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CAT MBA Prepration - Maths Shortcut

To find out if a number is divisible by seven:

Take the last digit, double it, and subtract it from the rest of the

number.

If the answer is more than a 2 digit number perform the above

again.

If the result is 0 or is divisible by 7 the original number is also

divisible by 7.

Example 1 ) 259

9*2= 18.

25-18 = 7 which is divisible by 7 so 259 is also divisible by 7.

Example 2 ) 2793

3*2= 6

279-6= 273

now 3*2=6

27-6= 21 which is divisible by 7 so 2793 is also divisible by 7 .

Now find out if following are divisible by 7

1) 2841

2) 3873

3) 1393

4) 2877

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TO FIND SQUARE OF A NUMBER BETWEEN 40 to 50

Sq (44) .

1) Subtract the number from 50 getting result A.

2) Square A getting result X.

3) Subtract A from 25 getting result Y

4) Answer is xy

EXAMPLE 1 : 44

50-44=6

Sq of 6 =36

25-6 = 19

So answer 1936

EXAMPLE 2 : 47

50-47=3

Sq 0f 3 = 09

25-3= 22

So answer = 2209

NOW TRY To Find Sq of 48 ,26 and 49

TO FIND SQUARE OF A 3 DIGIT NUMBER :

LET THE NUMBER BE XYZ

SQ (XYZ) is calculated like this

STEP 1. Last digit = last digit of SQ(Z)

STEP 2. Second Last Digit = 2*Y*Z + any carryover from STEP 1.

STEP 3. Third Last Digit 2*X*Z+ Sq(Y) + any carryover from STEP

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2.

STEP 4. Fourth last digit is 2*X*Y + any carryover from STEP 3.

STEP 5 . In the beginning of result will be Sq(X) + any carryover

from Step 4.

EXAMPLE :

SQ (431)

STEP 1. Last digit = last digit of SQ(1) =1

STEP 2. Second Last Digit = 2*3*1 + any carryover from STEP

1.= 6

STEP 3. Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP

2.= 2*4*1 +9= 17. so 7 and 1 carryover

STEP 4. Fourth last digit is 2*4*3 + any carryover (which is 1) . =

24+1=25. So 5 and carry over 2.

STEP 5 . In the beginning of result will be Sq(4) + any carryover

from Step 4. So 16+2 =18.

So the result will be 185761.

If the option provided to you are such that the last two digits are

different, then you need to carry out first two steps only , thus

saving time. You may save up to 30 seconds on each

calculations and if there are 4 such questions you save 2

minutes which may really affect UR Percentile score.

PYTHAGORAS THEROEM :

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In any given exam there are about 2 to 3 questions based on pythagoras theorem. Wouldn't it be nice that youremember some of the pythagoras triplets thus saving up to 30 seconds in each question. This saved time may beused to attempt other questions. Remember one more right question may make a lot of difference in URPERCENTILE score.

The unique set of pythagoras triplets with the Hypotenuse less than 100 or one of the side less than 20 are asfollows :

(3,4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65),(16, 63, 65), (48, 55, 73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97).

(15,112,113), (17,144,145), (19,180,181), (20,99,101)

If you multiply the digits of the above mentioned sets by any constant you will again get a pythagoras triplet .

Example : Take the set (3,4,5).

Multiply it by 2 you get (6,8,10) which is also a pythagoras triplet.

Multiply it by 3 you get ( 9,12,15) which is also a pythagoras triplet.

Multiply it by 4 you get (12,16,20) which is also a pythagoras triplet.

You may multiply by any constant you will get a pythagoras triplet

Take another example (5,12,13)

Multiply it by 5,6 and 7 and check if you get a pythagoras triplet.

TIPS FOR SMART GUESSING :

You will notice that in any case, whether it is a unique triplet or it is a derived triplet (derived by multiplying a constan tto a unique triplet), all the three numbers cannot be odd.

In case of unique triplet , the hypotenuse is always odd and one of the remaining side is odd the other one is even.

Below are the first few unique triplets with first number as Odd.

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3 4 5

5 12 13

7 24 25

9 40 41

11 60 61

You will notice following trend for unique triplets with first side as odd.

Hypotenuse = (Sq(first side) +1) / 2

Other side = Hypotenuse -1

Example : First side = 3 ,

so hypotenuse = (3*3+1)/2= 5 and other side = 5-1=4

Example 2: First side = 11

so hypotenuse = (9*9+1)/2= 41 and other side = 41-1=40

Please note that the above is not true for a derived triplet for example 9,12 and 15, which has been obtained from

multiplying 3 to the triplet of 3,4,5. You may check for other derived triplets.

Below are the first few unique triplets with first number as Even .

4 3 5

8 15 17

12 35 37

16 63 65

20 99 101

You will notice following trend for unique triplets with first side as Even.

Hypotenuse = Sq( first side/ 2)+1

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Other side = Hypotenuse-2

Example 1. First side =8

So hypotenuse = sq(8/2) +1= 17

Other side = 17-2=15

Example 2. First side = 16

So hypotenuse = Sq(16/2) +1 =65

Other side = 65-2= 63

PROFIT AND LOSS : In every exam there are from one to three

questions on profit and loss, stating that the cost was first

increased by certain % and then decreased by certain %. How

nice it would be if there was an easy way to calculate the final

change in % of the cost with just one formula. It would really help

you in saving time and improving UR Percentile. Here is the

formula for the same :

Suppose the price is first increase by X% and then decreased

by Y% , the final change % in the price is given by the following

formula

Final Difference % = X- Y - XY/100.

EXAMPLE 1. : The price of T.V set is increased by 40 % of the

cost price and then decreased by 25% of the new price . On

selling, the profit for the dealer was Rs.1,000 . At what price was

the T.V sold.

From the above mentioned formula you get :

Final difference % = 40-25-(40*25/100)= 5 %.

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So if 5 % = 1,000

then 100 % = 20,000.

C.P = 20,000

S.P = 20,000+ 1000= 21,000.

EXAMPLE 2 : The price of T.V set is increased by 25 % of cost

price and then decreased by 40% of the new price . On selling,

the loss for the dealer was Rs.5,000 . At what price was the T.V

sold.

From the above mentioned formula you get :

Final difference % = 25-40-(25*45/100)= -25 %.

So if 25 % = 5,000

then 100 % = 20,000.

C.P = 20,000

S.P = 20,000 - 5,000= 15,000.

Now find out the difference in % of a product which was :

First increased by 20 % and then decreased by 10 %.

First Increased by 25 % and then decrease by 20 %.




TIPS TO IMPROVE UR PERCENTILE :

HOW ABOUT SOLVING THE FOLLOWING QUESTION IN JUST

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10 SECONDS

Ajay can finish work in 21 days and Blake in 42 days. If Ajay,

Blake and Chandana work together they finish the work in 12

days. In how many days Blake and Chandana can finish the

work together ?

(21*12 )/(24-12) = (21*12)/9= 7*4= 28 days.

NOW CAREFULLY READ THE FOLLOWING TO SOLVE THE

TIME AND WORK PROBLEMS IN FEW SECONDS.

TIME AND WORK :

1. If A can finish work in X time and B can finish work in Y time

then both together can finish work in (X*Y)/ (X+Y) time.

2. If A can finish work in X time and A and B together can finish

work in S time then B can finish work in (XS)/(X-S) time.

3. If A can finish work in X time and B in Y time and C in Z time

then they all working together will finish the work in

(XYZ)/ (XY +YZ +XZ) time

4. If A can finish work in X time and B in Y time and A,B and C

together in S time then :

C can finish work alone in (XYS)/ (XY-SX-SY)

B+C can finish in (SX)/(X-S)

and A+ C can finish in (SY)/(Y-S)

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Here is another shortcut to improve URPERCENTILE.

TYPE 1 : Price of a commodity is increased by 60 %. By how

much % should the consumption be reduced so that the

expense remain the same.

TYPE 2 : Price of a commodity is decreased by 60 %. By how

much % can the consumption be increased so that the expense

remain the same.

Solution :

TYPE1 : (100* 60 ) / (100+60) = 37.5 %

TYPE 2 : (100* 60 ) / (100-60) = 150 %

Now do the following questions for UR Practice:

Q1. Price of a commodity is increased by 10 %. By how much %

should the consumption be reduced so that the expense remain

the same.

Q2. Price of a commodity is decreased by 10 %. By how much %

can the consumption be increased so that the expense remain

the same.



the same.



the same.

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the same.



the same.

Q7. Price of a commodity is increased by 50%. By how much %


the same.



the same.



the same.

Q10. Price of a commodity is decreased by 20 %. By how much

% can the consumption be increased so that the expense

remain the same.

CAT MBA Prepration - Maths Shortcut 2Numbers

1) 2^2n-1 is always divisible by 3

2^2n-1 = (3-1)^2n -1

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= 3M +1 -1

= 3M, thus divisible by 3

2) What is the sum of the divisors of 2^5.3^7.5^3.7^2?

ANS : (2^6-1)(3^8-1)(5^4-1)(7^3-1)/2.4.6

Funda : if a number 'n' is represented as

a^x * b^y * c^z ....

where, {a,b,c,.. } are prime numbers then

Quote:

(a) the total number of factors is (x+1)(y+1)(z+1) ....

(b) the total number of relatively prime numbers less than the number is n * (1-1/a) * (1-

1/b) * (1-1/c)....

(c) the sum of relatively prime numbers less than the number is n/2 * n * (1-1/a) * (1-1/b) *

(1-1/c)....

(d) the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x*y*...)

3) what is the highest power of 10 in 203!ANS : express 10 as product of primes; 10 = 2*5

divide 203 with 2 and 5 individually

203/2 = 101

101/2 = 50

50/2 = 25

25/2 = 12

12/2 = 6

6/2 = 3

3/2 = 1

thus power of 2 in 203! is, 101 + 50 + 25 + 12 + 6 + 3 + 1 = 198

divide 203 with 5

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203/5 = 40

40/5 = 8

8/5 = 1

thus power of 5 in 203! is, 49

so the power of 10 in 203! factorial is 49

4) x + y + z = 7 and xy + yz + zx = 10, then what is the maximum value of x? ( CAT 2002 has similar question )

ANS: 49-20 = 29, now if one of the y,z is zero, then the sum of other 2 squares shud be equal to 29, which means, xcan take a max value at 5

5) In how many ways can 2310 be expressed as a product of 3 factors?

ANS: 2310 = 2*3*5*7*11

When a number can be expressed as a product of n distinct primes,

then it can be expressed as a product of 3 numbers in (3^(n+1) + 1)/2 ways

6) In how many ways, 729 can be expressed as a difference of 2 squares?

ANS: 729 = a^2 - b^2

= (a-b)(a+b),

since 729 = 3^5,

total ways of getting 729 are, 1*729, 3*243, 9*81, 27*27.

So 4 ways

Funda is that, all four ways of expressing can be used to findout distinct a,b values,

for example take 9*81

now since 9*81 = (a-b)(a+b) by solving the system a-b = 9 and a+b = 81 we can have 45,36 as soln.

7) How many times the digit 0 will appear from 1 to 10000

ANS: In 2 digit numbers : 9,

In 3 digit numbers : 18 + 162 = 180,

In 4 digit numbers : 2187 + 486 + 27 = 2700,

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that is 302 + LCM(11,7,4) and thus the rems when the same number is divided by 4,7,11 respectively are,

302 mod 4 = 2

75 mod 7 = 5

10 mod 11 = 10

12) a^n - b^n is always divisible by a-b

13) if a+b+c = 0 then a^3 + b^3 + c^3 = 3abc

EXAMPLE: 40^3-17^3-23^3 is divisble by

since 40-23-17 = 0, 40^3-17^3-23^3 = 3*40*23*17 and thus, the number is divisible by 3,5,8,17,23 etc.

14) There is a seller of cigerette and match boxes who sits in the narrow lanes of cochin. He prices thecigerattes at 85 p, but found that there are no takers. So he reduced the price of cigarette and managed tosell all the cigerattes, realising Rs. 77.28 in all. What is the number of cigerattes?

a) 49

b) 81

c) 84

d) 92

ANS : (d)

since 77.28 = 92 * 84, and since price of cigarette is less than 85, we have (d) as answer

Quote:

i have given this question to make the funda clear

15) What does 100 stand for if 5 X 6 = 33

ANS : 81

SOLN : this is a number system question,

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30 in decimal system is 33 in some base 'n', by solving we will have n as 9

and thus, 100 will be 9^2 = 81

16) In any number system 121 is a perfect square,

SOLN: let the base be 'n'

then 121 can be written as n^2 + 2*n + 1 = (n+1)^2

hence proved

17) Most of you ppl know these, anyways, just in case

Quote:

(a) sum of first 'n' natural numbers - n*(n+1)/2

(b) sum of the squares of first 'n' natural numbers - n*(n+1)*(2n+1)/6

(c) sum of the cubes of first 'n' natural numbers - n^2*(n+1)^2/4

(d) total number of primes between 1 and 100 - 25

18 ) See Attachment to know how to find LCM, GCF of Fractions

Quote:

CAT 2002 has 2 questions on the above simple concept

19) Converting Recurring Decimals to Fractions

let the number x be 0.23434343434........

thus 1000 x = 234.3434343434......

and 10 x = 2.3434343434.........

thus, 990 x = 232

and hence, x = 232/990

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20) Reminder Funda

(a) (a + b + c) % n = (a%n + b%n + c%n) %n

EXAMPLE: The reminders when 3 numbers 1221, 1331, 1441 are divided by certain number 9 are 6, 8, 1respectively. What would be the reminder when you divide 3993 with

9?

the reminder would be (6 + 8 + 1) % 9 = 6

(b) (a*b*c) % n = (a%n * b%n * c%n) %n

EXAMPLE: What is the remainder left when 1073 * 1079 * 1087 is divided by 119 ? 1073 % 119 = ?

since 1190 is divisible by 119, 1073 mod 119 is 2

and thus, "the remainder left when 1073 * 1079 * 1087 is divided by 119 " is 2*8*16 mod 119 and that is 256 mod 119and that is (238 + 18 ) mod 119 and that is 18

Glossary : % stands for reminder operation

find the number of zeroes in 1^1* 2^2* 3^3* 4^4.............. 98 9̂8* 99^99* 100^100

the expresion can be rewritten as (100!)^100 / 0!* 1!* 2!* 3!....99!

Now the numerator has 2400 zeros

the formular for finding number of zeros in n! is

[n/5]+[n/5^2]...[n/5^r]

where r is such that 5^r<=n<5^(r+1)

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and [..] is the greastest integer function

for the numerator find the number of zeros using the above formulae..

for 0!...4! number of zeros ..0

5!...9!.number os zeros ..1

9!...14!... 2

15!..19!..................3

20!..24!..................4!

now at 25! the series makes a jump to 6

25!...29!.................6

30!...34!.................7

this goes on and again makes a jump at 50!

and then at 75!

so the number of zeros is...

5(1+2....19) + 25+ 50+ 75

the last 3 terms 25 50 and 75 are because of the jumps..

this gives numerator has 1100 zeros

now total number of zeros in expression is no of zeros in denominator - no of zeros in numerator

2400 - 1100

the Answer 1300

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To find out if a number is divisible by seven: Take the last digit, double it, and subtract it from the rest of thenumber. If the answer is more than a 2 digit number perform the above again. If the result is 0 or is divisible by 7 the original number is also divisible by 7. Example 1 ) 259 9*2= 18. 25-18 = 7 which is divisible by 7 so 259 is also divisible by 7. Example 2 ) 2793 3*2= 6 279-6= 273 now 3*2=6 27-6= 21 which is divisible by 7 so 2793 is also divisible by 7 . Now find out if following are divisible by 7 1) 2841 2) 3873 3) 1393 4) 2877

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TO FIND SQUARE OF A NUMBER BETWEEN 40 to 50 Sq (44) . 1) Subtract the number from 50 getting result A. 2) Square A getting result X. 3) Subtract A from 25 getting result Y 4) Answer is xy EXAMPLE 1 : 44 50-44=6 Sq of 6 =36 25-6 = 19 So answer 1936 EXAMPLE 2 : 47 50-47=3 Sq 0f 3 = 09 25-3= 22 So answer = 2209 NOW TRY To Find Sq of 48 ,26 and 49

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TO FIND SQUARE OF A 3 DIGIT NUMBER : LET THE NUMBER BE XYZ SQ (XYZ) is calculated like this STEP 1. Last digit = last digit of SQ(Z) STEP 2. Second Last Digit = 2*Y*Z + any carryover from STEP 1. STEP 3. Third Last Digit 2*X*Z+ Sq(Y) + any carryover from STEP 2. STEP 4. Fourth last digit is 2*X*Y + any carryover from STEP 3. STEP 5 . In the beginning of result will be Sq(X) + any carryover from Step 4. EXAMPLE : SQ (431) STEP 1. Last digit = last digit of SQ(1) =1 STEP 2. Second Last Digit = 2*3*1 + any carryover from STEP 1.= 6 STEP 3. Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP 2.= 2*4*1 +9= 17. so 7 and 1 carryover STEP 4. Fourth last digit is 2*4*3 + any carryover (which is 1) . = 24+1=25. So 5 and carry over 2. STEP 5 . In the beginning of result will be Sq(4) + any carryover from Step 4. So 16+2 =18. So the result will be 185761. If the option provided to you are such that the last two digits are different, then you need to carry out first two steps only , thus saving time. You may save up to 30 seconds on each calculations and if there are 4 such questions you save 2

minutes which may really affect UR Percentile score.

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PYTHAGORAS THEROEM : In any given exam there are about 2 to 3 questions based on pythagoras

theorem. Wouldn't it be nice that you remember some of the pythagoras

triplets thus saving up to 30 seconds in each question. This saved time may

be used to attempt other questions. Remember one more right question may make

a lot of difference in UR PERCENTILE score. The unique set of pythagoras triplets with the Hypotenuse less than 100 or

one of the side less than 20 are as follows : (3,4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37),

(9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65), (48, 55,

73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97). (15,112,113), (17,144,145), (19,180,181), (20,99,101) If you multiply the digits of the above mentioned sets by any constant you

will again get a pythagoras triplet . Example : Take the set (3,4,5).

Multiply it by 2 you get (6,8,10) which is also a pythagoras triplet. Multiply it by 3 you get ( 9,12,15) which is also a pythagoras triplet. Multiply it by 4 you get (12,16,20) which is also a pythagoras triplet. You may multiply by any constant you will get a pythagoras triplet Take another example (5,12,13)

Multiply it by 5,6 and 7 and check if you get a pythagoras triplet. TIPS FOR SMART GUESSING : You will notice that in any case, whether it is a unique triplet or it is a

derived triplet (derived by multiplying a constant to a unique triplet), all

the three numbers cannot be odd.

In case of unique triplet , the hypotenuse is always odd and one of the

remaining side is odd the other one is even. Below are the first few unique triplets with first number as Odd. 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61

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You will notice following trend for unique triplets with first side as odd. Hypotenuse = (Sq(first side) +1) / 2 Other side = Hypotenuse -1 Example : First side = 3 , so hypotenuse = (3*3+1)/2= 5 and other side = 5-1=4 Example 2: First side = 11 so hypotenuse = (9*9+1)/2= 41 and other side = 41-1=40 Please note that the above is not true for a derived triplet for example

9,12 and 15, which has been obtained from multiplying 3 to the triplet of

3,4,5. You may check for other derived triplets. Below are the first few unique triplets with first number as Even . 4 3 5 8 15 17 12 35 37 16 63 65 20 99 101 You will notice following trend for unique triplets with first side as Even. Hypotenuse = Sq( first side/ 2)+1 Other side = Hypotenuse-2 Example 1. First side =8 So hypotenuse = sq(8/2) +1= 17 Other side = 17-2=15 Example 2. First side = 16 So hypotenuse = Sq(16/2) +1 =65 Other side = 65-2= 63

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PROFIT AND LOSS : In every exam there are from one to three questions on profit and loss, stating that the cost was first increased by certain % and then decreased by certain %. How nice it would be if there was an easy way to calculate the final change in % of the cost with just one formula. It would really help

you in saving time and improving UR Percentile. Here is the formula for the same : <h

4>Suppose the price is first increase by X% and then decreased</h

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by Y% , the final change % in the price is given by the following formula Final Difference % = X- Y - XY/100. EXAMPLE 1. : The price of T.V set is increased by 40 % of the cost price and then decreased by 25% of the new price . On selling, the profit for the dealer was Rs.1,000 . At what price was the T.V sold. From the above mentioned formula you get : Final difference % = 40-25-(40*25/100)= 5 %. So if 5 % = 1,000 then 100 % = 20,000. C.P = 20,000 S.P = 20,000+ 1000= 21,000. EXAMPLE 2 : The price of T.V set is increased by 25 % of cost

price and then decreased by 40% of the new price . On selling, the loss for the dealer was Rs.5,000 . At what price was the T.V sold. From the above mentioned formula you get : Final difference % = 25-40-(25*45/100)= -25 %. So if 25 % = 5,000 then 100 % = 20,000. C.P = 20,000 S.P = 20,000 - 5,000= 15,000. Now find out the difference in % of a product which was : First increased by 20 % and then decreased by 10 %. First Increased by 25 % and then decrease by 20 %. First Increased by 20 % and then decrease by 25 %. First Increased by 10 % and then decrease by 10 %. First Increased by 20 % and then decrease by 15 %.

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TIPS TO IMPROVE UR PERCENTILE : HOW ABOUT SOLVING THE FOLLOWING QUESTION IN JUST 10 SECONDS

Ajay can finish work in 21 days and Blake in 42 days. If Ajay, Blake and Chandana work together they finish the work in 12 days. In how many days Blake and Chandana can finish the work together ? (21*12 )/(24-12) = (21*12)/9= 7*4= 28 days. NOW CAREFULLY READ THE FOLLOWING TO SOLVE THE TIME AND WORK PROBLEMS IN FEW SECONDS. TIME AND WORK : 1. If A can finish work in X time and B can finish work in Y time then both together can finish work in (X*Y)/ (X+Y) time. 2. If A can finish work in X time and A and B together can finish work in S time then B can finish work in (XS)/(X-S) time. 3. If A can finish work in X time and B in Y time and C in Z time then they all working together will finish the work in (XYZ)/ (XY +YZ +XZ) time 4. If A can finish work in X time and B in Y time and A,B and C together in S time then : C can finish work alone in (XYS)/ (XY-SX-SY) B+C can finish in (SX)/(X-S) and A+ C can finish in (SY)/(Y-S)

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Here is another shortcut to improve URPERCENTILE. TYPE 1 : Price of a commodity is increased by 60 %. By how

much % should the consumption be reduced so that the expense remain the same. TYPE 2 : Price of a commodity is decreased by 60 %. By how

much % can the consumption be increased so that the expense remain the same. Solution :

TYPE1 : (100* 60 ) / (100+60) = 37.5 % TYPE 2 : (100* 60 ) / (100-60) = 150 % Now do the following questions for UR Practice: http://groups.yahoo.com/group/urpercentile/

Q1. Price of a commodity is increased by 10 %. By how much % should the consumption be reduced so that the expense remain the same. Q2. Price of a commodity is decreased by 10 %. By how much % can the consumption be increased so that the expense remain the same. Q3. Price of a commodity is increased by 15 %. By how much % should the consumption be reduced so that the expense remain the same. Q4. Price of a commodity is decreased by 15 %. By how much % can the consumption be increased so that the expense remain the same. Q5. Price of a commodity is increased by 25 %. By how much % should the consumption be reduced so that the expense remain the same. Q6. Price of a commodity is decreased by 25 %. By how much % can the consumption be increased so that the expense remain the same. Q7. Price of a commodity is increased by 50%. By how much % should the consumption be reduced so that the expense remain the same.

http://groups.yahoo.com/group/urpercentile/



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Q8. Price of a commodity is decreased by 50 %. By how much % can the consumption be increased so that the expense remain the same. Q9. Price of a commodity is increased by 20 %. By how much % should the consumption be reduced so that the expense remain the same. Q10. Price of a commodity is decreased by 20 %. By how much % can the consumption be increased so that the expense remain the same.

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