math tutorial

1. If x to the 3/4 power equals 8, Find the value of x.

Ans: 16

Solution:

2. If solve for a.

Ans: 10,000

Solution:

3. Solve for x if log3

Ans: 4

Solution:

4. What is the value of

Ans: 3.79

Solution:

5. If log of 2 to the base 2 plus log of x to the base 2 is equal to 2, Find the value of x. Ans:2

Solution:

6. If (2 log x to the base 4) –(log 9 to the base 4) = 2, find x, Ans: 12

Solution:

7. Solve for x : log10 8=3-3 log10 x Ans: 5

Solution:

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8. If loga 10=0.25, what is the value of log10 a=?Ans. 4

Solution:

9. What is the value of log to the base 10 of 10003.3

Ans: 9.9

Solution:

10. Solve for x: log

Ans: 4

Solution:

11. Find the sum of the roots of 5x2 – 10x + 2 = 0

Ans: 2

Solution:

12. Compute the numerical coefficient of the 5th term of the

expansion of

Ans: 126720

Solution:

13. What is the sum of the coefficients of the expansion of

Ans: 0

Solution:

14. What is the sum of the coefficients in the expansion of

Ans: 1


Solution:

Substitute x = 1 y = 1 and z = 1

Sum of the coefficients =

Sum of the coefficients =

Sum of the coefficients = 1

15. Find the coefficient of the expansion of (x-y)15 containing the term x4y11

Ans: -1365

Solution:

16. Find the term involving x8 in the expansion of

Ans: 180 x8 y2

Solution:

Term involving x8 =

Term involving x8 = 90 x8 (-2y)2

Term involving x8 = 180 x8 y2

17. Find the 8th term of the sequence

Ans:

Solution:

18. What follows logically in these series of numbers 2,3,4,5,9,17…..Ans: 33

Solution:

19. Find the value of x if

Ans: 20

Solution: 20

20. Solve for x in the following equation.

Ans: 1

Solution:


21. Given f(x) = when f(x) is divided by

the remainder is K. Find the value of K.

Ans: 4

Solution:

22. Which of the following is a factor of 3x3 + 2x2 -32?Ans: x-2

Solution:

Note: If the remainder is zero, the number we assume is a factor. Therefore x-2 is a factor of 3x3+2x2 -32

23. Write a cubic equation whose roots are (-1,2,4)Ans: x3 – 5x2 + 2x + 8 = 0

Solution:

24. The mean proportion between 12 and x is equal to 6. Find the value of x. Ans: 3

Solution:

25. Find x if 7 is the fourth proportional to 36 and 28, and x. Ans: 9

Solution:

26. The force required to stretched a spring is proportional to the elongation. If 24 N stretches a spring 3 mm, find the force required to stretch the spring 2 mm. Ans: 16

Solution:

27. The volume of a hemisphere varies directly as the cube of it’s radius. The volume of the hemisphere with 2.54 cm. radius is 20.75 cm3. What is the volume of a sphere with3.25 cm. radius of the same kind of material? Ans:


Solution:

28. If x varies directly as y and inversely as z, and x=14, when y=7 and z=2, find x when y=16 and z =4. Ans: 16

Solution:

29. A tank is filled with 2 pipes. The first pipe can fill the tank

in 10 hours. But after it has been opened for 3 hours,

the second pipe is opened and the tank is filled up in 4 hours more. How long would it take the second pipe alone to fill the tank? The two pipes have different diameters. Ans: 15

Solution:

30. A and B working together can finish painting a house in six days. A working alone, can finish it in five days less than B. How long will it take each of them to finish the work alone? Ans: 10,15

Solution:

31. Crew no.1 can finish the installation of an antenna tower in 200 man-hours while Crew No.2 can finish the same job in 300 man-hours. How long will it take both crews to finish the same job, working together? Ans: 120

Solution:

32. An experienced statistical clerk submitted the following statistics to his manager on the average rate of production of transistorized radios in an assembly line, 1.5 workers produced 3 radios in 2 hours. How many workers are employed in the assembly line working 40 hrs each per week if weekly production is 480 radios? Ans: 12

Solution:

No. of man-hours to produced 3 radios

No. of man-hours to produce 480 radios

By proportion:

33. A certain job can be done by 72 men in 100 days. There were 80 men at the start of the project but after 40 days, 30 of them had to be transferred to another project. How long will it take the remaining workforce to complete the job?


Ans: 80

Solution: Let x =no. of days it will take for the remaining workforce to complete the job

Then,

34. Twenty (20) men can finish the job in 30 days. Twenty five (25) men were hired at the start and 10 quit after 20 days. How many days will it take to finish the job? Ans: 27

Solution: No. of man-days:

Total number of days to finish the job

35. Given two numbers such that the difference of twice the first and the second number is 12. If the sum of the first and the second number is 36, find the number. Ans: 16,20

Solution:

36. The product of of a number is 500. What is the

number. Ans: 100

Solution:

37. The square of the number increased by 16 is the same as 10 times the number. Find the numbers.Ans: 2,8

Solution:

38. The sides of the right triangle are 8, 15 and 17 units. If each side is doubled, how many square units will the area of the new triangle. Ans: 240

Solution:

39. Compute the median of the following set of numbers 4,5,7,10,14,22,25,30. Ans: 12

Solution: The middle number is 10 and 14Therefore the median is the average of 10 and 14 = 12

40. To get an A in a course a student’s average must be at least 90(and at most 100). If a particular student’s grade so far are 84,88 and 93. What scores must that student get on the last test to earn an A if all tests court equally?Ans: 95

Solution:

41. A students has test scores of 75, 83 and 78. The final tests counts half the total grade. What must be the minimum (integer) score on the final so that the average will be 80? Ans: 82

Solution:


42. Which of the following is a prime number?Ans: 487

Solution:Factors:

Note: A prime number is a positive integer that has exactly two factors, the number itself and 1.

43. The boat travels downstream in 2/3 the time as it does going upstream, if the velocity of the river current is 8 kph, determine the velocity of the boat in the still water.Ans: 40 kph

Solution:

44. The velocity of an airplane in still air is 125 kph. The velocity of the wind due east is 25 kph. If the plane travels east and returns back to its base again after 4 hours. At what distance does the plane travel due east?Ans: 240 km.

Solution:

45. Roberto is 25 years younger than his father, However his father will be twice his age in 10 years. Find the ages of Roberto and his father. Ans: 15,40

Solution: x = age of Roberto’s fatherx – 25 = age of Roberto x + 10= age of Roberto’s father 10 years from now

46. The sum of the ages of Maria and Anna is 35. When Maria was two thirds her present age and Anna was 3/4 of her present age, the sum of their ages was 25. How old is Maria now?Ans: 15

Solution:

47. Ten (10) yrs from now the sum of the ages of A and B is equal to 50. Six (6) yrs ago, the difference of their ages is equal to 6. How old is A and B? Ans: A=18, B=12

Solution: Past(6 yrs. Ago) Present(10 yrs. Later)A-6 A A + 10B-6 B B + 10

48. Two gallons of 20% salt solution is mixed with 4 gallons of 50% salt solution. Determine the percentage of salt solution in the new mixture. Ans: 40%

Solution:20% 50% x%

=


4 62

49. Two alcohol solution consists of a 40 gallons of 35% alcohol and other solution containing 50% alcohol. If the two solutions are combined together, they will have a mixture of 40% alcohol. How many gallons of solutions containing 50% alcohol? Ans: 20

Solution: 35% 50% 40%

50. If you owned a sari-sari store in Kuwait, at what price will you mark a small camera for sale the cost P600 in order that you may offer 20% discount on the marked price and still makes a profit of 25 % on the selling price? Ans: P1000

Solution:

51. A mechanical Engineer bought 24 boxes of screws for P2200.00. There were three types of screws bought. Screw A costs P300 per box, screw B costs P150 and screw C costs P50 per box. How many boxes of screw A did he buy? Ans: 2

Solution:x=no. of boxes of screw A

y=no. of boxes of screw B

z= no. of boxes of screw C

52. Dalisay Corporation’s gross margin is 45% of sales. Operating expenses such as sales and administration are 15% of sales. Dalisay is in 40% tax bracket. What percent of sales is their profit after taxes? Ans: 18%

Solution: Gross margin=45% of sales

Operating expenses = 15% of sales

Net profit = 45-15

Net profit =30% of sales

Tax =40%

Profit = 0.60 of 30% of sales

Profit = 18% of sales

53. What is the sum of the progression 4,9,14,19… up to the 20th term? Ans: 1030

Solution:

54. The are 9 arithmetic mean between 11 and 51. Compute the sum of the progression. Ans: 341


x40 40+x

Solution:

55. An arithmetic progression starts with 1, has 9 terms and the middle term is 21. Determine the sum of the first 9 terms. Ans: 189

Solution:

Middle term = a + 4d

56. If the sum is 220 and the first term is 10, find the common difference if the last term is 30.Ans: 2

Solution:

57. Determine the sum S of the following series

Ans: 15050

Solution:

58. Find the sum of the first n positive multiples of 4. Ans: 2n(n+1)

Solution:

59. The sum of the three numbers in AP is 33, if the sum of their squares is 461, find the numbers. Ans: 4,11,18

Solution:


60. Find the value of x if the following forms a harmonic

progression.

Ans: 7

Solution:

61. Determine the positive value of x so that x,x2-5 and 2x will be a harmonic progression. Ans: 3

Solution:

62. There are 4 geometric mean between 3 and 729, Find the sum of the G.P. Ans: 1092

Solution:

63. The number x, 2x + 7, 10x – 7 form a G.P. Find the value of x: Ans: 7

Solution:

64. Find the value of x from the given Geometric Progression

Ans: 15

Solution:

65. The arithmetic mean of 6 numbers is 17. If two numbers are added to the progression, the new set of number will have an arithmetic mean of 19. What are the two numbers is their difference is 4? Ans: 23 and 27


Solution:

let x = one number x + 4 = other number

66. If one third of the air in a tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes? Ans: 0.9122

Solution:

67. If a stroke of a vacuum pump removed 15% of the air from the container, how many strokes are required to removed 95%of the air? Ans: 19

Solution:

68. The sum of a geometric series is as follows:

Ans: 1163.91

Solution:

69. x and y are positive numbers. If x,-3,y forms a G.P and -3,y,x forms an A>P. Find the value of x. Ans: 6,-3

Solution:

70. The sum of a geometric series is as follows:


Ans: 1163.91

Solution:

71. Find the total distance traveled by the tip of a pendulum if the distance of the first swing is 6 cm. and the distance of each succeeding swing is 0.98 of the distance of the previous swing. Ans: 300

Solution:

72. Suppose a ball rebounds one half the distance if falls. If it is dropped from a height of 40 fet, how far does it travel before coming to stop?Ans: 120 feet

Solution:

Total distance the ball has traveled = 80 + 40 = 120 feet

73. A policeman is pursuing a thief who is ahead by 72 of his own leaps. The thief takes 6 leaps while the policeman is taking 5 leaps, but 4 leaps of the thief are as long as 3 leaps of the policeman. How many leaps will each make before the thief is caught? Ans: 648 leaps & 540 leaps

Solution:Let:

additional leaps of the thief

number of leaps of the policeman

By proportion:

74. Find the angle whose supplement exceeds 5 times its complement.

Ans: 67.5o

Solution:Let:

Then,

75. Find the angle whose supplement exceeds 6 times its complement by 20o.

Ans: 76o

Solution:Let:

Then,

76. How many sides has an equiangular polygon if each of its interior angles is 165 degrees?

Ans: 24 sides

Let:

Then,


77. Find the largest angle of a triangle if the sum and difference of two angles are 100 and 20 degrees, respectively.

Ans: 80 degrees

Solution:Let:

Then,

Solving the two equations simultaneously, we get:

Thus, the largest angle is the third angle:

78. The two corresponding sides of two similar polygon is 4:6 If the perimeter of the larger polygon is 48 cm, what is the perimeter if the smaller polygon? Ans: 32 cm.

Solution:

79. The corresponding sides of two similar polygon is 4:5. If the area of the smaller polygon is 56 sq. cm., what is the area of the bigger polygon? Ans: 87.5 sq.cm.

Solution:

80. The sides of a pentagon measures 6,5,2,8 and 4 m. respectively. The shortest side of a similar pentagon is 1 meter, find the perimeter of this pentagon. Ans: 12.5 m.

Solution:Perimeter of given pentagon

Perimeter of given pentagon = 25 m.

Ratio of corresponding smaller sides = ratio of perimeter

(perimeter of the smaller pentagon)

81. The corresponding sides of two similar polygons is 2:4. Determine the ratio of the area to the perimeter of the biggest polygon if the smallest polygon has a perimeter of 24 cm. and an area of 36 sq. cm. Ans: 3

Solution:

82. Seven regular hexagons, each with 6 cm. sides are arranged so that they share the same sides and the centers of the six hexagons are equidistant from the seventh central hexagon. Determine the ratio of the total area of the hexagons to the total outer perimeter enclosing the hexagons. Ans: 6.06

Solution:

83. A chord is 36 cm. long and its mid point is 36 cm. from the midpoint of the longer arc. Find the radius of the circle.


Ans: 22.5

Solution:

84. A circular play area is being laid out in a field. The point P,R and Q have been found such that PS=24 m., PQ=66 m., and RS=12 m. If T is to be another point on the circle, how far from R will it lie? Ans: 96 m.

Solution:

85. Two secants AC and AE have lengths of 80 m. and 100 m. respectively are drawn from point A outside a circle and intersects the circle at points C,B, D and E. If the angle between the two secants is 25o, and AB is equal to 40 m. compute the length of AD. Ans: 32 m.

Solution:

86. An engineer places his transit along the line tangent to the circle at point A such that PA = 200 m. He locates another point B on the circle and finds PB = 80 m. If a third point C, on the circle lies along PB, how far from point B will it be? Ans: 420 m.

Solution:

87. A circle having a center at point O has a radius of 20 m. A triangle ABC is inscribe in a circle. If the angle BCA = 30o

and BAC = 50o compute the area of the triangle. Ans: 301.73 m2

Solution:

88. The central circle has 10 cm. radius. Six equal smaller circles are to be arranged so that they are externally tangent to the central circle and each tangent to the adjacent small circle. What should be the radius in cm. of each small circle? Ans: 10

Solution:

89. The distance between the center of the circles which are mutually tangent to each other externally are 10,12 and 14 units, The area of the largest circle is: Ans: 64

Solution:

Area of largest circle =

Area of largest circle = 64

90. The area of a circular sector is 800 m2. If the radius is equal to 16 m, compute for the length of arc Ans: 100 m.Solution:


x

91. A sector is bent to form a cone. If the angle of a sector is 30 degrees and radius of 6 cm., what is the altitude of the cone? Ans: 5.98 cm.

Solution:

92. Find the volume of a cone to be constructed from a sector having a diameter of 72 cm. and a central angle of 150o. Ans: 7711.82

Solution:

93. The lateral area of a right circular cone is 40 The base radius is 4 m. What is the slant height? Ans: 10

Solution:

94. The ratio of the volume to the lateral area of a right circular cone is 2:1. if the altitude is 15 cm., what is the ratio of the slant height to the radius.Ans: 5:2

Solution:


Ratio of slant height to radius = 5:2

95. Given a solid right circular cone having a height of 8 cm. has a volume equal to 4 times the volume of the smaller cone that could be cut from the same cone having the same axis. Compute the height of the smaller cone. Ans: 5.04 cm.

Solution:

96. A circular cone having an altitude of 9 m. is divided into 2 segments having the same vertex. If the smaller altitude is 6 m., find the ratio of the volume of small cone to the big cone. Ans: 0.296

Solution:

97. The height of a right circular cone is “h”. If it contains oil and water at equal depth of h/2, what is the ratio of the volume of the volume of oil to that of the water.Ans: 7

Solution:

98. A plane is passed parallel to the base of a triangular pyramid of altitude of 9 m. such that the area of the base is 9 times the area of the triangle of intersection. How far from the vertex does the plane intersect the altitude?Ans: 3 m.

Solution:

99. The volume of a cone having an inclined axis at an angle of 60o with the base is equal to 1884.96 cu.m., find the length of the axis of the cone, if the radius at the base is 10 m. Ans: 20.78 m.

Solution:


100. The radius of the base of a cone of revolution is 32 inches and its altitude is 54 inches. What is the altitude of a cylinder of the same volume whose diameter of the base is 48 inches? Ans: 32 in.

Solution:

101.A pyramid has a triangular base whose sides are 10 cm., 14 cm. and 18 cm. respectively. Compute the volume of the inscribed cone if it has an altitude of 28 cm. Ans: 323.19

Solution:

102.A solid gold in the form of a frustum of a pyramid has smaller base area of 20cm2 and a bigger base area of 60 cm2 . It has an altitude of 24 cm. If the gold is melted to form a 9 spherical balls, find the radius of each balls. Ans: 2.90 cm.

Solution:

103.The bases of a right prism is a hexagon with one of each side equal to 6 cm. If the volume of the right prism is 500 cu. cm., find the distance between the bases. Ans: 5.35 cm.

Solution:

104.The bases of a right prism is a hexagon with one of each side equal to 6 cm. The bases are 12 cm. apart. What is the volume of the right pris. Ans: 1122.4 cu. cm.

Solution:


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