is

(a) 10 Km/hr (b) 50 Km/hr (c) 70 Km/hr (d) none of these

25. The fourth proportional to 4, 6, 8 is


26. The third proportional to 12, 18 is


27. The mean proportional between 25, 81 is


28. The number which has the same ratio to 26 that 6 has to 13 is


29. The fourth proportional to 2a, a2, c is

(a) ac/2 (b) ac (c) 2/ac (d) none of these

30. If four numbers 1/2, 1/3, 1/5, 1/x are proportional then x is

(a) 6/5 (b) 5/6 (c) 15/2 (d) none of these

31. The mean proportional between 12x2 and 27y2 is

(a) 18xy (b) 81xy (c) 8xy (d) none of these



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32. If A = B/2 = C/5, then A : B : C is

(a) 3 : 5 : 2 (b) 2 : 5 : 3 (c) 1 : 2 : 5 (d) none of these

33. If a/3 = b/4 = c/7, then a + b + c/c is


34. If p/q = r/s = 2.5/1.5, the value of ps : qr is

(a) 3/5 (b) 1:1 (c) 5/3 (d) none of these

35. If x : y = z : w = 2.5 : 1.5, the value of (x + z)/(y + w) is

(a) 1 (b) 3/5 (c) 5/3 (d) none of these

36. If (5x – 3y)/(5y – 3x) = 3/4, the value of x : y is


37. If A : B = 3 : 2 and B : C = 3 : 5, then A : B : C is


38. If x/2 = y/3 = z/7, then the value of (2x – 5y + 4z)/2y is

(a) 6/23 (b) 23/6 (c) 3/2 (d) 17/6

39. If x : y = 2 : 3, y : z = 4 : 3 then x : y : z is


40. Division of Rs. 750 into 3 parts in the ratio 4 : 5 : 6 is

(a) (200, 250, 300) (b) (250, 250, 250) (c) (350, 250, 150) (d) 8 : 12 : 9

41. The sum of the ages of 3 persons is 150 years. 10 years ago their ages were in the ratio

7 : 8 : 9. Their present ages are

(a) (45, 50, 55) (b) (40, 60, 50) (c) (35, 45, 70) (d) none of these

42. The numbers 14, 16, 35, 42 are not in proportion. The fourth term for which they will

be in proportion is


43. If x/y = z/w, implies y/x = w/z, then the process is called

(a) Dividendo (b) Componendo (c) Alternendo (d) none of these

44. If p/q = r/s = p – r/q – s, the process is called

(a) Subtrahendo (b) Addendo (c) Invertendo (d) none of these

45. If a/b = c/d, implies (a + b)/(a – b) = (c + d)/(c – d), the process is called

(a) Componendo (b) Dividendo (c) Componendo and Dividendo (d) none of these

46. If u/v = w/p, then (u – v)/(u + v) = (w – p)/(w + p). The process is called

(a) Invertendo (b) Alternendo (c) Addendo (d) none of these

47. 12, 16, *, 20 are in proportion. Then * is


48. 4, *, 9, 13½ are in proportion. Then * is




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49. The mean proportional between 1.4 gms and 5.6 gms is

(a) 28 gms (b) 2.8 gms (c) 3.2 gms (d) none of these

50. If a/4 = b/5 = c/9 then (a+b+c)/c is


51. Two numbers are in the ratio 3 : 4; if 6 be added to each terms of the ratio, then the

new ratio will be 4 : 5, then the numbers are:

(a) 14, 20 (b) 17, 19 (c) 18 and 24 (d) none of these

52. If a/4 = b/5 then

(a) (a+4) / (a-4) = (b-5) / (b+5) (b) (a+4) / (a-4) = (b+5) / (b-5)

(c) (a-4) / (a+4) = (b+5) / (b-5) (d) none of these

53. If a: b = 4: 1 then √a/b + √b/a is:

(a) 5/2 (b) 4 (c) 5 (d) none of these

54. If x/(b+c-a) = y/(c+a-b) = z/(a+b-c) then (b-c)x + (c-a)y + (a-b)z is


55. 4x–1/4 is expressed as (a) –4x1/4 (b) x–1 (c) 4/x1/4 (d) none of these

56. The value of 81/3 is (a) 3√2 (b) 4 (c) 2 (d) none of these

57. The value of 2 × (32) 1/5 is (a) 2 (b) 10 (c) 4 (d) none of these

58. The value of 4/(32)1/5 is (a) 8 (b) 2 (c) 4 (d) none of these

59. The value of (8/27)1/3 is (a) 2/3 (b) 3/2 (c) 2/9 (d) none of these

60. The value of 2(256)–1/8 is (a) 1 (b) 2 (c) 1/2 (d) none of these

61. 2½.4¾ is equal to (a) a fraction (b) a positive integer (c) a negative integer (d) none of these

62. (81x4/y-8)1/4 has simplified value equal to (a) xy2 (b) x2y (c) 9xy2 (d) none of these

63. xa–b × xb–c × xc–a is equal to (a) x (b) 1 (c) 0 (d) none of these

64. The value of (2p2q3/3xy)0 where p, q, x, y ≠ 0 is equal to

(a) 0 (b) 2/3 (c) 1 (d) none of these

65. {(33)2 × (42)3 × (53)2} / {(32)3 × (43)2 × (52)3} is (a) 3/4 (b) 4/5 (c) 4/7 (d) 1

66. Which is True? (a) 20 > (1/2)0 (b) 20 < (1/2)0 (c) 20 = (1/2)0 (d) none of these

67. If x1/p = y1/q = z1/r and xyz = 1, then the value of p + q + r is

(a) 1 (b) 0 (c) 1/2 (d) none of these

68. The value of ya–b × yb–c × yc–a × y–a–b is (a) ya+b (b) y (c) 1 (d) 1/ya+b

69. The True option is

(a) x2/3 = 3√x2 (b) x2/3 = √x3 (c) x2/3 > 3√x2 (d) x2/3 < 3√x2



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70. The simplified value of 16x–3y2 × 8–1x3y–2 is

(a) 2xy (b) xy/2 (c) 2 (d) none of these

71. The value of (8/27)–1/3 × (32/243)–1/5 is (a) 9/4 (b) 4/9 (c) 2/3 (d) none of these

72. The value of {(x + y)2/3 (x – y)3/2/√x+y × √(x – y)3}6 is

(a) (x + y)2 (b) (x – y) (c) x + y (d) none of these

73. Simplified value of (125)2/3 × √25 × 3√53 × 51/2 is

(a) 5 (b) 1/5 (c) 1 (d) none of these

74. [{(2)1/2 . (4)3/4 . (8)5/6 . (16)7/8 . (32)9/10}4]3/25 is

(a) A fraction (b) an integer (c) 1 (d) none of these

75. [1–{1–(1–x2)–1}–1]–1/2 is equal to (a) x (b) 1/x (c) 1 (d) none of these

76. [(xn)n-1/n]1/(n+1) is equal to (a) xn (b) xn+1 (c) xn–1 (d) none of these

77. If a3–b3 = (a–b) (a2 + ab + b2), then the simplified form of

[xl/xm]l2+lm+m2x[xm/xn]m2+mn+n2x[xn/xl]l2+ln+n2 (a) 0 (b) 1 (c) x (d) none of these

75. Using (a–b)3 = a3–b3–3ab(a–b) tick the correct of these when x = p1/3 – p–1/3

(a) x3+3x = p + 1/p (b) x3 + 3x = p – 1/p (c) x3 + 3x = p + 1 (d) none of these

76. On simplification, 1/(1+am–n+am–p) + 1/(1+an–m+an–p) + 1/(1+ap–m+ap–n) is equal to

(a) 0 (b) a (c) 1 (d) 1/a

77. The value of (xa/xb)a+b x (xb/xc)b+c x (xc/xa)c+a (a) 1 (b) 0 (c) 2 (d) none of these

78. If x = 3 1/3 + 3 -1/3, then 3x3 -9x is (a) 15 (b) 10 (c) 12 (d) none of these

79. If ax = b, by = c, cz = a then xyz is (a) 1 (b) 2 (c) 3 (d) none of these

80. The value of (xa/xb)(a2+ab+b2) x (xa/xc)(b2+bc+c2) x (xc/xa)(c2+ca+a2)

(a) 1 (b) 0 (c) -1 (d) none of these

81. If 2x = 3y = 6-z, 1/x + 1/y + 1/z is (a) 1 (b) 0 (c) 2 (d) none of these

82. log 6 + log 5 is expressed as (a) log 11 (b) log 30 (c) log 5/6 (d) none of these

83. log 28 is equal to (a) 2 (b) 8 (c) 3 (d) none of these

84. log 32/4 is equal to (a) log 32/log 4 (b) log 32-log 4 (c) 23 (d) none of these

85. log (1 x 2 x 3) is equal to (a) log 1 + log 2 + log 3 (b) log 3 (c) log 2 (d) none

86. The value of log 0.0001 to the base 0.1 is (a) -4 (b) 4 (c) ¼ (d) none

87. If 2 log x = 4 log 3, then x is equal to (a) 3 (b) 9 (c) 2 (d) none

88. log √2 64 is equal to (a) 12 (b) 6 (c) 1 (d) none of these

89. log 2√3 1728 is equal to (a) 2√3 (b) 2 (c) 6 (d) none of these



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90. log (1/81) to the base 9 is equal to (a) 2 (b) ½ (c) –2 (d) none

91. log 0.0625 to the base 2 is equal (a) 4 (b) 5 (c) 1 (d) none

92. Given log2 = 0.3010 and log3 = 0.4771 the value of log 6 is

(a) 0.9030 (b) 0.9542 (c) 0.7781 (d) none of these

93. The value of log2 log2 log2 16 (a) 0 (b) 2 (c) 1 (d) none

94. The value of log 1/3 to the base 9 is (a) – ½ (b) ½ (c) 1 (d) none

95. If log x + log y = log (x+y), y can be expressed as

(a) x–1 (b) x (c) x/x–1 (d) none of these

96. The value of log2 [log2 {log3 (log3273)}] is equal to


97. If log2x + log4x + log16x = 21/4, these x is equal to


98. Given that log102 = x and log103 = y, the value of log1060 is expressed as

(a) x – y + 1 (b) x + y + 1 (c) x – y – 1 (d) none of these

99. Given that log102 = x, log103 = y, then log101.2 is expressed in terms of x and y as

(a) x + 2y – 1 (b) x + y – 1 (c) 2x + y – 1 (d) none of these

100. Given that log x = m + n and log y = m – n, the value of log 10x/y2 is expressed in

terms of m and n as (a) 1 – m + 3n (b) m – 1 + 3n (c) m + 3n + 1 (d) none

101. The simplified value of 2 log105 + log108 – ½ log104 is (a) 1/2 (b) 4 (c) 2 (d) none

102. log [1 – {1 – (1 – x2)–1}–1]–1/2 can be written as (a) log x2 (b) log x (c) log 1/x (d) none

103. The simplified value of log 4√7293√9-1.27-4/3 is (a) log 3 (b) log 2 (c) log ½ (d) none

104. The value of (logba × logcb × logac)3 is equal to (a) 3 (b) 0 (c) 1 (d) none

105. The logarithm of 64 to the base 2√2 is (a) 2 (b) √2 (c) 1/2 (d) none

106. The value of log825 given log 2 = 0.3010 is (a) 1 (b) 2 (c) 1.5482 (d) none

CHAPTER 02 A: EQUATIONS

VSmart Academy | CA Karan Chandwani | +91 7887 7887 10 Page19

INTRODUCTION: Equation is defined to be a mathematical statement of equality. If the equality is true for certain value of the variable involved, the equation is often called a conditional equation and equality sign ‘=’ is used; while if the equality is true for all values of the variable involved, the equation is called an identity. For Example: (x + 2)/3 + (x + 3)/2 = 3 holds true only for x =1. So, it is a conditional. On the other hand, (x +2)/3 + (x + 3)/2 = (5 x +13)/6 is an identity since it holds for all values of the variable x. Determination of value of the variable which satisfies an equation is called solution of the equation or root of the equation. An equation in which highest power of the variable is 1 is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or more linear equations involving two or more variables are called Simultaneous Linear Equations. An equation of degree 2 (highest Power of the variable is 2) is called Quadratic Equation and the equation of degree 3 is called Cubic Equation. For Example: 8x+17(x–3) = 4 (4x–9) + 12 is a Linear equation. 3x2 + 5x +6 = 0 is a Quadratic equation. 4x3 + 3x2 + x–7 = 1 is a Cubic equation. x + 2y = 1, 2x + 3y = 2 are jointly called Simultaneous equations.

SIMPLE EQUATION: A simple equation in one unknown x is in the form ax + b = 0. Note: A simple equation has only one root. Example: (4x/3) -1 = (14/15) x + (19/5). Solution: Example: The denominator of a fraction exceeds the numerator by 5 and if 3 be added to both the 3 fraction becomes 4. Find the fraction. Solution: Example: If thrice of A’s age 6 years ago be subtracted from twice his present age, the result would be equal to his present age. Find A’s present age. Solution:



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Example: A number consists of two digits the digit in the ten’s place is twice the digit in the unit’s place. If 18 be subtracted from the number the digits are reversed. Find the number. Solution: Example: For a certain commodity the demand equation giving demand ‘d’ in kg, for a price ‘p’ in rupees per kg. is d = 100 (10 – p). The supply equation giving the supply s in kg. for a price p in rupees per kg. is s=75(p – 3). The market price is such at which demand equals supply. Find the market price and quantity that will be bought and sold. Solution:

SIMULTANEOUS LINEAR EQUATIONS IN TWO UNKNOWNS: The general form of a linear equations in two unknowns x and y is ax + by + c = 0 where a, b are non-zero coefficients and c is a constant. Two such equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 form a pair of simultaneous equations in x and y. A value for each unknown which satisfies simultaneously both the equations will give the roots of the equations. METHOD OF SOLUTION: 1. ELIMINATION METHOD: In this method two given linear equations are reduced to a linear equation in one unknown by eliminating one of the unknowns and then solving for the other unknown. Example: Solve: 2x + 5y = 9 and 3x – y = 5. Solution: 2. CROSS MULTIPLICATION METHOD: Let two equations be: a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0 We write the coefficients of x, y and constant terms and two more columns by repeating the coefficients of x and y as follows:



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Example: Solve 3x + 2y + 17 = 0, 5x – 6y – 9 = 0 Solution:

METHOD OF SOLVING SIMULTANEOUS LINEAR EQUATION WITH THREE VARIABLES: Example: Solve for x, y and z: 2x – y + z = 3, x + 3y – 2z = 11, 3x – 2y + 4z = 1 Solution: Example: Solve for x, y and z: 1/x + 1/y + 1/z = 5; 2/x – 3/y – 4/z = -11; 3/x + 2/y – 1/z=-6 Solution:



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Example: Solve for x, y and z: xy/(x+y) = 70; xz/(x+z) = 84; yz/(y+z) = 140 Solution:

PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS: Example: If the numerator of a fraction is increased by 2 and the denominator by 1 it becomes 1. Again, if the numerator is decreased by 4 and the denominator by 2 it becomes 1/2. Find the fraction. Solution: Example: The age of a man is three times the sum of the ages of his two sons and 5 years hence his age will be double the sum of their ages. Find the present age of the man? Solution:



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Example: A number consist of three digits of which the middle one is zero and the sum of the other digits is 9. The number formed by interchanging the first and third digits is more than the original number by 297 find the number. Solution:

QUADRATIC EQUATION: HOW TO FIND OUT THE ROOTS OF A QUADRATIC EQUATION:

HOW TO CONSTRUCT A QUADRATIC EQUATION: For the equation ax2 + bx + c = 0 we have, or x2 + b/ax + c/a = 0 or x2 – (-b/a)x + c/a = 0 or x2 – (sum of roots)x + (product of roots) = 0 NATURE OF THE ROOTS:

Example: Solve x2 – 5x + 6 = 0 Solution:



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Example: Examine the nature of the roots of the following equations: i) x2 – 8x + 16 = 0 ii) 3x2 – 8x + 4 = 0 iii) 5x2 – 4x + 2 = 0 iv) 2x2 – 6x – 3 = 0 Solution: Example: If α & β be roots of x2 + 7x + 12 = 0 find equation whose roots are (α + β)2 (α - β)2 Solution: Example: If α, β be the roots of 2x2 – 4x – 1 = 0 find the value of α2/β + β2/α Solution: Example: Solve x: 4x – 3.2x+2 + 25 = 0 Solution:



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Example: Solve (x-1/x)2 + 2(x+1/x) = 7 1/4 Solution: Example: Solve 2x–2 + 23–x = 3 Solution: Example: If 1 root of equation is 2 -√3 from equation given that roots are irrational. Solution: Example: If α β are 2 roots of equation x2 – px + q = 0 form equation whose roots α/β & β/α Solution:



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Example: If the roots of the equation p(q – r)x2 + q(r – p)x + r(p – q) = 0 Solution:

PROBLEMS ON QUADRATIC EQUATION: Example: Difference between a number & its positive square root is 12; find numbers? Solution: Example: A piece of iron rod costs Rs. 60. If the rod was 2 metre shorter and each metre costs Re. 1.00 more, the cost would remain unchanged. What is the length of the rod? Solution: Example: Divide 25 into two parts so that sum of their reciprocals is 1/6. Solution:



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SOLUTION OF CUBIC EQUATION: On trial basis putting if some value of x stratifies the equation then we get a factor. This is a trial and error method. With this factor to factorise the LHS and then other get values of x. Example: Solve x3 – 7x + 6 = 0 Solution: Example: Solve for real x: x3 + x + 2 = 0 Solution:




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1. The equation –7 x + 1 = 5 – 3 x will be satisfied for x equal to: a) 2 b) –1 c) 1 d) none of these 2. The root of the equation (x+4)/4 + (x-5)/3 = 11 is: a) 20 b) 10 c) 2 d) none of these 3. Pick up the correct value of x for x/30 = 2/45: a) x = 5 b) x = 7 c) x =1 1/3 d) none of these 4. The solution of the equation (x + 24)/5 = 4 + x/4: a) 6 b) 10 c) 16 d) none of these 5. 8 is the solution of the equation: a) (x+4)/4 + (x-5)/3 = 11 b) (x+4)/2 + (x+10)/9 = 8 c) (x+24)/5 = 4 + x/4 d) (x-15)/10 + (x+5)/5 = 4 6. The value of y that satisfies the equation (y+11)/6 – (y+1)/9 = (y+7)/4 is: a) –1 b) 7 c) 1 d) -1/7 7. The solution of the equation (p+2) (p-3) + (p+3) (p-4) = p(2p-5) is: a) 6 b) 7 c) 5 d) none of these 8. The equation (12x+1)/4 = (15x – 1)/5 + (2x – 5)/(3x - 1) is true for: a) x=1 b) x=2 c) x=5 d) x=7 9. Pick up the correct value x for which x/0.5 – 1/0.05 + x/0.005 – 1/0.0005 = 0: a) x=0 b) x = 1 c) x = 10 d) none of these 10. The sum of two numbers is 52 and their difference is 2. The numbers are: a) 17 and 15 b) 12 and 10 c) 27 and 25 d) none of these 11. The diagonal of a rectangle is 5 cm and one of at sides is 4 cm. Its area is: a) 20 sq.cm. b) 12 sq.cm. c) 10 sq.cm. d) none of these 12. Divide 56 into two parts such that three times the first part exceeds one third of the second by 48. The parts are: a) (20, 36) b) (25, 31) c) (24, 32) d) none of these 13. The sum of the digits of a two digits number is 10. If 18 be subtracted from it the digits in the resulting number will be equal. The number is: a) 37 b) 73 c) 75 d) none of these numbers. 14. The fourth part of a number exceeds the sixth part by 4. The number is: a) 84 b) 44 c) 48 d) none of these 15. Ten years ago the age of a father was four times of his son. Ten years hence the age of the father will be twice that of his son. The present ages of the father and the son are: a) (50, 20) b) (60, 20) c) (55, 25) d) none of these 16. The product of two numbers is 3200 and the quotient when the larger number is divided by the smaller is 2. The numbers are: a) (16, 200) b) (160, 20) c) (60, 30) d) (80, 40)




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17. The denominator of a fraction exceeds the numerator by 2. If 5 be added to the numerator the fraction increases by unity. The fraction is: a) 5/7 b) 1/3 c) 7/9 d) 3/5 18. Three persons Mr. Roy, Mr. Paul and Mr. Singh together have Rs. 51. Mr. Paul has Rs. 4 less than Mr. Roy and Mr. Singh has got Rs. 5 less than Mr. Roy. They have the money as: a) (Rs. 20, Rs. 16, Rs. 15) b) (Rs. 15, Rs. 20, Rs. 16) c) (Rs. 25, Rs. 11, Rs. 15) d) none of these 19. A number consists of two digits. The digits in the ten’s place is 3 times the digit in the unit’s place. If 54 is subtracted from the number the digits are reversed. The number is: a) 39 b) 92 c) 93 d) 94 20. One student is asked to divide a half of a number by 6 and other half by 4 and then to add the two quantities. Instead of doing so the student divides the given number by 5. If the answer is 4 short of the correct answer then the number was: a) 320 b) 400 c) 480 d) none of these 21. If a number of which the half is greater than 1/5th of the number by 15 then the number is: a) 50 b) 40 c) 80 d) none of these 22. The solution of the set of equations 3x + 4y = 7, 4x – y = 3 is: a) (1, –1) b) (1, 1) c) (2, 1) d) (1, –2) 23. The values of x and y satisfying the equations x/2 + y/3 = 2; x + 2y = 8 are given by the pair: a) (3, 2) b) (–2, –3) c) (2, 3) d) none of these

24. x/p + y/p = 2, x + y = p + q are satisfied by the values given by the pair: a) (x=p, y=q) b) (x=q, y=p) c) (x=1, y=1) d) none of these 25. The solution for the pair of equations 1/16x + 1/15y = 9/20, 1/20x – 1/27y = 4/45 is given by: a) (1/4, 1/3) b) (1/3, ¼) c) (3,4) d) (4,3) 26. Solve for x and y: 4/x – 5/y = (x+y)/xy + 3/10 and 3xy = 10 (y-x): a) (5, 2) b) (–2, –5) c) (2, –5) d) (2, 5) 27. The pair satisfying the equations x + 5y = 36, (x+y)/(x-y) = 5/3 is given by: a) (16, 4) b) (4, 16) c) (4, 8) d) none of these 28. Solve for x and y: x–3y = 0; x+2y = 20: a) x = 4, y = 12 b) x = 12, y = 4 c) x = 5, y = 4 d) none of these 29. The simultaneous equations 7x–3y = 31; 9x–5y = 41 have solutions given by: a) (–4, –1) b) (–1, 4) c) (4, –1) d) (3, 7) 30. 1.5x + 2.4 y = 1.8, 2.5(x+1) = 7y have solutions as: a) (0.5, 0.4) b) (0.4, 0.5) c) (1/2, 5/2) d) (2, 5) 31. The values of x and y satisfying the equations 3/(x+y) + 2/(x-y) = 3; 2/(x+y) + 3/(x-y) = 3 2/3 are given by: a) (1, 2) b) (-1, -2) c) (1, ½) d) (2, 1) 32. 1.5x + 3.6y = 2.1, 2.5 (x+1) = 6y: a) (0.2, 0.5) b) (0.5, 0.2) c) (2, 5) d) (-2, -5) 33. x/5 + y/6 +1 = x/6 + y/5 = 28: a) (6, 9) b) (9, 6) c) (60, 90) d) (90, 60)



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34. x/4 = y/3 = z/2; 7x + 8y + 5z= 62 a) (4, 3, 2) b) (2, 3, 4) c) (3, 4, 2) d) (4, 2, 3) 35. xy/(x+y) =20, yz/(y+z) =40, zx/(z+x) = 24: a) (120, 60, 30) b) (60, 30, 120) c) (30, 120, 60) d) (30, 60, 120) 36. 2x + 3y + 4z = 0; x + 2y – 5z = 0; 10x + 16y – 6z = 0: a) (0, 0, 0) b) (1, –1, 1) c) (3, 2, –1) d) (1, 0, 2) 37. 1/3 (x+y) + 2z = 21; 3x – 1/2 (y+z) = 65; x + 1/2 (x+y–z) = 38: a) (4, 9, 5) b) (2, 9, 5) c) (24, 9, 5) d) (5, 24, 9) 38. 4/x – 5/y = (x+y)/xy + 3/10 3xy = 10 (y-x): a) (2, 5) b) (5, 2) c) (2, 7) d) (3, 4) 39. x/0.01 + (y+0.03)/0.05 = y/0.02 + (x+0.03)/0.04 = 2: a) (1, 2) b) (0.1, 0.2) c) (0.01, 0.02) d) (0.02, 0.01) 40. xy/(y-x) = 110, yz/(z-y) = 132, zx/(z+x) = 60/11: a) (12, 11, 10) b) (10, 11, 12) c) 11, 10, 12) d) (12, 10, 11) 41. 3x – 4y + 70z = 0; 2x + 3y – 10= 0; x + 2y + 3z = 13: a) (1, 3, 7) b) (1, 7, 3) c) (2, 4, 3) d) (-10, 10, 1) 42. Monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenses are in the ratio 7 : 9. If each saves Rs. 50 per month find their monthly incomes: a) (500, 400) b) (400, 500) c) (300, 600) d) (350, 550) 43. Find the fraction which is equal to 1/2 when both its numerator and denominator are increased by 2. It is equal to 3/4 when both are increased by 12. a) 3/8 b) 5/8 c) 2/8 d) 2/3 44. The age of a person is twice the sum of the ages of his two sons and five years ago his age was thrice the sum of their ages. Find his present age: a) 60 years b) 52 years c) 51 years d) 50 years 45. A number between 10 and 100 is five times the sum of its digits. If 9 be added to it the digits are reversed find the number. a) 54 b) 53 c) 45 d) 55 46. The wages of 8 men and 6 boys amount to Rs. 33. If 4 men earn Rs. 4.50 more than 5 boys determine the wages of each man and boy. a) (Rs. 1.50, Rs. 3) b) (Rs. 3, Rs. 1.50) c) (Rs. 2.50, Rs. 2) d) (Rs. 2, Rs. 2.50) 47. A number consisting of two digits is four times the sum of its digits and if 27 be added to it the digits are reversed. The number is: a) 63 b) 35 c) 36 d) 60 48. Of two numbers, 1/5th of the greater is equal to 1/3rd of the smaller and their sum is 16. The numbers are: a) (6, 10) b) (9, 7) c) (12, 4) d) (11, 5) 49. y is older than x by 7 years 15 years back x’s age was 3/4 of y’s age. Their present ages are: a) (x=36, y=43) b) (x=50, y=43) c) (x=43, y=50) d) (x=40, y=47) 50. The sum of the digits in a three digit number is 12. If the digits are reversed the number is increased by 495 but reversing only of the ten’s and unit digits increases the number by 36. The number is: a) 327 b) 372 c) 237 d) 273



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51. Two numbers are such that twice the greater number exceeds twice the smaller one by 18 and 1/3rd of the smaller and 1/5th of the greater number are together 21. The numbers are: a) (36, 45) b) (45, 36) c) (50, 41) d) (55, 46) 52. The demand and supply equations for a certain commodity are 4q + 7p = 17 and p = q/3 + 7/4 respectively where p is the market price and q is the quantity then the equilibrium price and quantity are: (a) 2, 3/4 (b) 3, 1/2 (c) 5, 3/5 (d) None of these. 53. If the roots of the equation 2x2 + 8x – m3= 0 are equal then value of m is: a) -3 b) -1 c) 1 d) -2 54. If 22x=3 – 32.2x + 1 = 0 then values of x are: a) 0, 1 b) 1, 2 c) 0, 3 d) 0, -3 55. The values of 4+ ________1__________ 4+ _______1_______ 4+ _____1_____ 4+……..∞

a) 1±√2 b) 2 + √5 c) 2 ± √5 d) none of these 56. If α β be the roots of the equation 2x2 – 4x – 3 = 0 the value of α2 + β2 is: a) 5 b) 7 c) 3 d) – 4 57. If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals then a2/ac + bc/a2 is equal to a) 2 b) –2 c) 1 d) –1 58. The equation x2 – (p + 4)x + 2p + 5 = 0 has equal roots the values of p will be. a) ± 1 b) 2 c) ± 2 d) –2 59. The roots of the equation x2 + (2p – 1)x + p2 = 0 are real if. a) p > 1 b) p < 4 c) p > 1/4 d) p < ¼ 60. If x = m is one of the solutions of the equation 2x2 + 5x – m = 0 the possible values of m are: a) (0, 2) b) (0, –2) c) (0, 1) d) (1, –1) 61. If p and q are the roots of x2 + 2x + 1 = 0 then the values of p3 + q3 becomes: a) 2 b) –2 c) 4 d) – 4 62. If L + M + N = 0 and L, M, N are rationals the roots of the equation (M + N – L)x2 + (N + L – M)x + (L + M – N) = 0 are: a) real and irrational b) real and rational c) imaginary and equal d) real and equal 63. If α and β are the roots of x2 = x + 1 then value of α2/β – β2/α is: a) 2√5 b) √5 c) 3√5 d) –2√5 64. If p ≠ q and p2 = 5p – 3 and q2 = 5q – 3 the equation having roots as p/q and q/p is: a) x2 – 19x + 3 = 0 b) 3x2 – 19x – 3 = 0 c) 3x2 – 19x + 3 = 0 d) 3x2 + 19x + 3 = 0 65. If one root of 5x2 + 13x + p = 0 be reciprocal of the other then the value of p is: a) –5 b) 5 c) 1/5 d) –1/5 66. A solution of the quadratic equation (a+b–2c)x2 + (2a–b–c)x + (c+a–2b) = 0 is: a) x = 1 b) x = –1 c) x = 2 d) x = – 2 67. If the root of the equation x2–8x+m = 0 exceeds the other by 4 then the value of m is: a) m = 10 b) m = 11 c) m = 9 d) m = 12



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68. The values of x in the equation 7(x+2p)2 + 5p2 = 35xp + 117p2 are: a) (4p, –3p) b) (4p, 3p) c) (–4p, 3p) d) (–4p, –3p) 69. The solutions of the equation 6x/(x+1) + 6(x+1)/x =13 are: a) (2, 3) b) (3, –2) c) (–2, –3) d) (2, –3) 70. The satisfying values of x for the equation: 1/(x+p+q) = 1/x + 1/p + 1/q are: a) (p, q) b) (–p, –q) c) (p, –p) d) (–p, q) 71. The values of x for the equation: x2 +9x + 18 = 6 – 4x are: a) (1, 12) b) (–1, –12) c) (1, –12) d) (–1, 12) 72. The values of x satisfying the equation: √(2x2 + 5x - 2) - √(2x2 + 5x – 9) = 1 are: a) (2, –9/2) b) (4, –9) c) (2, 9/2) d) (-2, 9/2) 73. The solution of the equation 3x2 – 17x + 24 = 0 are: a) (2, 3) b) (2, 3 2/3) c) (3, 2 2/3) d) (3, 2/3) 74. The equation 3(3x2 + 15)/6 + 2x2 + 9 = (2x2 + 96)/7 + 6 has got the solution as: a) (1, 1) b) (1/2, –1) c) (1, –1) d) (2, –1) 75. The equation [(l-m)/2]x2 – [(l+m)/2]x + m = 0 has got 2 values of x to satisfy the equation given as: a) (1, 2m/l-m) b) (1, m/l-m) c) (1, 2l/l-m) d) (1, l/l-m) 76. The sum of two numbers is 8 and the sum of their squares is 34. Taking one number as x form an equation in x and hence find the numbers. The numbers are: a) (7, 10) b) (4, 4) c) (3, 5) d) (2, 6) 77. The difference of two positive integers is 3 and the sum of their squares is 89. Taking the smaller integer as x form a quadratic equation and solve it to find the integers. The integers are: a) (7, 4) b) (5, 8) c) (3, 6) d) (2, 5) 78. Five times of a positive whole number is 3 less than twice the square of the number. The number is: a) 3 b) 4 c) –3 d) 2 79. The area of a rectangular field is 2000 sq.m and its perimeter is 180m. Form a quadratic equation by taking the length of the field as x and solve it to find the length and breadth of the field. The length and breadth are: a) (205m, 80m) b) (50m, 40m) c) (60m, 50m) d) none 80. Two squares have sides p cm and (p + 5) cms. The sum of their squares is 625 sq. cm. The sides of the squares are: a) (10 cm, 30 cm) b) (12 cm, 25 cm) c) (15 cm, 20 cm) d) none of these 81. Divide 50 into two parts such that the sum of their reciprocals is 1/12. The numbers are: a) (24, 26) b) (28, 22) c) (27, 23) d) (20, 30) 82. There are two consecutive numbers such that the difference of their reciprocals is 1/240. The numbers are: a) (15, 16) b) (17, 18) c) (13, 14) d) (12, 13) 83. The hypotenuse of a right–angled triangle is 20cm. The difference between its other two sides be 4cm. The sides are: a) (11cm, 15cm) b) (12cm, 16cm) c) (20cm, 24cm) d) none of these 84. The sum of two numbers is 45 and the mean proportional between them is 18. The numbers are: a) (15, 30) b) (32, 13) c) (36, 9) d) (25, 20)



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85. The sides of an equilateral triangle are shortened by 12 units 13 units and 14 units respectively and a right angle triangle is formed. The side of the equilateral triangle is: a) 17 units b) 16 units c) 15 units d) 18 units 86. A distributor of apple Juice has 5000 bottle in the store that it wishes to distribute in a month. From experience it is known that demand D (in number of bottles) is given by D = –2000p2 + 2000p + 17000. The price per bottle that will result zero inventory is: a) Rs. 3 b) Rs. 5 c) Rs. 2 d) none of these 87. The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is 12; the two numbers are: a) 3√2, 2√3 b) 5√2, 3√5 c) 2√2, 5√2 d) none of these 88. The solution of the cubic equation x3–6x2+11x–6 = 0 is given by the triplet: a) (–1, 1 –2) b) (1, 2, 3) c) (–2, 2, 3) d) (0, 4, –5) 89. The cubic equation x3 + 2x2 – x – 2 = 0 has 3 roots namely: a) (1, –1, 2) b) (–1, 1, –2) c) (–1, 2, –2) d) (1, 2, 2) 90. x, x – 4, x + 5 are the factors of the left–hand side of the equation: a) x3 + 2x2 – x – 2 = 0 b) x3 + x2 – 20x = 0 c) x3 – 3x2 – 4x + 12 = 0 d) x3 – 6x2 + 11x – 6 = 0 91. The equation 3x3 + 5x2 = 3x + 5 has got 3 roots and hence the factors of the left–hand side of the equation 3x3 + 5x2 – 3x – 5 = 0 are: a) x – 1, x – 2, x – 5/3 b) x – 1, x + 1, 3x + 5 c) x+1, x-1, 3x-5 d) x-1, x+1, x-2 92. The roots of the equation x3 + 7x2 – 21x – 27 = 0 are: a) (– 3, – 9, – 1) b) (3, – 9, – 1) c) (3, 9, 1) d) (– 3, 9, 1) 93. The roots of x3 + x2 – x – 1 = 0 are: a) (– 1, – 1, 1) b) (1, 1, – 1) c) (– 1, – 1, – 1) d) (1, 1, 1) 94. The satisfying value of x3 + x2 – 20x = 0 are: a) (1, 4, – 5) b) (2, 4, – 5) c) (0, – 4, 5) d) (0, 4, – 5) 95. The roots of the cubic equation x3 + 7x2 – 21x – 27 = 0 are: a) (–3, –9, –1) b) (3, –9, –1) c) (3, 9, 1) d) (–3, 9, 1) 96. If 4x3 + 8x2 – x – 2 = 0 then value of (2x + 3) is given by: a) 4, –1, 2 b) –4, 2, 1 c) 2, –4, –1 d) none of these 97. The rational root of the equation 2x3 – x2 – 4x + 2 = 0 is: a) 1/2 b) – 1/2 c) 2 d) – 2

CHAPTER 04: TIME VALUE OF MONEY

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INTRODUCTION:

People earn money for spending it on housing, food, clothing, education, entertainment etc. Sometimes extra expenditures have also to be met with. For example: there might be a marriage in the family; one may want to buy house, one may want to set up his or her business, one may want to buy a car and so on. Some people can manage to put aside some money for such expected and unexpected expenditures. But most people have to borrow money for such contingencies. From where they can borrow money? Money can be borrowed from friends or money lenders or Banks. If you can arrange a loan from your friend it might be interest free but if you borrow money from lenders or Banks you will have to pay some charge periodically for using money of money lenders or Banks. This charge is called interest. Let us take another view. People earn money for satisfying their various needs as discussed above. After satisfying those needs some people may have some savings. People may invest their savings in debentures or lend to other people or simply deposit it into bank. In this way they can earn interest on their investment. Interest can be defined as the price paid by a borrower for the use of a lender’s money.

WHY IS INTEREST PAID?

TIME VALUE OF MONEY: Value of a unity of money is different in different time periods. The sum of money received in future is less valuable than it is today. We can say that the present worth of money received after some time will be less than a money received today. Since money received today has more value rational investors would prefer current receipts to future receipts. If they postpone their receipts, they will certainly charge some money i.e. interest. OPPORTUNITY COST: The lender has a choice between using his money in different investments. If he chooses one, he forgoes the return from all others. We can say, lending incurs an opportunity cost due to the possible alternative uses of the lent money. INFLATION: Inflation is a fall in the purchasing power of money. Due to inflation a given amount of money buys fewer goods in the future than it will now. The borrower needs to compensate the lender for this. LIQUIDITY PREFERENCE: People prefer to have their resources available in a form that can immediately be converted into cash rather than a form that takes time or money to realize. RISK FACTOR: There is always a risk that the borrower will go bankrupt or otherwise default on the loan. A lender generally charges more interest rate (risk premium) for taking more risk.

DEFINITION OF INTEREST AND SOME OTHER RELATED TERMS:

INTEREST: Price paid by borrower for use of lender’s money. If you borrow (or lend) some money from (or to) a person for a particular period you would pay (or receive) more money than your initial borrowing (or lending). This excess money paid (or received) is called interest. Suppose you borrow (or lend) Rs. 50,000 for a year and you pay (or receive) Rs. 55,000 after one year, the difference between initial borrowing (or lending) Rs. 50,000 and end payment (or receipts) Rs. 55,000 i.e. Rs. 5,000 is the amount of interest you paid (or earned). PRINCIPAL: Principal is initial value of lending (or borrowing). If you invest your money the value of initial investment is also called principal. Suppose you borrow (or lend) Rs. 50,000 from a person for one year. Rs. 50,000 in this example is the ‘Principal.’ Or suppose you deposit Rs. 20,000 in your bank account for one year. In this example Rs. 20,000 is the principal. RATE OF INTEREST: Rate at which interest is charged for defined length of time for use of principal generally on yearly basis is known to be the rate of interest. Rate of interest is usually expressed in



percentages. Suppose you invest Rs. 20,000 in your bank account for 1 year with interest rate of 5% per annum. It means you earn Rs. 5 as interest every Rs. 100 of principal amount in a year. Per annum means for a year.

ACCUMULATED AMOUNT (BALANCE): Final value of an investment. It is the sum total of principal & interest earned. Suppose you deposit Rs. 50,000 in bank for 1 year with interest rate of 5% p.a., you earn interest of Rs. 2,500 after 1 year. After 1 year you get Rs. 52,500 (principal+ interest).

SIMPLE INTEREST AND COMPOUND INTEREST:

SIMPLE INTEREST: The interest paid for keeping Rs. 100 for 1 year is known as the rate percent per annum. Thus, if money is borrowed at the rate of 8% per annum then the interest paid for keeping Rs. 100 for 1 year is Rs. 8. Clearly the interest you pay is proportionate to the money that you borrow & also to the period of time for which you keep the money; the more the money & the time, the more the interest. Interest is also proportionate to the rate of interest agreed upon by the lending & the borrowing parties. Thus, interest varies directly with principal, time and rate. Simple interest is the interest computed on the principal for the entire period of borrowing. It is calculated on the outstanding principal balance & not on interest previously earned. It means no interest is paid on interest earned during the term of loan. Example 1: How much interest will be earned on Rs. 2,000 at 6% simple interest for 2 years? Solution: Example 2: Aditya Sharma deposited Rs. 50,000 in a bank for 2 years with interest rate of 5.5% p.a. How much interest would she earn? What will be the final value of investment? Solution: Example 4: Dishu Patani deposited Rs. 1,00,000 in his bank for 2 years at simple interest rate of 6%. How much interest would he earn? How much would be the final value of deposit? Solution:



Example 5: Find interest rate if amount owed after 6 months is Rs. 1,050, borrowed amount being Rs. 1,000. Solution: Example 6: Pokemon invested Rs. 70,000 in bank at rate of 6.5% p.a. simple interest rate. He received Rs. 85,925 after end of term. Find out period for which sum was invested by Pokemon. Solution: Example 7: Saheb deposited some amount in bank for 7 ½ years at rate of 6% p.a. simple interest. Saheb received Rs. 1,01,500 at the end of the term. Compute initial deposit of Saheb. Solution: Example 8: A sum of Rs. 46,875 was lent out at simple interest & at the end of 1 year 8 months, the total amount was Rs. 50,000. Find the rate of interest percent per annum. Solution: Example 9: What sum of money will produce Rs. 28,600 as an interest in 3 years and 3 months at 2.5% p.a. simple interest? Solution:



Example 10: In what time will Rs. 85,000 amounts to Rs. 1,57,675 at 4.5 % p.a.? Solution: COMPOUND INTEREST: If principal remains the same for the entire period or time then interest is called as simple interest. However, in practice the method according to which banks, insurance corporations and other money lending and deposit taking companies calculate interest is different. Suppose you deposit Rs. 50,000 in ICICI bank for 2 years at 7% p.a. compounded annually. Interest will be calculated in the following way: This interest is Rs. 245 more than the simple interest on Rs. 50,000 for 2 years at 7% p.a. This excess interest is due to fact that the principal for 2nd year was more than the principal for 1st year. The interest calculated in this manner is called compound interest. The interest that accrues when earnings for each specified period of time added to the principal thus increasing the principal base on which subsequent interest is computed. For a given principal, rate and time the compound interest is generally more than the simple interest. Example 11: Malini Hema deposited Rs. 1,00,000 in a nationalized bank for 3 years. If the rate of interest is 7% p.a., calculate the interest that bank has to pay to Malini Hema after 3 years if interest is compounded annually. Also calculate the amount at the end of 3rd year. Solution:



CONVERSION PERIOD: In the example discussed above the interest was calculated on yearly basis i.e. the interest was compounded annually. However, in practice it is not necessary that the interest be compounded annually. For example, in banks the interest is often compounded twice a year (half yearly or semi-annually) i.e. interest is calculated and added to the principal after every six months. In some financial institutions interest is compounded quarterly i.e. four times a year. The period at the end of which the interest is compounded is called conversion period. When the interest is calculated & added to the principal every six months the conversion period is six months. In this case number of conversion periods per year would be two. If the loan or deposit was for five years then the number of conversion period would be ten. TYPICAL CONVERSION PERIODS ARE GIVEN BELOW:

Conversion period Description No. of conversion period in a year

1 day Compounded daily 365

1 month Compounded monthly 12

3 months Compounded quarterly 4

6 months Compounded semi-annually 2

12 months Compounded annually 1

FORMULA FOR COMPOUND INTEREST: Note: Computation of ‘A’ shall be quite simple with a calculator. However compound interest table and tables for at various rates per annum with (a) annual compounding; (b) monthly compounding & (c) daily compounding are available. Example 12: Rs. 2,000 is invested at annual rate of interest of 10%. What is the amount after 2 years if compounding is done (a) Annually (b) Semi-annually (c) Quarterly (d) Monthly. Solution:



Example 13: Determine the compound amount and compound interest on Rs. 1,000 at 6% compounded semi-annually for 6 years. Given that (1 + i)n = 1.42576 for i = 3% and n = 12. Solution: Example 14: Compute the compound interest on Rs. 4,000 for 1½ years at 10% per annum compounded half- yearly. Solution: Examples 15: On what sum will the compound interest at 5% per annum for 2 years compounded annually be Rs. 1,640? Solution: Example 16: What annual rate of interest compounded annually doubles an investment in 7 years? Given that 21/7 = 1.104090 Solution: Example 17: In what time will Rs. 8,000 amounts to Rs. 8,820 at 10% per annum interest compounded half-yearly? Solution:



Example 18: Find the rate percent per annum if Rs. 2,00,000 amounts to Rs. 2,31,525 in 1½ year interest being compounded half-yearly. Solution: Example 19: A certain sum invested at 4% per annum compounded semi-annually amounts to Rs. 78,030 at the end of one year. Find the sum. Solution: Example 20: Rs. 16,000 invested at 10% p.a. compounded semi-annually amounts to Rs. 18,522. Find the time period of investment. Solution: Example 21: A person opened an account on April, 2001 with a deposit of Rs. 800. The account paid 6% interest compounded quarterly. On October 1, 2011 he closed the account and added enough additional money to invest in a 6-month time-deposit for Rs. 1,000, earning 6% compounded monthly. (a) How much additional amount did the person invest on October 1? (b) What was the maturity value of his time deposit on April 1 2012? (c) How much total interest was earned? Given that (1 + i)n is 1.03022500 for i=1½% n=2 and (1+ i)n is 1.03037751 for i=½% and n=6. Solution:



EFFECTIVE RATE OF INTEREST:

If interest is compounded more than once a year the effective interest rate for a year exceeds the per annum interest rate. But if interest is compounded annually effective interest rate for the year will be equal to actual interest rate per annum. So effective interest rate can be defined as the equivalent annual rate of interest compounded annually if interest is compounded more than once a year. Example 22: Rs. 5,000 is invested in a Term Deposit Scheme that fetches interest 6% per annum compounded quarterly. What will be the interest after one year? What is effective rate of interest? Solution: Example 23: Find the amount of compound interest and effective rate of interest if an amount of Rs. 20,000 is deposited in a bank for one year at the rate of 8% per annum compounded semi-annually. Solution:



Example 24: Which is a better investment 3% per year compounded monthly or 3.2% per year simple

interest? Given that (1+0.0025)12 =1.0304. Solution:

ANNUITY:

In many cases you must have noted that your parents have to pay an equal amount of money regularly like every month or every year. For example, payment of life insurance premium, rent of your house (if you stay in a rented house), payment of housing loan, vehicle loan etc. In all these cases they pay a constant amount of money regularly. Sometimes some people received a fixed amount of money regularly like pension rent of house etc. In all these cases annuity comes into the picture. When we pay (or receive) a fixed amount of money periodically over a specified time period we create an annuity. Thus, annuity can be defined as a sequence of periodic payments (or receipts) regularly over a specified period of time. There is a special kind of annuity also that is called Perpetuity. It is one where the receipt or payment takes place forever. Since the payment is forever, we cannot compute a future value of perpetuity. However, we can compute the present value of the perpetuity. To be called annuity a series of payments (or receipts) must have following features: (1) Amount paid (or received) must be constant over the period of annuity and (2) Time interval between two consecutive payments (or receipts) must be the same. ANNUITY REGULAR AND ANNUITY DUE / IMMEDIATE:

Annuity regular: First payment/ receipt takes place at the end of first period. Annuity Due or Annuity Immediate: First receipt/payment is made today (at the beginning of the annuity).

FUTURE VALUE:

Future value is the cash value of an investment at some time in the future. It is tomorrow’s value of today’s money compounded at the rate of interest.



Example 25: You invest Rs. 3000 in a two-year investment that pays you 12% per annum. Calculate the future value of the investment. Solution: FUTURE VALUE OF ANNUITY REGULAR: Example 26: Find the future value of an annuity of Rs. 500 made annually for 7 years at interest rate of 14% compounded annually. Given that (1.14)7 = 2.5023. Solution: Example 27: Rs. 200 is invested at the end of each month in an account paying interest 6% per year compounded monthly. What is the future value of this annuity after 10th payment? Given that (1.005)10=1.0511 Solution:



FUTURE VALUE OF ANNUITY DUE OR ANNUITY IMMEDIATE: The relationship between the value of an annuity due and an ordinary annuity in case of future value is: Example 28: Z invests Rs. 10,000 every year starting from today for next 10 years. Suppose interest rate is 8% per annum compounded annually. Calculate future value of the annuity. Given that (1+0.08)10 = 2.15892500. Solution:

PRESENT VALUE:

Present value is today’s value of tomorrow’s money discounted at the interest rate. Future value and present value are related to each other in fact they are the reciprocal of each other.

• Computation of P may be simple if we make use of either the calculator or the present value table showing values of 1/(1+i)n for various time periods/per annum interest rates.

• For positive i the factor 1/(1+i)n is always less than 1 indicating thereby future amount has smaller present value.

Example 29: What is present value of Re. 1 to be received after 2 years compounded annually @10% interest rate? Solution: Example 30: Find the present value of Rs. 10,000 to be required after 5 years if the interest rate be 9%. Given that (1.09)5=1.5386. Solution:



PRESENT VALUE OF AN ANNUITY REGULAR: The present value (V) of an annuity (A) is the sum of the present values of the payments. A = V / P (n, i) can be used to compute the amount of annuity if we have present value (V), ‘n’ the number of time period & the rate of interest in decimal. Example 31: S borrows Rs. 5,00,000 to buy a house. If he pays equal instalments for 20 years and 10% interest on outstanding balance what will be the equal annual instalment? Solution: Example 32: Rs. 5,000 is paid every year for 10 years to pay off a loan. What is the loan amount if interest rate be 14% per annum compounded annually? Solution: Example 33: Y bought a TV costing Rs. 13,000 by making a down payment of Rs. 3,000 and agreeing to make equal annual payment for 4 years. How much would be each payment if the interest on unpaid amount be 14% compounded annually? Solution: PRESENT VALUE OF ANNUITY DUE OR ANNUITY IMMEDIATE:

Example 34: Suppose your mom decides to gift you Rs. 10,000 every year starting from today for the next five years. You deposit this amount in a bank as and when you receive and get 10% per annum interest rate compounded annually. What is the present value of this annuity? Solution:



SINKING FUND:

It is the fund created for a specified purpose by way of sequence of periodic payments over a time period at a specified interest rate. Interest is compounded at the end of every period. Size of the sinking fund deposits is computed from A = P.A (n, i) where A is the amount to be saved, P is the periodic payment, n the payment period. Example 35: How much amount is required to be invested every year so as to accumulate Rs. 3,00,000 at the end of 10 years if interest is compounded annually at 10%? Solution:

APPLICATIONS:

LEASING: Leasing is a financial arrangement under which the owner of the asset (lessor) allows the user of the asset (lessee) to use the asset for a defined period of time (lease period) for a consideration (lease rental) payable over a given period of time. This is a kind of taking an asset on rent. Example 36: ABC Ltd. wants to lease out an asset costing Rs. 3,60,000 for a five-year period. It has fixed a rental of Rs. 1,05,000 per annum payable annually starting from the end of 1st year. Suppose rate of interest is 14% per annum compounded annually on which money can be invested by the company. Is this agreement favourable to the company? Solution: Example 37: A company is considering proposal of purchasing a machine either by making full payment of Rs. 4,000 or by leasing it for four years at an annual rate of Rs. 1,250. Which course of action is preferable if the company can borrow money at 14% compounded annually? Solution:



CAPITAL EXPENDITURE (INVESTMENT DECISION): Capital Expenditure means purchasing an asset (which results in outflow of money) today in anticipation of benefits (cash inflow) which would flow across the life of an investment. For taking investment decision we compare the Present Value of cash outflow & present value of cash inflows. If Present Value of cash inflows is greater than Present value of cash outflows decision should be in favour of investment. Example 38: A machine can be purchased for Rs. 50,000. Machine will contribute Rs. 12,000 per year for the next 5 years. Assume borrowing cost is 10% per annum compounded annually. Determine whether machine should be purchased or not. Solution: Example 39: A machine with useful life of 7 years costs Rs. 10,000 while another machine with useful life of five years costs Rs. 8,000. The first machine saves labour expenses of Rs. 1,900 annually and the 2nd one saves labour expenses of Rs. 2,200 annually. Determine the preferred course of action. Assume cost of borrowing as 10% compounded per annum. Solution: VALUATION OF BOND: A bond is a debt security in which the issuer owes the holder a debt & is obliged to repay the principal & interest. Bonds are generally issued for a fixed term longer than 1 year. Example 40: An investor intends purchasing a three-year Rs. 1,000 par value bond having nominal interest rate of 10%. At what price the bond may be purchased now if it matures at par and the investor requires a rate of return of 14%? Solution:

PERPETUITY: Perpetuity is an annuity in which the periodic payments or receipts begin on a fixed date and continue indefinitely or perpetually. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. The formula for evaluating perpetuity is relatively straight forward. Two points which are important to understand in this regard are: (a) The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low

present value (today’s value of the future cash flows). (b) Additionally, because the principal is never repaid, there is no present value for the principal. Therefore, the price of perpetuity is simply the coupon amount over the appropriate discount rate or yield.



CALCULATION OF MULTI PERIOD PERPETUITY: The formula for determining the present value of multi-period perpetuity is as follows:

Example 41: Ramesh wants to retire and receive Rs. 3,000 a month. He wants to pass this monthly payment to future generations after his death. He can earn an interest of 8% compounded annually. How much will he need to set aside to achieve his perpetuity goal? Solution: CALCULATION OF GROWING PERPETUITY: A stream of cash flows that grows at a constant rate forever is known as growing perpetuity. The formula for determining the present value of growing perpetuity is as follows: Example 42: Assuming that the discount rate is 7% per annum, how much would you pay to receive Rs. 50, growing at 5%, annually, forever? Solution: CALCULATING RATE OF RETURN: 1) Calculating the rate of return provides important information that can be used for future

investments. For example, if you invested in a stock that showed a substantial gain after several months of performance, you may decide to purchase more of that stock. If the stock showed a continual loss, it may be wise to conduct research to find a better-performing stock.

2) Calculating the rate of return is that it allows you to gauge your investment and decision- making skills. Investments that create a gain or profit are great. However, if you continually make investments at a loss, then you may want to change your investment strategies. A great attribute of successful business people is knowing how and when to make investments, as is knowing when to change strategies. With a firm grasp of calculating the rate of return, you can manage and monitor your investments at various stages to determine the outcome of your investments. This leads to a higher level of confidence and the skills necessary to be a savvy investor.

Net Present Value Technique (NPV): The net present value technique is a discounted cash flow method that considers the time value of money in evaluating capital investments. An investment has cash flows



throughout its life, and it is assumed that a rupee of cash flow in the early years of an investment is worth more than a rupee of cash flow in a later year. The net present value method uses a specified discount rate to bring all subsequent net cash inflows after the initial investment to their present values (the time of the initial investment is year 0). DETERMINING DISCOUNT RATE: Theoretically, the discount rate or desired rate of return on an investment is the rate of return the firm would have earned by investing the same funds in the best available alternative investment that has the same risk. Determining the best alternative opportunity available is difficult in practical terms so rather than using the true opportunity cost, organizations often use an alternative measure for the desired rate of return. An organization may establish a minimum rate of return that all capital projects must meet; this minimum could be based on an industry average or the cost of other investment opportunities. Many organizations choose to use the overall cost of capital or Weighted Average Cost of Capital (WACC) that an organization has incurred in raising funds or expects to incur in raising the funds needed for an investment. The net present value of a project is the amount, in current value of rupees, the investment earns after paying cost of capital in each period.

NET PRESENT VALUE:

Net present value = Present value of net cash inflow – Total net initial investment Since it might be possible that some additional investment may also be required during the life time of the project then appropriate formula shall be: Net present value = Present value of cash inflow – Present value of cash outflow The steps to calculating net present value are: 1. Determine the net cash inflow in each year of the investment. 2. Select the desired rate of return or discounting rate or Weighted Average Cost of Capital. 3. Find the discount factor for each year based on the desired rate of return selected. 4. Determine the present values of the net cash flows by multiplying the cash flows by respective the

discount factors of respective period called Present Value (PV) of Cash flows 5. Total the amounts of all PVs of Cash Flows Decision Rule: If NPV > 0 Accept the Proposal If NPV < 0 Reject the Proposal Example 43: Compute the net present value for a project with a net investment of Rs. 1,00,000 and net cash flows year one is Rs. 55,000; for year two is Rs. 80,000 and for year 3 is Rs. 15,000. Further, the company’s cost of capital is 10%? [PVIF @ 10% for 3 years are 0.909, 0.826 and 0.751] Solution:



NOMINAL RATE OF RETURN:

The nominal rate is the stated interest rate. If a bank pays 5% annually on a savings account, then 5% is the nominal interest rate. So if you deposit Rs. 100 for 1 year, you will receive Rs. 5 in interest. However, that Rs. 5 will probably be worth less at the end of the year than it would have been at the beginning. This is because inflation lowers the value of money. As goods, services, and assets, such as real estate, rise in price. The nominal interest rate is conceptually the simplest type of interest rate. It is quite simply the stated interest rate of a given bond or loan. It is also defined as a stated interest rate. This interest works according to the simple interest and does not take into account the compounding periods. Real Rate of Return: The real interest rate is so named because it states the “real” rate that the lender or investor receives after inflation is factored in; that is, the interest rate that exceeds the inflation rate. A comparison of real and nominal interest rates can therefore be summed up in this equation: Nominal Rate of Return – Inflation = Real Rate of Return Nominal Interest Rate = Real Interest Rate + Inflation Effective Rate: It is the actual equivalent annual rate of interest at which an investment grows in value when interest is credited more often than once a year. If interest is paid m times in a year it can be found by calculating: The chief advantage to knowing the difference between nominal, real and effective rates is that it allows consumers to make better decisions about their loans and investments. A loan with frequent compounding periods will be more expensive than one that compounds annually. A bond that only pays a 1% real interest rate may not be worth investors’ time if they seek to grow their assets over time. These rates effectively reveal the true return that will be posted by a fixed-income investment and the true cost of borrowing for an individual or business. Effective and nominal interest rates allow banks to use the number that looks most advantageous to the consumer. When banks are charging interest, they advertise the nominal rate, which is lower and does not reflect how much interest the consumer would owe on the balance after a full year of compounding. On the other hand, with deposit accounts where banks are paying interest, they generally advertise the effective rate because it is higher than the nominal rate.

COMPOUND ANNUAL GROWTH RATE (CAGR): Compounded Annual Growth Rate (CAGR) is a business and investing specific term for the smoothed annualized gain of an investment over a given time periodic is not an accounting term, but remains widely used, particularly in growth industries or to compare the growth rates of two investments because CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. CAGR is often used to describe the growth over a period of time of some element of the business, for example revenue, units delivered, registered users, etc. Applications: These are some of the common CAGR applications: • Calculating average returns of investment funds. • Demonstrating and comparing the performance of investment advisors. • Comparing the historical returns of stocks with bonds or with a savings account. • Forecasting future values based on the CAGR of a data series. • Analysing and communicating the behaviour, over a series of years, of different business measures such as

sales, market share, costs, customer satisfaction, and performance.



Choose the most appropriate option (a) (b) (c) or (d): 1. S.I on Rs. 3,500 for 3 years at 12% per annum is: (a) Rs. 1,200 (b) Rs. 1,260 (c) Rs. 2,260 (d) none of these 2. P = 5,000, R = 15, T = 4 ½ using I = PRT/100, I will be: (a) Rs. 3,375 (b) Rs. 3,300 (c) Rs. 3,735 (d) none of these 3. If P = 5,000, T = 1, I = Rs. 300, R will be: (a) 5% (b) 4% (c) 6% (d) none of these 4. If P = Rs. 4,500, A = Rs. 7,200, then Simple interest i.e. I will be: (a) Rs. 2,000 (b) Rs. 3,000 (c) Rs. 2,500 (d) Rs. 2,700 5. P=Rs. 12,000, A=Rs. 16,500, T=2½ years. Rate of percent per annum simple interest will be: (a) 15% (b) 12% (c) 10% (d) none of these 6. P=Rs. 10,000, I=Rs. 2,500, R=12½% SI. The number of years T will be: (a) 1½ years (b) 2 years (c) 3 years (d) none of these 7. P=Rs. 8,500, A=Rs. 10,200, R=121/2% SI. T will be: (a) 1 yr. 7 mth. (b) 2 yrs. (c) 11/2 yr. (d) none of these 8. The sum required to earn a monthly interest of Rs. 1,200 at 18% per annum SI is: (a) Rs. 50,000 (b) Rs. 60,000 (c) Rs. 80,000 (d) none of these 9. A sum of money amount to Rs. 6,200 in 2 years and Rs. 7,400 in 3 years. The principal and rate of interest are: (a) Rs. 3,800, 31.57% (b) Rs. 3,000, 20% (c) Rs. 3,500, 15% (d) none of these 10. A sum of money doubles itself in 10 years. The number of years it would triple itself is: (a) 25 years (b) 15 years (c) 20 years (d) none of these 11. If P=Rs. 1,000, R=5% p.a., n=4; What is Amount and C.I. is: (a) Rs. 1,215.50, Rs. 215.50 (b) Rs. 1,125, Rs. 125 (c) Rs. 2,115, Rs. 115 (d) none of these 12. Rs. 100 will become after 20 years at 5% p.a. compound interest amount of: (a) Rs. 250 (b) Rs. 205 (c) Rs. 265.50 (d) none of these 13. The effective rate of interest corresponding to a nominal rate 3% p.a. payable half yearly is: (a) 3.2% p.a. (b) 3.25% p.a. (c) 3.0225% p.a. (d) none of these 14. A machine is depreciated at the rate of 20% on reducing balance. The original cost of the machine was Rs. 1,00,000 and its ultimate scrap value was Rs. 30,000. The effective life of the machine is: (a) 4.5 years (appx.) (b) 5.4 years (appx.) (c) 5 years (appx.) (d) none of these 15. If A = Rs. 1,000, n = 2 years, R = 6% p.a. compound interest payable half-yearly, then principal (P) is: (a) Rs. 888.80 (b) Rs. 885 (c) 800 (d) none of these 16. The population of town increases every year by 2% of the population at the beginning of that year. The number of years by which the total increase of population be 40% is: (a) 7 years (b) 10 years (c) 17 years (app.) (d) none of these




17. The difference between C.I and S.I on a certain sum of money invested for 3 years at 6% p.a. is Rs. 110.16. The sum is: (a) Rs. 3,000 (b) Rs. 3,700 (c) Rs. 12,000 (d) Rs. 10,000 18. The useful life of a machine is estimated to be 10 years and cost Rs. 10,000. Rate of depreciation is 10% p.a. The scrap value at the end of its life is: (a) Rs. 3,486.78 (b) Rs. 4,383 (c) Rs. 3,400 (d) none of these 19. The effective rate of interest corresponding a nominal rate of 7% p.a. convertible quarterly is: (a) 7% (b) 7.5% (c) 5% (d) 7.18% 20. The C.I. on Rs. 16,000 for 1 ½ years at 10% p.a. payable half -yearly is: (a) Rs. 2,222 (b) Rs. 2,522 (c) Rs. 2,500 (d) none of these 21. The C.I. on Rs. 40,000 at 10% p.a. for 1 year when the interest is payable quarterly is: (a) Rs. 4,000 (b) Rs. 4,100 (c) Rs. 4,152.51 (d) none of these 22. The difference between the S.I. and the C.I. on Rs. 2,400 for 2 years at 5% p.a. is: (a) Rs. 5 (b) Rs. 10 (c) Rs. 16 (d) Rs. 6 23. The annual birth and death rates per 1,000 are 39.4 and 19.4 respectively. The number of years in which the population will be doubled assuming there is no immigration or emigration is: (a) 35 years (b) 30 years (c) 25 years (d) none 24. The C.I. on Rs. 4,000 for 6 months at 12% p.a. payable quarterly is: (a) Rs. 243.60 (b) Rs. 240 (c) Rs. 243 (d) none of these 25. The present value of an annuity of Rs. 3,000 for 15 years at 4.5% p.a. C.I. is: (a) Rs. 23,809.41 (b) Rs. 32,218.63 (c) Rs. 32,908.41 (d) none of these 26. The amount of an annuity certain of Rs. 150 for 12 years at 3.5% p.a. C.I. is: (a) Rs. 2,190.28 (b) Rs. 1,290.28 (c) Rs. 2,180.28 (d) none of these 27. A loan of Rs. 10,000 is to be paid back in 30 equal instalments. The amount of each instalment to cover the principal and at 4% p.a. C.I. is: (a) Rs. 587.87 (b) Rs. 587 (c) Rs. 578.87 (d) none of these 28. A = Rs. 1,200 n = 12 years i = 0.08, V=? Using the formula: value of v will be: (a) Rs. 3,039 (b) Rs. 3,990 (c) Rs. 9,930 (d) none of these 29. a = Rs. 100 n = 10, i = 5% find the FV of annuity. Using the formula: , FV is equal to: (a) Rs. 1,258 (b) Rs. 2,581 (c) Rs. 1,528 (d) none of these 30. If the amount of an annuity after 25 years at 5% p.a. C.I. is Rs. 50,000 the annuity will be: (a) Rs. 1,406.90 (b) Rs. 1,046.90 (c) Rs. 1,146.90 (d) none of these 31. Given annuity of Rs. 100 amounts to Rs. 3,137.12 at 4.5% p.a. C.I. The number of years will be:(a) 25 years (appx.) (b) 20 years (appx.) (c) 22 years (d) none of these 32. A company borrows Rs. 10,000 on condition to repay it with compound interest at 5% p.a. by annual instalments of Rs. 1,000 each. The number of years by which the debt will be clear is: (a) 14.2 years (b) 10 years (c) 12 years (d) none of these



33. Mr. X borrowed Rs. 5,120 at 12 ½ % p.a. C.I. At the end of 3 years, the money was repaid along with the interest accrued. The amount of interest paid by him is: (a) Rs. 2,100 (b) Rs. 2,170 (c) Rs. 2,000 (d) none of these 34. Mr. Paul borrows Rs. 20,000 on condition to repay it with C.I. at 5% p.a. in annual instalments of Rs. 2000 each. The number of years for the debt to be paid off is: (a) 10 years (b) 12 years (c) 11 years (d) none of these 35. A person invests Rs. 500 at the end of each year with a bank which pays interest at 10% p.a. C.I. annually. The amount standing to his credit one year after he has made his yearly investment for the 12th time is: (a) Rs. 11,764.50 (b) Rs. 10,000 (c) Rs. 12,000 (d) none of these 36. The present value of annuity of Rs. 5,000 per annum for 12 years at 4% p.a. C.I. annually is: (a) Rs. 46,000 (b) Rs. 46,850 (c) Rs. 15,000 (d) none of these 37. A person desires to create a fund to be invested at 10% C.I. per annum to provide for a prize of Rs. 300 every year. Using V = a/I, find V and V will be: (a) Rs. 2,000 (b) Rs. 2,500 (c) Rs. 3,000 (d) none of these 38. A = Rs. 5,200, R = 5% p.a., T = 6 years, P will be: (a) Rs. 2,000 (b) Rs. 3,880 (c) Rs. 3,000 (d) none of these 39. If P = 1,000, n = 4 years, R = 5% p.a. then C.I. will be: (a) Rs. 215.50 (b) Rs. 210 (c) Rs. 220 (d) none of these 40. The time in which a sum of money will be double at 5% p.a. C.I. is: (a) Rs. 10 years (b) 12 years (c) 14.2 years (d) none of these 41. If A = Rs. 10,000, n = 18yrs., R = 4% p.a. C.I., P will be: (a) Rs. 4,000 (b) Rs. 4,900 (c) Rs. 4,500 (d) none of these 42. The time by which a sum of money would treble itself at 8% p.a. C.I. is: (a) 14.28 years (b) 14 years (c) 12 years (d) none of these 43. The present value of an annuity of Rs. 80 years for 20 years at 5% p.a. is: (a) Rs. 997 (appx.) (b) Rs. 900 (c) Rs. 1,000 (d) none of these 44. A person bought a house paying Rs. 20,000 cash down and Rs. 4,000 at the end of each year for 25 yrs. at 5% p.a. C.I. The cash down price is: (a) Rs. 75,000 (b) Rs. 76,000 (c) Rs. 76,392 (d) none of these 45. A man purchased a house valued at Rs. 3,00,000. He paid Rs. 2,00,000 at the time of purchase and agreed to pay the balance with interest at 12% per annum compounded half yearly in 20 equal half yearly instalments. If the first instalment is paid after six months from the date of purchase then the amount of each instalment is: [Given log 10.6 = 1.0253 and log 31.19 = 1.494] (a) Rs. 8,719.66 (b) Rs. 8,769.21 (c) Rs. 7,893.13 (d) none of these Choose the most appropriate option (a) (b) (c) or (d): 1. The difference between compound and simple interest at 5% per annum for 4 years on Rs. 20,000 is Rs.: (a) 250 (b) 277 (c) 300 (d) 310

ADDITIONAL QUESTIONS FOR SELF PRACTICE



2. The compound interest on half-yearly rests on Rs. 10,000 the rate for the first and second years being 6% and for the third year 9% p.a. is Rs.: (a) 2,200 (b) 2,287 (c) 2,285 (d) None 3. The present value of Rs. 10,000 due in 2 years at 5% p.a. compound interest when the interest is paid on yearly basis is Rs.: (a) 9,070 (b) 9,000 (c) 9,061 (d) None 4. The present value of Rs. 10,000 due in 2 years at 5% p.a. compound interest when the interest is paid on half-yearly basis is Rs.: (a) 9,070 (b) 9,069 (c) 9,061 (d) None 5. Johnson left Rs. 1,00,000 with the direction that it should be divided in such a way that his minor sons Tom, Dick and Harry aged 9, 12 and 15 years should each receive equally after attaining the age 25 years. The rate of interest being 3.5%, how much each son receive after getting 25 years old? (a) 50,000 (b) 51,994 (c) 52,000 (d) None 6. In how many years will a sum of money double at 5% p.a. compound interest? (a) 15 years 3 months (b) 14 years 2 months (c) 14 years 3 months (d) 15 years 2 months 7. In how many years a sum of money trebles at 5% p.a. compound interest payable on half- yearly basis? (a) 18 yrs 7 mnths (b) 18 yrs 6 mnths (c) 18 yrs 8 mnths (d) 22 yrs 3 mnths 8. A machine depreciates at 10% of its value at the beginning of a year. The cost and scrap value realized at the time of sale being Rs. 23,240 and Rs. 9,000 respectively. For how many years the machine was put to use? (a) 7 yrs (b) 8 yrs (c) 9 yrs (d) 10 yrs 9. A machine worth Rs. 4,90,740 is depreciated at 15% on its opening value each year. When its value would reduce to Rs. 2,00,000? (a) 4 yrs 6 mnths (b) 4 yrs 7 mnths (c) 4 yrs 5 mnths (d) 5 yrs 7 mnths approximately 10. A machine worth Rs. 4,90,740 is depreciated at 15% of its opening value each year. When its value would reduce by 90%? (a) 11 years 6 months (b) 11 years 7 months (c) 11 years 8 months (d) 14 years 2 months 11. Alibaba borrows Rs. 6 lakhs Housing Loan at 6% repayable in 20 annual instalments commencing at the end of the first year. How much annual payment is necessary: (a) 52,420 (b) 52,419 (c) 52,310 (d) 52,320 12. A sinking fund is created for redeeming debentures worth Rs. 5 lakhs at the end of 25 years. How much provision needs to be made out of profits each year provided sinking fund investments can earn interest at 4% p.a.? (a) 12,006 (b) 12,040 (c) 12,039 (d) 12,035 13. A machine costs Rs. 5,20,000 with an estimated life of 25 years. A sinking fund is created to replace it by a new model at 25% higher cost after 25 years with a scrap value realization of Rs. 25000. what amount should be set aside every year if the sinking fund investments accumulate at 3.5% compound interest p.a.? (a) 16,000 (b) 16,500 (c) 16,050 (d) 16,005 14. Raja aged 40 wishes his wife Rani to have Rs. 40 lakhs at his death. If his expectation of life is another 30 years and he starts making equal annual investments commencing now at 3% compound interest p.a. how much should he invest annually? (a) 84,448 (b) 84,450 (c) 84,449 (d) 84,077 15. Appu retires at 60 years receiving a pension of 14,400 a year paid in half-yearly instalments for rest of his life after reckoning his life expectation to be 13 years and that interest at 4% p.a. is payable half-yearly. What single sum is equivalent to his pension? (a) 1,45,000 (b) 1,44,900 (c) 1,44,800 (d) 1,44,700

TABLE 1(a)Compound Interest

Annual Compounding

No. of Periods (1 + i)n

n 10% per Annum 14% per Annum 18% per Annumi = 0.10 i = 0.14 i = 0.18

1 1.1 1.14 1.182 1.21 1.2996 1.39243 1.331 1.48154 1.643034 1.4641 1.68896 1.938785 1.61051 1.92541 2.287766 1.77156 2.19497 2.699557 1.94872 2.50227 3.185478 2.14359 2.85258 3.758869 2.35795 3.25194 4.4354610 2.59374 3.70722 5.2338411 2.85312 4.22622 6.1759312 3.13843 4.8179 7.2875913 3.45227 5.4924 8.5993614 3.7975 6.26,133 10.147215 4.17725 7.13792 11.973816 4.59497 8.13723 14.12917 5.05447 9.27644 16.672318 5.55992 10.5751 19.673319 6.11591 12.0557 23.214420 6.7275 12.7435 27.393

TABLE 1(b)Present Value of Re. 1

Annual Compounding

No. of Periods (1 + i)–n

n 10% per Annum 14% per Annum 18% per Annum1 .909091 .877193 .8474582 .826446 .769468 .7181843 .751315 .674972 .6086314 .683014 .592081 .5157895 .620921 .519369 .4371096 .564474 .455587 .3704327 .513158 .399638 .3139258 .466507 .35056 .2660389 .424098 .307508 .22545610 .385543 .269744 .19106411 .350494 .236618 .16191912 .318631 .20756 .13721913 .289664 .18207 .11628814 .263331 .15971 .098548915 .239392 .140097 .08351616 .217629 .122892 .070776317 .197845 .1078 .059979918 .179859 .0945614 .050830419 .163508 .0829486 .043076620 .148644 0.72762 .0365056

APPENDICES

© The Institute of Chartered Accountants of India

BUSINESS MATHEMATICSA.2

TABLE 2(a)Present Value of an Annuity

Annual Compounding

No. of 10% per Annum 14% per Annum 18% per AnnumPeriods P(n, i) 1/P(n, i) P(n, i) 1/P(n, i) P(n, i) 1/P(n, i)

n

1 .909091 1.1 .877192 1.14 .847458 1.182 1.73554 .576191 1.64666 .60729 1.56564 .6387163 2.48685 .402115 2.32163 .430732 2.17427 .4599244 3.16987 .315471 2.91371 .343205 2.69006 .3717395 3.79079 .263798 3.43308 .291284 3.12717 .3197786 4.35526 .229607 3.88867 .257158 3.4976 .285917 4.86842 .205406 4.2883 .233193 3.81153 .2623628 5.33493 .187444 4.63886 .21557 4.07757 .2452449 5.75902 .173641 4.94637 .202169 4.30302 .232395

10 6.14457 .162745 5.21611 .191714 4.49409 .22251511 6.49506 .153963 5.45273 .183394 4.65601 .21477612 6.81369 .146763 5.66029 .176669 4.79323 .20862813 7.10336 .140779 5.84236 .171164 4.90951 .20368614 7.36669 .135746 6.00207 .166609 5.00806 .19967815 7.60608 .131474 6.14217 .162809 5.09158 .19640316 7.82371 .127817 6.26506 .159615 5.16236 .1937117 8.02155 .124664 6.37286 .156915 5.22233 .19148518 8.20141 .12193 6.46742 .154621 5.27316 .18963919 8.36492 .119547 6.55037 .152663 5.31624 .18810320 8.51356 .11746 6.62313 .150986 5.35275 .18682

TABLE 2(b)Amount of an Annuity

Annual Computing

No. of 10% per Annum 14% per Annum 18% per AnnumPeriods A(n, i) 1/A(n, i) A(n, i) 1/A(n, i) A(n, i) 1/A(n, i)

n

1 1,000000 .999999994 1.00000001 999999993 1 .9999999962 2.100000 .476190473 2.14000001 .467289717 2.18000001 .4587155953 3.310000 .302114802 3.43960003 .290731478 3.57240001 .279923864 4,641000 .215470802 4.92114404 .203204782 5.21543202 .191738675 6.105100 .16379748 6.61010421 .151283545 7.15420979 .1397778416 7.71561006 .129607379 8.53551881 .117157495 9.44196755 .1059101297 9,48717108 .105405499 10.7304915 .0931923765 12.1415217 .0823619998 11.4358882 .0874440168 13.2327603 0.755700232 15.3269956 .06524435869 13.579477 0.736405385 16.0853467 .0621683833 19.0858549 .0523948237

10 15.9374248 .0627453949 19.3372953 0.517135403 23.5213088 .042514641111 18.5311672 0.539631415 23.0445166 .043394271 28.7551443 .03477638612 21.384284 .0467633146 27.270749 .0366693265 34.9310704 .028627808713 24.5227124 .0407785234 32.0886539 .0311636631 42.218663 .023686207214 27.9749837 .0357462229 37.5810655 .0266091445 50.8180224 0.19678058215 31.772482 .0314737765 43.8424147 .0228089627 60.9652664 .016402782416 35.9497303 .0278166204 50.9803528 .0196153998 72.9390144 .013710083817 40.5447033 .0246641341 59.1176022 .0169154357 87.0680371 .01148527118 45.5991737 .021930222 68.3940666 .0146211514 103.740284 .0096394569619 51.1590911 .019546868 78.969236 .0126631591 123.413535 .0081028389720 57.274999 .0174596250 91.0249291 .0109860014 146.627971 .00681998115


A.3APPENDICES

i =1

%4

i =1

%2

i =3

%4

n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0025 0000 0.9975 0623 1.0050 0000 0.9950 2488 1.0075 0000 0.9925 55832 1.0050 0625 0.9950 1869 1.0100 2500 0.9900 7450 1.0150 5625 0.9851 67083 1.0075 1877 0.9925 3734 1.0150 7513 0.9851 4876 1.0226 6917 0.9778 33334 1.0100 3756 0.9900 6219 1.0201 5050 0.9802 4752 1.0303 3919 0.9705 54175 1.0125 6266 0.9875 9321 1.0252 5125 0.9753 7067 1.0380 6673 0.9633 2920

6 1.0150 9406 0.9851 3038 1.0303 7751 0.9705 1808 1.0458 5224 0.9561 58027 1.0176 3180 0.9826 7370 1.0355 2940 0.9656 8963 1.0536 9613 0.9490 40228 1.0201 7588 0.9802 2314 1.0407 0704 0.9608 8520 1.0615 9885 0.9419 75409 1.0227 2632 0.9777 7869 1.0459 1058 0.9561 0468 1.0695 6084 0.9349 631810 1.0252 8313 0.9753 4034 1.0511 4013 0.9513 4794 1.0775 8255 0.9280 0315

11 1.0278 4634 0.9729 0807 1.0563 9583 0.9466 1487 1.0856 6441 0.9210 949412 1.0304 1596 0.9704 8187 1.0616 7781 0.9419 0534 1.0938 0690 0.9142 381513 1.0329 9200 0.9680 6171 1.0669 8620 0.9372 1924 1.1020 1045 0.9074 324114 1.0355 7448 0.9656 4759 1.0723 2113 0.9325 5646 1.1102 7553 0.9006 773315 1.0381 6341 0.9632 3949 1.0776 8274 0.9279 1688 1.1186 0259 0.8939 7254

16 1.0407 5882 0.9608 3740 1.0830 7115 0.9233 0037 1.1269 9211 0.8873 176617 1.0433 6072 0.9584 4130 1.0884 8651 0.9187 0684 1.1354 4455 0.8307 123118 1.0459 6912 0.9560 5117 1.0939 2894 0.9141 3616 1.1439 6039 0.8741 561419 1.0485 8404 0.9536 6700 1.0993 9858 0.9095 8822 1.1525 4009 0.8676 487820 1.0512 0550 0.9512 8878 1.1048 9558 0.9050 6290 1.1611 8414 0.8611 8985

21 1.0538 3352 0.9489 1649 1.1104 2006 0.9005 6010 1.1698 9302 0.8547 790122 1.0564 6810 0.9465 5011 1.1159 7216 0.8960 7971 1.1786 6722 0.8484 158923 1.0591 0927 0.9441 8964 1.1215 5202 0.8916 2160 1.1875 0723 0.8421 001424 1.0617 5704 0.9418 3505 1.1271 5978 0.8871 8567 1.1964 1353 0.8358 314025 1.0644 1144 0.9394 8634 1.1327 9558 0.8827 7181 1.2053 8663 0.8296 0933

26 1.0670 7247 0.9371 4348 1.1384 5955 0.8783 7991 1.2144 2703 0.8234 335827 1.0697 4015 0.9348 0646 1.1441 5185 0.8740 0986 1.2235 3523 0.8173 038028 1.0724 1450 0.9324 7527 1.1498 7261 0.8696 6155 1.2327 1175 0.8112 196629 1.0750 9553 0.9301 4990 1.1556 2197 0.8653 3488 1.2419 5709 0.8051 808030 1.0777 8327 0.9278 3032 1.1614 0008 0.8610 2973 1.2512 7176 0.7991 8690

31 1.0804 7773 0.9255 1653 1.1672 0708 0.8567 4600 1.2606 5630 0.7932 376232 1.0831 7892 0.9232 0851 1.1730 4312 0.8524 8358 1.2701 1122 0.7873 326233 1.0858 8687 0.9209 0624 1.1789 0833 0.8482 4237 1.2796 3706 0.7814 715834 1.0886 0159 0.9186 0972 1.1848 0288 0.8440 2226 1.2892 3434 0.7756 541835 1.0913 2309 0.9163 1892 1.1907 2689 0.8398 2314 1.2989 0359 0.7698 8008

36 1.0940 5140 0.9140 3384 1.1966 8052 0.8356 4492 1.3086 4537 0.7641 489637 1.0967 8653 0.9117 5445 1.2026 6393 0.8314 8748 1.3184 6021 0.7584 605138 1.0995 2850 0.9094 8075 1.2086 7725 0.8273 5073 1.3283 4866 0.7528 144039 1.1022 7732 0.9072 1272 1.2147 2063 0.8232 3455 1.3383 1128 0.7472 103240 1.1050 3301 0.9049 5034 1.2207 9424 0.8191 3886 1.3483 4861 0.7416 4796

41 1.1077 9559 0.9026 9361 1.2268 9821 0.8150 6354 1.3584 6123 0.7361 270142 1.1105 6508 0.9004 4250 1.2330 3270 0.8110 0850 1.3686 4969 0.7306 471643 1.1133 4149 0.8981 9701 1.2391 9786 0.8069 7363 1.3789 1456 0.7252 080944 1.1161 2485 0.8959 5712 1.2453 9385 0.8029 5884 1.3892 5642 0.7198 095245 1.1189 1516 0.8937 2281 1.2516 2082 0.7989 6402 1.3996 7584 0.7144 5114

46 1.1217 1245 0.8914 9407 1.2578 7892 0.7949 8907 1.4101 7341 0.7091 326447 1.1245 1673 0.8892 7090 1.2641 6832 0.7910 3390 1.4207 4971 0.7038 537448 1.1273 2802 0.8870 5326 1.2704 8916 0.7870 9841 1.4314 0533 0.6986 141449 1.1301 4634 0.8848 4116 1.2768 4161 0.7831 8250 1.4421 4087 0.6934 135350 1.1329 7171 0.8826 3457 1.2832 2581 0.7792 8607 1.4529 5693 0.6882 5165

TABLE 3Future Value and Present Value

i = rate of interest per period, n = number of periods



n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0100 0000 0.9900 9901 1.0125 0000 0.9876 5432 1.0150 0000 0.9852 21672 1.0201 0000 0.9802 9605 1.0251 5625 0.9754 6106 1.0302 2500 0.9706 61753 1.0303 0100 0.9705 9015 1.0379 7070 0.9634 1833 1.0456 7838 0.9563 16994 1.0406 0401 0.9609 8034 1.0509 4534 0.9515 2428 1.0613 6355 0.9421 84235 1.0510 1005 0.9514 6569 1.0640 8215 0.9397 7706 1.0772 8400 0.9282 6033

6 1.0615 2015 0.9420 4524 1.0773 8318 0.9281 7488 1.0934 4326 0.9145 42197 1.0721 3535 0.9327 1805 1.0908 5047 0.9167 1593 1.1098 4491 0.9010 26798 1.0828 5671 0.9234 8322 1.1044 8610 0.9053 9845 1.1264 9259 0.8877 11129 1.0936 8527 0.9143 3982 1.1182 9218 0.8942 2069 1.1433 8998 0.8745 922410 1.1046 2213 0.9052 8695 1.1322 7083 0.8831 8093 1.1605 4083 0.8616 6723

11 1,1156 6835 0.8963 2372 1,1464 2422 0.8722 7746 1.1779 4894 0.8489 332312 1.1268 2503 0.8874 4923 1.1607 5452 0.8615 0860 1.1956 1817 0.8363 874213 1.1380 9328 0.8786 6260 1.1752 6395 0.8508 7269 1.2135 5244 0.8240 270214 1.1494 7421 0.8699 6297 1.1899 5475 0.8403 6809 1.2317 5573 0.8118 492815 1.1609 6896 0.8613 4947 1.2048 2918 0.8299 9318 1.2502 3207 0.7998 5150

16 1.1725 7864 0.8528 2126 1.2198 8955 0.8197 4635 1.2689 8555 0.7880 310417 1.1843 0443 0.8443 7749 1.2351 3817 0.8096 2602 1.2880 2033 0.7763 852618 1.1961 4748 0.8360 1731 1.2505 7739 0.7996 3064 1.3073 4064 0.7649 115919 1.2081 0895 0.8277 3992 1.2662 0961 0.7897 5866 1.3269 5075 0.7536 074720 1.2201 9004 0.8195 4447 1.2820 3723 0.7800 0855 1.3468 5501 0.7424 7042

21 1.2323 9194 0.8114 3017 1.2980 6270 0.7703 7881 1.3670 5783 0.7314 979522 1.2447 1586 0.8033 9621 1.3142 8848 0.7608 6796 1.3875 6370 0.7206 876323 1.2571 6302 0.7954 4179 1.3307 1709 0.7514 7453 1.4083 7715 0.7100 370824 1.2697 3465 0.7875 6613 1.3473 5105 0.7421 9707 1.4295 0281 0.6995 439225 1.2824 3200 0.7797 6844 1.3641 9294 0.7330 3414 1.4509 4535 0.6892 0583

26 1.2952 5631 0.7720 4796 1.3812 4535 0.7239 8434 1.4727 0953 0.6790 205227 1.3082 0888 0.7644 0392 1.3985 1092 0.7150 4626 1.4948 0018 0.6689 857428 1.3212 9097 0.7568 3557 1.4159 9230 0.7062 1853 1.5172 2218 0.6590 992529 1.3345 0388 0.7493 4215 1.4336 9221 0.6974 9978 1.5399 8051 0.6493 588730 1.3478 4892 0.7419 2292 1.4516 1336 0.6888 8867 1.5630 8022 0.6397 6243

31 1.3613 2740 0.7345 7715 1.4697 5853 0.6803 8387 1.5865 2642 0.6303 078132 1.3749 4068 0.7273 0411 1.4881 3051 0.6719 8407 1.6103 2432 0.6209 929233 1.3886 9009 0.7201 0307 1.5067 3214 0.6636 8797 1.6344 7918 0.6118 156834 1.4025 7699 0.7129 7334 1.5255 6629 0.6554 9429 1.6589 9637 0.6027 740735 1.4166 0276 0.7059 1420 1.5446 3587 0.6474 0177 1.6838 8132 0.5938 6608

36 1.4307 6878 0.6989 2495 1.5639 4382 0.6394 0916 1.7091 3954 0.5850 897437 1.4450 7647 0.6920 0490 1.5834 9312 0.6315 1522 1.7347 7663 0.5764 430938 1.4595 2724 0.6851 5337 1.6032 8678 0.6237 1873 1.7607 9828 0.5679 242339 1.4741 2251 0.6783 6967 16233 2787 06160 1850 1.7872 1025 0.5595 312640 1.4888 6373 0.6716 5314 1.6436 1946 0.6084 1334 1.8140 1841 0.5512 6232

41 1.5037 5237 0.6650 0311 1.6641 6471 0.6009 0206 1.8412 2868 0.5431 155942 1.5187 8989 0.6584 1892 1.6849 6677 0.5934 8352 1.8688 4712 0.5350 892543 1.5339 7779 0.6518 9992 1.7060 2885 0.5861 5656 1.8968 7982 0.5271 815344 1.5493 1757 0.6454 4546 1.7273 5421 0.5789 2006 1.9253 3302 0.5193 906745 1.5648 1075 0.6390 5492 1.7489 4614 0.5717 7290 1.9542 1301 0.5117 1494

46 1.5804 5885 0.6327 2764 1.7708 0797 0.5647 1397 1.9835 2621 0.5041 526547 1.5962 6344 0.6264 6301 1.7929 4306 0.5577 4219 2.0132 7910 0.4967 021248 1.6122 2608 0.6202 6041 1.8153 5485 0.5508 5649 2.0434 7829 0.4893 617049 1.6283 4834 0.6141 1921 1.8380 4679 0.5440 5579 2.0741 3046 0.4821 297550 1.6446 3182 0.6080 3882 1.8610 2237 0.5373 3905 2.1052 4242 0.4750 0468

i =1% i =11

%4

i =11

%2


A.5APPENDICES

n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0175 0000 0.9828 0098 1.0200 0000 0.9803 9216 1.0225 0000 0.9779 95112 1.0353 0625 0.9658 9777 1.0404 0000 0.9611 6878 1.0455 0625 0.9564 74443 1.0534 2411 0.9492 8528 1.0612 0800 0.9423 2233 1.0690 3014 0.9354 27324 1.0718 5903 0.9329 5851 1.0824 3216 0.9238 4543 1.0930 8332 0.9148 43355 1.0906 1656 0.9169 1254 1.1040 8080 0.9057 3081 1.1176 7769 0.8947 1232

6 1.1097 0235 0.9011 4254 1.1261 6242 0.8879 7138 1.1428 2544 0.8750 24277 1.1291 2215 0.8856 4378 1.1486 8567 0.8705 6018 1.1685 3901 0.8557 69468 1.1488 8178 0.8704 1157 1.1716 5938 0.8534 9037 1.1948 3114 0.8369 38359 1.1689 8721 0.8554 4135 1.1950 9257 0.8367 5527 1.2217 1484 0.8185 216110 1.1894 4449 0.8407 2860 1.2189 9442 0.8203 4830 1.2492 0343 0.8005 1013

11 1.2102 5977 0.8262 6889 1.2433 7431 0.8042 6304 1.2773 1050 0.7828 949912 1.2314 3931 0.8120 5788 1.2682 4179 0.7884 9318 1.3060 4999 0.7656 674813 1.2529 8950 0.7980 9128 1.2936 0663 0.7730 3253 1.3354 3611 0.7488 190514 1.2749 1682 0.7843 6490 1.3194 7876 0.7578 7502 1.3654 8343 0.7323 413715 1.2972 2786 0.7708 7459 1.3458 6834 0.7430 1473 1.3962 0680 0.7162 2628

16 1.3199 2935 0.7576 1631 1.3727 8571 0.7284 4581 1.4276 2146 0.7004 658017 1.3430 2811 0.7445 8605 1.4002 4142 0.7141 6256 1.4597 4294 0.6850 521218 1.3665 3111 0.7317 7990 1.4282 4625 0.7001 5937 1.4925 8716 0.6699 776319 1.3904 4540 0.7191 9401 1.4568 1117 0.6864 3076 1.5261 7037 0.6552 348420 1.4147 7820 0.7068 2458 1.4859 4740 0.6729 7133 1.5605 0920 0.6408 1647

21 1.4395 3681 0.6946 6789 1.5156 6634 0.6597 7582 1.5956 2066 0.6267 153822 1.4647 2871 0.6827 2028 1.5459 7967 0.6468 3904 1.6315 2212 0.6129 245723 1.4903 6146 0.6709 7817 1.5768 9926 0.6341 5592 1.6682 3137 0.5994 372424 1.5164 4279 0.6594 3800 1.6084 3725 0.6217 2149 1.7057 6658 0.5862 466825 1.5429 8054 0.6480 9632 1.6406 0599 0.6095 3087 1.7441 4632 0.5733 4639

26 1.5699 8269 0.6369 4970 1.6734 1811 0.5975 7928 1.7833 8962 0.5607 299727 1.5974 5739 0.6259 9479 1.7068 8648 0.5858 6204 1.8235 1588 0.5483 911728 1.6254 1290 0.6152 2829 1.7410 2421 0.5743 7455 1.8645 4499 0.5363 238829 1.6538 5762 0.6046 4697 1.7758 4469 0.5631 1231 1.9064 9725 0.5245 221330 1.6828 0013 0.5942 4764 1.8113 6158 0.5520 7089 1.9493 9344 0.5129 8008

31 1.7122 4913 0.5840 2716 1.8475 8882 0.5412 4597 1.9932 5479 0.5016 920132 1.7422 1349 0.5739 8247 1.8845 4059 0.5306 3330 2.0381 0303 0.4906 523333 1.7727 0223 0.5641 1053 1.9222 3140 0.5205 2873 2.0839 6034 0.4798 555834 1,8037 2452 0.5544 0839 1.9606 7603 0.5100 2817 2.1308 4945 0.4692 964135 1.8352 8970 0.5448 7311 1.9998 8955 0.5000 2761 2.1787 9356 0.4589 6960

36 1.8674 0727 0.5355 0183 2.0398 8734 0.4902 2315 2.2278 1642 0.4488 700237 1.9000 8689 0.5262 9172 2.0806 8509 0.4806 1093 2.2779 4229 0.4389 926838 1.9333 3841 0.5172 4002 2.1222 9879 0.4711 8719 2.3291 9599 0.4293 327039 1.9671 7184 0.5083 4400 2.1647 4477 0.4619 4822 2.3816 0290 0.4198 852840 2.0015 9734 0.4996 0098 2.2080 3966 0.4528 9042 2.4351 8897 0.4106 4575

41 2.0366 2530 0.4910 0834 2.2522 0046 0.4440 1021 2.4899 8072 0.4016 095442 2.0722 6624 0.4825 6348 2.2972 4447 0.4353 0413 2.5460 0528 0.3927 721643 2.1085 3090 0.4742 6386 2.3431 8936 0.4267 6875 2.6032 9040 0.3841 292544 2.1454 3019 0.4661 0699 2.3900 5314 0.4184 0074 2.6618 6444 0.3756 765345 2.1829 7522 0.4580 9040 2.4378 5421 0.4101 9680 2.7217 5639 0.3674 0981

46 2.2211 7728 0.4502 1170 2.4866 1129 0.4021 5373 2.7829 9590 0.3593 250047 2.2600 4789 0.4424 6850 2.5363 4352 0.3942 6836 2.8456 1331 0.3514 180948 2.2995 9872 0.4348 5848 2.5870 7039 0.3865 3761 2.9096 3961 0.3436 851849 2.3398 4170 0.4273 7934 2.6388 1179 0.3789 5844 2.9751 0650 0.3361 224250 2.3807 8893 0.4200 2883 2.6915 8803 0.3715 2788 3.0420 4640 0.3287 2608

i = 21

%4

i = 13

%4

i = 2%



n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0250 0000 0.9756 0976 1.0300 0000 0.9708 7379 1.0350 0000 0.9661 83572 1.0506 2500 0.9518 1440 1.0609 0000 0.9425 9591 1.0712 2500 0.9335 10703 1.0768 9063 0.9285 9941 1.0927 2700 0.9151 4166 1.1087 1788 0.9019 42714 1.1038 1289 0.9059 5064 1.1255 0881 0.8884 8705 1.1475 2300 0.8714 42235 1.1314 0821 0.8838 5429 1.1592 7407 0.8626 0878 1.1876 8631 0.8419 7317

6 1.1596 9342 0.8622 9687 1.1940 5230 0.8374 8426 1.2292 5533 0.8135 00647 1.1886 8575 0.8412 6524 1.2298 7387 0.8130 9151 1.2722 7926 0.7859 90968 1.2184 0290 0.8207 4657 1.2667 7008 0.7894 0923 1.3168 0904 0.7594 11569 1.2488 6297 0.8007 2836 1.3047 7318 0.7664 1673 1.3628 9735 0.7337 309710 1.2800 8454 0.7811 9840 1.3439 1638 0.7440 9391 1.4105 9876 0.7089 1881

11 1.3120 8666 0.7621 4478 1.3842 3387 0.7224 2128 1.4599 6972 0.6849 457112 1.3448 8882 0.7435 5589 1.4257 6089 0.7013 7988 1.5110 6866 0.6617 833013 1.3785 1104 0.7254 2038 1.4685 3371 0.6809 5134 1.5639 5606 0.6394 041514 1.4129 7382 0.7077 2720 1.5125 8972 0.6611 1781 1.6186 9452 0.6177 817915 1.4482 9817 0.6904 6556 1.5579 6742 0.6418 6195 1.6753 4883 0.5968 9062

16 1.4845 0562 0.6736 2493 1.6047 0644 0.6231 6694 1.7339 8604 0.5767 059117 1.5216 1826 0.6571 9506 1.6528 4763 0.6050 1645 1.7946 7555 0.5572 037818 1.5596 5872 0.6411 6591 1.7024 3306 0.5873 9461 1.8574 8920 0.5383 611419 1.5986 5019 0.6255 2772 1.7535 0605 0.5702 8603 1.9225 0132 0.5201 556920 1.6386 1644 0.6102 7094 1.8061 1123 0.5536 7575 1.9897 8886 0.5025 6588

21 1.6795 8185 0.5953 8629 1.8602 9457 0.5375 4928 2.0594 3147 0.4855 709022 1,7215 7140 0.5808 6467 1,9161 0341 0.5218 9250 2,1315 1158 0.4691 506323 1.7646 1068 0.5666 9724 1.9735 8651 0.5066 9175 2.2061 1448 0.4532 856324 1.8087 2595 0.5528 7535 2.0327 9411 0.4919 3374 2.2833 2849 0.4379 571325 1.8539 4410 0.5393 9059 2.0937 7793 0.4776 0557 2.3632 4498 0.4231 4699

26 1.9002 9270 0.5262 3472 2.1565 9127 0.4636 9473 2.4459 5856 0.4088 376727 1.9478 0002 0.5133 9973 2.2212 8901 0.4501 8906 2.5315 6711 0.3950 122428 1.9964 9502 0.5008 7778 2.2879 2768 0.4370 7675 2.6201 7196 0.3816 543429 2.0464 0739 0.4886 6125 2.3565 6551 0.4243 4636 2.7118 7798 0.3687 481530 2.0975 6758 0.4767 4269 2.4272 6247 0.4119 8676 2.8067 9370 0.3562 7841

31 2.1500 0677 0.4651 1481 2.5000 8035 0.3999 8715 2.9050 3148 0.3442 303532 2.2037 5694 0.4537 7055 2.5750 8276 0.3883 3703 3.0067 0759 0.3325 897133 2.2588 5086 0.4427 0298 2.6523 3524 0.3770 2625 3.1119 4235 0.3213 427134 2.3153 2213 0.4319 0534 2.7319 0530 0.3660 4490 3.2208 6033 0.3104 760535 2.3732 0519 0.4213 7107 2.8138 6245 0.3553 8340 3.3335 9045 0.2999 7686

36 2.4325 3532 0.4110 9372 2.8982 7833 0.3450 3243 3.4502 6611 0.2898 327237 2.4933 4870 0.4010 6705 2.9852 2668 0.3349 8294 3.5710 2543 0.2800 316138 2.5556 8242 0.3912 8492 3.0747 8348 0.3252 2615 3.6960 1132 0.2705 619439 2.6195 7448 0.3817 4139 3.1670 2698 0.3157 5355 3.8253 7171 0.2614 125040 2.6850 6384 0.3724 3062 3.2620 3779 0.3065 5684 3.9592 5972 0.2525 7247

41 2.7521 9043 0.3633 4695 3.3598 9893 0.2976 2800 4.0978 3381 0.2440 313742 2.8209 9520 0.3544 8483 3.4606 9589 0.2889 5922 4.2412 5799 0.2357 791043 2.8915 2008 0.3458 3886 3.5645 1677 0.2805 4294 4.3897 0202 0.2278 059044 2.9638 0808 03374 0376 3.6714 5227 0.2723 7178 4.5433 4160 0.2201 023145 3.0379 0328 0.3291 7440 3.7815 9584 0.2644 3862 4.7023 5855 0.2126 5924

46 3.1138 5086 0.3211 4576 3.8950 4372 0.2567 3653 4.8669 4110 0.2054 678747 3.1916 9713 0.3133 1294 4.0118 9503 0.2492 5876 5.0372 8404 0.1985 196848 3.2714 8956 0.3056 7116 4.1322 5188 0.2419 9880 5.2135 8898 0.1918 064549 3.3532 7680 0.2982 1576 4.2562 1944 0.2349 5029 5.3960 6459 0.1853 202450 3.4371 0872 0.2909 4221 4.3839 0602 0.2281 0708 5.5849 2686 0.1790 5337

i = 3% i = 3 1 % 2i = 2 1 % 2


A.7APPENDICES

n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0400 0000 0.9615 3846 1.0450 0000 0.9569 3780 1.0500 0000 0.9523 80952 1.0816 0000 0.9245 5621 1.0920 2500 0.9157 2995 1.1025 0000 0.9070 29483 1.1248 6400 0.8889 9636 1.1411 6613 0.8762 9660 1.1576 2500 0.8638 37604 1.1698 5856 0.8548 0419 1.1925 1860 0.8382 6134 1.2155 0625 0.8227 02475 1.2166 5290 0.8219 2711 1.2461 8194 0.8024 5105 1.2762 8156 0.7835 2617

6 1.2653 1902 0.7903 1453 1.3022 6012 0.7678 9574 1.3400 9564 0.7462 15407 1.3159 3178 0.7599 1781 1.3608 6183 0.7348 2846 1.4071 0042 0.7106 81338 1.3685 6905 0.7306 9021 1.4221 0061 0.7031 8513 1.4774 5544 0.6768 39369 1.4233 1181 0.7025 8674 1.4860 9514 0.6729 0443 1.5513 2822 0.6446 089210 1.4802 4428 0.6755 6417 1.5529 6942 0.6439 2768 1.6288 9463 0.6139 1325

11 1.5394 5406 0.6495 8093 1.6228 5305 0.6161 9874 1.7103 3936 0.5846 792912 1.6010 3222 0.6245 9705 1.6958 8143 0.5896 6386 1.7958 5633 0.5568 374213 1.6650 7351 0.6005 7409 1.7721 9610 0.5642 7164 1.8856 4914 0.5303 213514 1.7316 7645 0.5774 7508 1.8519 4492 0.5399 7286 1.9799 3160 0.5050 679515 1.8009 4351 0.5552 6450 1.9352 8244 0.5167 2044 2.0789 2818 0.4810 1710

16 1.8729 8125 0.5339 0818 2.0223 7015 0.4944 6932 2.1828 7459 0.4581 115217 1.9479 0050 0.5133 7325 2.1133 7681 0.4731 7639 2.2920 1832 0.4362 966918 2.0258 1652 0.4936 2812 2.2084 7877 0.4528 0037 2.4066 1923 0.4155 206519 2.1068 4918 0.4746 4242 2.3078 6031 0.4333 0179 2.5269 5020 0.3957 339620 2.1911 2314 0.4563 8695 2.4117 1402 0.4146 4286 2.6532 9771 0.3768 8948

21 2.2787 6807 0.4388 3360 2.5202 4116 0.3967 8743 2.7859 6259 0.3589 423622 2.3699 1879 0.4219 5539 2.6336 5201 0.3797 0089 2.9252 6072 0.3418 498723 2.4647 1554 0.4057 2633 2.7521 6635 0.3633 5013 3.0715 2376 0.3255 713124 2.5633 0416 0.3901 2147 2.8760 1383 0.3477 0347 3.2250 9994 0.3100 679125 2.6658 3633 0.3751 1680 3.0054 3446 0.3327 3060 3.3863 5494 0.2953 0277

26 2.7724 6978 0.3606 8923 3.1406 7901 0.3184 0248 3.5556 7269 0.2812 407327 2.8833 6858 0.3468 1657 3.2820 0956 0.3046 9137 3.7334 5632 0.2678 483228 2.9987 0332 0.3334 7747 3.4296 9999 0.2915 7069 3.9201 2914 0.2550 936429 3.1186 5145 0.3206 5141 3.5840 3649 0.2790 1502 4.1161 3560 0.2429 463230 3.2433 9751 0.3083 1867 3.7453 1813 0.2670 0002 4.3219 4238 0.2313 7745

31 3.3731 3341 0.2964 6026 3.9138 5745 0.2555 0241 4.5380 3949 0.2203 594732 3.5080 5875 0.2850 5794 4.0899 8104 0.2444 9991 4.7649 4147 0.2098 661733 3.6483 8110 0.2740 9417 4.2740 3018 0.2339 7121 5.0031 8854 0.1998 725434 3.7943 1634 0.2635 5209 4.4663 6154 0.2238 9589 5.2533 4797 0.1903 548035 3.9460 8899 0.2534 1547 4.6673 4781 0.2142 5444 5.5160 1537 0.1812 9029

36 4.1039 3255 0.2436 6872 4.8773 7846 0.2050 2817 5.7918 1614 0.1726 574137 4.2680 8986 0.2342 9685 5.0968 6049 0.1961 9921 6.0814 0694 0.1644 356338 4.4388 1345 0.2252 8543 5.3262 1921 0.1877 5044 6.3854 7729 0.1566 053639 4.6163 6599 0.2166 2061 5.5658 9908 0.1796 6549 6.7047 5115 0.1491 479740 4.8010 2063 0.2082 8904 5.8163 6454 0.1719 2870 7.0399 8871 0.1420 4568

41 4.9930 6145 0.2002 7793 6.0781 0094 0.1645 2507 7.3919 8815 0.1352 816042 5.1927 8391 0.1925 7493 6.3516 1548 0.1574 4026 7.7615 8756 0.1288 396243 5.4004 9527 0.1851 6820 6.6374 3818 0.1506 6054 8.1496 6693 0.1227 044044 5.6165 1508 0.1780 4635 6.9361 2290 0.1441 7276 8.5571 5028 0.1168 613345 5.8411 7568 0.1711 9841 7.2482 4843 0.1379 6437 8.9850 0779 0.1112 9651

46 6.0748 2271 0.1646 1386 7.5744 1961 0.1320 2332 9.4342 5818 0.1059 966847 6.3178 1562 0.1582 8256 7.9152 6849 0.1263 3810 9.9059 7109 0.1009 492148 6.5705 2824 0.1521 9476 8.2714 5557 0.1208 9771 10.4012 6965 0.0961 421149 6.8333 4937 0.1463 4112 8.6436 7107 0.1156 9158 10.9213 3313 0.0915 639150 7.1066 8335 0.1407 1262 9.0326 3627 0.1107 0965 11.4673 9979 0.0872 0373

i = 41

% 2i = 4% i = 5%



n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n (1 + i)n (1 + i)–n

1 1.0600 0000 0.9433 9623 1.0700 0000 0.9345 7944 1.0800 0000 0.9259 25932 1.1236 0000 0.8899 9644 1.1449 0000 0.8734 3873 1.1664 0000 0.8573 38823 1.1910 1600 0.8396 1928 1.2250 4300 0.8162 9788 1.2597 1200 0.7938 32244 1.2624 7696 0.7920 9366 1.3107 9601 0.7628 9521 1.3604 8896 0.7350 29855 1.3382 2558 0.7472 5817 1.4025 5173 0.7129 8618 1.4693 2808 0.6805 8320

6 1.4185 1911 0.7049 6054 1.5007 3035 0.6663 4222 1.5868 7432 0.6301 69637 1.5036 3026 0.6650 5711 1.6057 8148 0.6227 4974 1.7138 2427 0.5834 90408 1.5938 4807 0.6274 1237 1.7181 8618 0.5820 0910 1.8509 3021 0.5402 68889 1.6894 7896 0.5918 9846 1.8384 5921 0.5439 3374 1.9990 0463 0.5002 489710 1.7908 4770 0.5583 9478 1.9671 5136 0.5083 4929 2.1589 2500 0.4631 9349

11 1.8982 9856 0.5267 8753 2.1048 5195 0.4750 9280 2.3316 3900 0.4288 828612 2.0121 9647 0.4969 6936 2.2521 9159 0.4440 1196 2.5181 7012 0.3971 137613 2.1329 2826 0.4688 3902 2.4098 4500 0.4149 6445 2.7196 2373 0.3676 979214 2.2609 0396 04423 0096 2.5785 3415 0.3878 1724 2.9371 9362 0.3404 610415 2.3965 5819 0.4172 6506 2.7590 3154 0.3624 4602 3.1721 6911 0.3152 4170

16 2.5403 5168 0.3936 4628 2.9521 6375 0.3387 3460 3.4259 4264 0.2918 904717 2.6927 7279 0.3713 6442 3.1588 1521 0.3165 7439 3.7000 1805 0.2702 689518 2.8543 3915 0.3503 4379 3.3799 3228 0.2958 6392 3.9960 1950 0.2502 490319 3.0255 9950 0.3305 1301 3.6165 2754 0.2765 0833 4.3157 0106 0.2317 120620 3.2071 3547 0.3118 0473 3.8696 8446 0.2584 1900 4.6609 5714 0.2145 4821

21 3.3995 6360 0.2941 5540 4.1405 6237 0.2415 1309 5.0338 3372 0.1986 557522 3.6035 3742 0.2775 0510 4.4304 0174 0.2257 1317 5.4365 4041 0.1839 405123 3.8197 4966 0.2617 9726 4.7405 2986 0.2109 4688 5.8714 6365 0.1703 152824 4.0489 3464 0.2469 7855 5.0723 6695 0.1971 4662 6.3411 8074 0.1576 993425 4.2918 7072 0.2329 9863 5.4274 3264 0.1842 4918 6.8484 7520 0.1460 1790

26 4.5493 8296 0.2198 1003 5.8073 5292 0.1721 9549 7.3963 5321 0.1352 017627 4.8223 4594 0.2073 6795 6.2138 6763 0.1609 3037 7.9880 6147 0.1251 868228 5.1116 8670 0.1956 3014 6.6488 3836 0.1504 0221 8.6271 0639 0.1159 137229 5.4183 8790 0.1845 5674 7.1142 5705 0.1405 6282 9.3172 7490 0.1073 275230 5.7434 9117 0.1741 1013 7.6122 5504 0.1313 6712 10.0626 5689 0.0993 7733

31 6.0881 0064 0.1642 5484 8.1451 1290 0.1227 7301 10.8676 6944 0.0920 160532 6.4533 8668 0.1549 5740 8.7152 7080 0.1147 4113 11.7370 8300 0.0852 000533 6.8405 8988 0.1461 8622 9.3253 3975 0.1072 3470 12.6760 4964 0.0788 889334 7.2510 2528 0.1379 1153 9.9781 1354 0.1002 1934 13.6901 3361 0.0730 453135 7.6860 8679 0.1301 0522 10.6765 8148 0.0936 6294 14.7853 4429 0.0676 3454

36 8.1472 5200 0.1227 4077 11.4239 4219 0.0875 3546 15.9681 7184 0.0626 245837 8.6360 8712 0.1157 9318 12.2236 1814 0.0818 0884 17.2456 2558 0.0579 857238 9.1542 5235 0.1092 3885 13.0792 7141 0.0764 5686 18.6252 7563 0.0536 904839 9.7035 0749 0.1030 5552 13.9948 2041 0.0714 5501 20.1152 9768 0.0497 134140 10.2857 1794 0.0972 2219 14.9744 5784 0.0667 8038 21.7245 2150 0.0460 3093

41 10.9028 6101 0.0917 1905 16.0226 6989 0.0624 1157 23.4624 8322 0.0426 212342 11.5570 3267 0.0865 2740 17.1442 5678 0.0583 2857 25.3394 8187 0.0394 641143 12.2504 5463 0.0816 2962 18.3443 5475 0.0545 1268 27.3666 4042 0.0365 408444 12.9854 8191 0.0770 0908 19.6284 5959 0.0509 4643 29.5559 7166 0.0338 341145 13.7646 1083 0.0726 5007 21.0024 5176 0.0476 1349 31.9204 4939 0.0313 2788

46 14.5904 8748 0.0685 3781 22.4726 2338 0.0444 9859 34.4740 8534 0.0290 073047 15.4659 1673 0.0646 5831 24.0457 0702 0.0415 8747 37.2320 1217 0.0268 586148 16.3938 7173 0.0609 9840 25.7289 0651 0.0388 6679 40.2105 7314 0.0248 690849 17.3775 0403 0.0575 4566 27.5299 2997 0.0363 2410 43.4274 1899 0.0230 269350 18.4201 5427 0.0542 8836 29.4570 2506 0.0339 4776 46.9016 1251 0.0213 2123

i = 6% i = 7% i = 8%


A.9APPENDICES

TABLE 4Log-Tables

LOGARITHAMS



LOG-TABLES


A.11APPENDICES

TABLE 5ANTILOGARITHMS

LOG-TABLES



LOG-TABLES


A.13APPENDICES

Exam

ple:

To fi

nd th

ear

ea u

nder

the

curv

e be

twee

nth

e m

ean

and

a po

int 2

.24

stan

dard

devi

atio

ns to

the

right

of t

hem

ean,

look

up

the

valu

eop

posi

te 2

.2an

d un

der

0.04

in th

eta

ble;

0.4

875

of th

e ar

eaun

der t

hecu

rve

lies

betw

een

the

mea

n an

d a

zVa

lue

of 2

.24

Tabl

e 6

Area

s un

der t

he S

tand

ard

Nor

mal

Prob

abili

ty D

istri

butio

n be

twee

n th

e M

ean

and

Posi

tive

Valu

es o

f z

x0.

000.

010.

020.

030.

040.

050.

060.

070.

080.

09

0.0

0.00

000.

0040

0.00

800.

0120

0.01

600.

0199

0.02

390.

0279

0.03

190.

0359

0.1

0.03

980.

0438

0.04

780.

0517

0.05

570.

0596

0.06

360.

0675

0.07

140.

0753

0.2

0.07

930.

0832

0.08

710.

0910

0.09

480.

0987

0.10

260.

1064

0.11

030.

1141

0.3

0.11

790.

1217

0.12

550.

1293

0.13

310.

1368

0.14

060.

1443

0.14

800.

1517

0.4

0.15

540.

1591

0.16

280.

1664

0.17

000.

1736

0.17

720.

1808

0.18

440.

1879

0.5

0.19

150.

1950

0.19

850.

2019

0.20

540.

2088

0.21

230.

2157

0.21

900.

2224

0.6

0.22

570.

2291

0.23

240.

2357

0.23

890.

2422

0.24

540.

2486

0.25

170.

2549

0.7

0.25

800.

2611

0.26

420.

2673

0.27

040.

2734

0.27

640.

2794

0.28

230.

2852

0.8

0.28

810.

2910

0.29

390.

2967

0.29

950.

3023

0.30

510.

3078

0.31

060.

3133

0.9

0.31

590.

3186

0.32

120.

3238

0.32

640.

3289

0.33

150.

3340

0.33

650.

3389

1.0

0.34

130.

3438

0.34

610.

3485

0.35

080.

3531

0.35

540.

3577

0.35

990.

3621

1.1

0.36

430.

3665

0.36

860.

3708

0.37

290.

3749

0.37

700.

3790

0.38

100.

3830

1.2

0.38

490.

3869

0.38

880.

3907

0.39

250.

3944

0.39

620.

3980

0.39

970.

4015

1.3

0.40

320.

4049

0.40

660.

4082

0.40

990.

4115

0.41

310.

4147

0.41

620.

4177

1.4

0.41

920.

4207

0.42

220.

4236

0.42

510.

4265

0.42

790.

4292

0.43

060.

4319

1.5

0.43

320.

4345

0.43

570.

4370

0.43

820.

4394

0.44

060.

4418

0.44

290.

4441

1.6

0.44

520.

4463

0.44

740.

4484

0.44

950.

4505

0.45

150.

4525

0.45

350.

4545

1.7

0.45

540.

4564

0.45

730.

4582

0.45

910.

4599

0.46

080.

4616

0.46

250.

4633

1.8

0.46

410.

4649

0.46

560.

4664

0.46

710.

4678

0.46

860.

4693

0.46

990.

4706

1.9

0.47

130.

4719

0.47

260.

4732

0.47

380.

4744

0.47

500.

4756

0.47

610.

4767

2.0

0.47

720.

4778

0.47

830.

4788

0.47

930.

4798

0.48

030.

4808

0.48

120.

4817

2.1

0.48

210.

4826

0.48

300.

4834

0.48

380.

4842

0.48

460.

4850

0.48

540.

4857

2.2

0.48

610.

4864

0.48

680.

4871

0.48

750.

4878

0.48

810.

4884

0.48

870.

4890

2.3

0.48

930.

4896

0.48

980.

4901

0.49

040.

4906

0.49

090.

4911

0.49

130.

4916

2.4

0.49

180.

4920

0.49

220.

4925

0.49

270.

4929

0.49

310.

4932

0.49

340.

4936

2.5

0.49

380.

4940

0.49

410.

4943

0.49

450.

4946

0.49

480.

4949

0.49

510.

4952

2.6

0.49

530.

4955

0495

60.

4957

0.49

590.

4960

0.49

610.

4962

0.49

630.

4964

2.7

0.49

650.

4966

0.49

670.

4968

0.49

690.

4970

0.49

710.

4972

0.49

730.

4974

2.8

0.49

740.

4975

0.49

760.

4977

0.49

770.

4978

0.49

790.

4979

0.49

800.

4981

2.9

0.49

810.

4982

0.49

820.

4983

0.49

840.

4984

0.49

850.

4985

0.49

860.

4986

3.0

0.49

870.

4987

0.49

870.

4988

0.49

880.

4989

0.49

890.

4989

0.49

900.

4990



Tabl

e 7

Area

s in

Bot

h Ta

ils C

ombi

ned

for

Stud

ent’s

t D

istri

butio

n

Are

a in

Bot

h Ta

ils C

ombi

ned

Deg

ree

ofFr

eedo

m0.

100.

050.

020.

01

16.

314

12.7

0631

.821

63.6

572

2.92

04.

303

6.96

59.

925

32.

353

3.18

24.

541

5.84

14

2.13

22.

776

3.74

74.

604

52.

015

2.57

13.

365

4.03

26

1.94

32.

447

3.14

33.

707

71.

895

2.36

52.

998

3.49

98

1.86

02.

306

2.89

63.

355

91.

833

2.26

22.

821

3.25

010

1.81

22.

228

2.76

43.

169

111.

796

2.20

12.

718

3.10

612

1.78

22.

179

2.68

13.

055

131.

771

2.16

02.

650

3.01

214

1.76

12.

145

2.62

42.

977

151.

753

2.13

12.

602

2.94

716

1.74

62.

120

2.58

32.

921

171.

740

2.11

02.

567

2.89

818

1.73

42.

101

2.55

22.

878

191.

729

2.09

32.

539

2.86

120

1.72

52.

086

2.52

82.

845

211.

721

2.08

02.

518

2.83

122

1.71

72.

074

2.50

82.

819

231.

714

2.06

92.

500

2.80

724

1.71

12.

064

2.49

22.

797

251.

708

2.06

02.

485

2.78

726

1.70

62.

056

2.47

92.

779

271.

703

2.05

22.

473

2.77

128

1.70

12.

048

2.46

72.

763

291.

699

2.04

52.

462

2.75

630

1.69

72.

042

2.45

72.

750

401.

684

2.02

12.

423

2.70

460

1.67

12.

000

2.39

02.

660

120

1.65

81.

980

2.35

82.

617

Nor

mal

Dis

tribu

tion

1.64

51.

960

2.32

62.

576

Exam

ple:

To fi

nd th

e va

lue

of t

that

corre

spon

ds to

an

area

of 0

.10

inbo

th ta

ils o

f the

dist

ribut

ion

com

bine

d, w

hen

ther

e ar

e 19

degr

ess

offre

edom

, lo

okun

der t

he 0

.10

colu

mn,

and

proc

eed

dow

n to

the

19 d

egre

es o

ffre

edom

row

; the

appr

opria

te t

val

ueth

ere

is 1.

729


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