1. This document contains a 50-question multiple choice test on variation equations and concepts. Questions test understanding of direct, inverse, and joint variation, as well as applying formulas to word problems involving variations. 2. Key concepts covered include direct and inverse variation, finding constants of variation, and applying variation equations to physics concepts like Boyle's law, Charles' law, and Newton's law of universal gravitation. 3. Questions range from identifying variation equations to applying equations to solve for unknown variables in word problems involving variations.

1. Second Periodical Test
PART I.
Multiple Choice. Write the best answer before each number. Write E if the answer is
not in the choices.
1. What is y varies inversely as x in variation equation?
a. y = k/x b. y = kx c. y = x/k d. none of these
2. r varies directly as the cube of s and inversely as t can be written as _______.
a. 𝑟 = 𝑘𝑠3
/𝑡 b. . 𝑟 = 𝑘𝑠2
/𝑡 c. . 𝑟 = 𝑘𝑠3
𝑡 d. a and c
3. What is the formula to find the constant of variation when x varies directly as a?
a. k= x/a c. k = a/x
b. k= ax d. none of these
4. b varies jointly as a, t and h can be written as ______.
a. 𝑎 = 𝑘
𝑏
𝑡ℎ
b. 𝑎 = 𝑏
𝑘
𝑡ℎ
c. b = kath d. none of these
5. What is “twice the sum of x and y varies directly with the sum of a andh and inversely with the
cube of c” in variation statement?
a. 2(𝑥 + 𝑦) =
𝑘(𝑎+ℎ)
𝑐2 c. 2(𝑥 + 𝑦) =
𝑘(𝑎+ℎ)
𝑐3
b. 2(𝑥 − 𝑦) =
𝑘(𝑎+ℎ)
𝑐2
d. 2(𝑥 − 𝑦) =
𝑘(𝑎+ℎ)
𝑐3
6. Which of the following equations best corresponds the statement “the area of triangle varies
jointly as the base and the height”?
a. A = kbh b. A = bh c. A = kb/h d. A = kh/b
7. Boyle’s law states that if the temperature is constant, the volume of a mass of gas varies inversely
as the pressure to which it is subjected. If V represents the volume and p is the pressure, which of the
following is the correct equation of the law?
a. V = kp b. V = k(1/p) c. V = P/k d. none of these
8. Charles’ law states that if the pressure is constant, the volume of a given mass of gas varies
directly as the absolute temperature. Let V and t represent the volume and temperature, respectively.
Which of the following is an applicable formula of Charles; law?
a. V = k/t b. V = kt c. V = t/k d. none of these
9. The energy that an item possesses due to its motion is called kinetic energy. The kinetic energy of
an object varies jointly with the mass of the object and the square of its velocity. Which of the
following is NOT a formula of the statement?
a. 𝐾 = 𝑘𝑚𝑣2
b. 𝑘 = 𝐾
𝑚
𝑣2
c. 𝑚 = 𝐾
𝑘
𝑣2
d. 𝑣 = √
𝐾
𝑘𝑚
10. What phrase best describes the equation =
𝑘𝑦2
(𝑧−2)
?
a. x varies directly as the square root of y and inversely as z less than 2
b. x varies directly as k and the square of y and inversely as z plus 2
c. x varies directly as the square of y and inversely as z, less than 2
d. x varies directly as the square of y and inversely as z less than 2
11. The acceleration a of a moving object varies directly as the distance d it travels and inversely as
the square of the time t it travels. What is used to solve for an object’s acceleration?
a. 𝑎 = 𝑘𝑑
√𝑡
⁄ c. 𝑎 = 𝑘𝑑
𝑡
⁄
b. 𝑎 = 𝑘𝑑
𝑡2
⁄ d. 𝑎 = 𝑘𝑡
𝑑2
⁄

2. 12. Newton’s law of gravitation states that the gravitational attraction between two bodies varies
directly as the product of their masses and inversely as the square of the distance between their
centers of gravity. Let G, M, m and drespectively represent the gravitational attraction, the two
masses, and the distance, then the law states that ________.
a. G = k (
𝑀𝑚
𝑑2
) b. G =
𝑘𝑀𝑚
𝑑
c. G =
𝑘𝑚2
𝑑2
d. 𝐺 = 𝑘𝑀𝑚𝑑2
13. If a varies directly as b, what is the formula to find the constant of variation?
a. a = k/b b. a = kb c. k = a/b d. k = b/a
14. If g is inversely proportional to h, what is the equation in finding the value of the constant of
variation?
a. g = k/h b. g = h/k c. k = gh d. none of these
15. The equation used to find the constant of variation in the statement “r varies jointly as s and t” is
_______.
a. r = kst b. k = (st)/r c. k = r/(st) d. k = (rs)t
For items 16-18, use the values in the box.
16. What is the constant of variation if h varies directly as t+m?
a. 4 b. 3 c. 2 d. 1
17. a varies inversely as m. What is a if m is halved?
a. 4 b. 3 c. 2 d. 1
18. t varies directly as the sum of a and h and inversely with half of m. Find t when h, a and m were
increased by 8.
a. 8 b. 6 c. 3 d. 3⁄2
19. If a varies directly as b and a = 32 when b = 4. Find the constant of variation.
a. 8 b. 36 c. 28 d. 128
20. If a varies directly as b and a = 32 when b = 4. Find b when a is 968.
a. 28 b. 36 c. 121 d. 128
21. Given that h varies directly as d. If d is 9 when h is 108, what is h when d is 8?
a. 69 b. 72 c. 96 d. 112
22. If y varies inversely as x and when x = 8, y is the reciprocal of 32. Find k.
a. 6 b. 4 c. 1/2 d. 1/4
23. If v is inversely proportional to w and v is 65 when w is 3, what is w if v is 45?
a. 5 b. 10 c. 15 d. 20
24. Given that y varies inversely as x. If y = 3 when x = 4, find y when x = 6.
a. 12 b. 6 c. 4 d. 2
25. If w varies inversely as y and w is equal to 2 when y = 3, find the value of w if y = 3.
a. 1 b. 2 c. 3 d. 4
26. Given that x varies jointly as a, z and y. If x = 2016 when a = 7, z = 8, and y = 9, what is the
value of k?
a. 3 b. 4 c. 5 d. 6
27. Given that x varies jointly as w and y and x = 24 when w = 3 and y = 4.
m = 4
a = 1
t = 3
h = 7

3. Find the value of w if x = 40 and y = 5.
a. 2 b. 4 c. 8 d. 16
28. If b varies jointly as a and t. If b=60 when a = 4 and t =5, find b when a = 8 and t = 13.
a. 123 b. 213 c. 302 d. 312
29. z varies jointly as x and y. If z =3 when x =3 and y =15, find z when x = 6 and y = 9.
a. 3 b. 3.6 c. 6 d. 6.3
30. If f varies directly as g and inversely as the square of h, and f = 20 when g = 50 and h = 5, find f
when g = 18 and h = 6.
a. 3 b. 5 c. 10 d. 15
31. Given that w varies directly as the product of x and y and inversely as the square of z. If w = 9
when x = 6, y = 27, and z = 3, find the value of w when x = 4, y = 7, and z = 2.
a. 7 b. 7
2
⁄ c. 7
3
⁄ d. 7
4
⁄
32. Given that w varies directly as the product of x and y and inversely as the square of z. If w = 9
when x = 6, y = 27, and z = 3, find the value of z when w, x and y were divided by 3.
a. 9 b. 3 c. √3 d. √
1
3
33. If (x-4) varies inversely as (y+3) and x = 8 when y = -1. Find the variation constant.
a. 4 b. 6 c. 8 d. 10
34.j varies directly as the square of l and inversely proportional to twice the m. Ifj is 1 when l is 3
and m = 9, what is j when l=4 and m=8?
a. 1 b. 2 c. 3 d. 4
35.d varies jointly as e and l. if d = 1.2 when e = 0.3 and l = 0.4. Find d when e = 0.8 and l = 0.005.
a. 0.4 b. 0.04 c. 0.004 d. 0.0004
36. If c varies directly as l and inversely as t and c is 8 when l is 16 and t is 4. What is c if l and t is
equal to 8?
a. 1 b. 2 c. 3 d. 4
37. Given that a varies directly as the product of two consecutive numbers and inversely as b.
5 and 6 were the numbers when a = 60 and b = 18. Which of the following are the numbers when
a = 112 and b = 18?
a. 4 and 5 b. 6 and 7 c. 7 and 8 d. 8 and 9
38.The cost varies directly as the number of pencils bought. P27 was spent for 3 pencils, how many
pencils cost P432?
a. 6 b. 18 c. 36 d. 48
39. What happens to t when h is doubled in the equation t = 4h?
a. halved b. tripled c. doubled d. can’t be determined
40. In comparing centigrade and Fahrenheit thermometers, it has been found that the centigrade
reading varies directly as the difference between the Fahrenheit reading and 32℉.If a centigrade
thermometer reads 100℃ when a Fahrenheit reads 212℉, what will be the centigrade read when the
Fahrenheit reads 100℉?
a. 132/9 b. 132/5 c. 340/9 d. 340/5
41. The pressure on the bottom of a swimming pool varies directly as the depth. If the pressure is
624,000 lb. when the water is 2 ft., find the pressure when the pool is 4 feet and 6 inches deep.
a. 1404000 lb. b. 1044000 lb. c. 1400400 lb d. 1040400 lb

4. 42. The area of triangle varies jointly as the base and the height. A triangle with a base of 8 cm. and
a height of 9 cm. has an area of 36 square centimeters. Find the area when the base is 10 cm. and the
height is 7 cm.
a. 35 sq. cm. b. 70 sq. cm. c. 105 sq. cm. d. 140 sq. cm.
43. The amount of coal used by a steamship traveling at a uniform speed varies jointly as the
distance traveled and the square of the speed. If a steamship was 45 tons of coal traveling 80 miles at
15 miles per hour, how many tons will it use if it travels 120 miles at 20 miles per hour?
a. 6 b. 12 c. 120 d. 1200
44. The number of days d needed in repairing a house varies inversely as the number of men m
working. It takes 15 days for 2 men to repair the house. How many men are needed to complete the
job in 6 days?
a. 2 b. 3 c. 4 d. 5
45. The number of minutes needed to solve an exercise set of variation problems varies directly
as the number of problems and inversely as the number of people working on the solutions. It
takes 4 people to solve 18 problems in 36 minutes. How many minutes will it take 6 people to
solve 42 problems?
a. 50 b. 53 c. 56 d. 58
46. Boyle’s law states that if the temperature is constant, the volume of a mass of gas varies
inversely as the pressure to which it is subjected. If V represents the volume and p is the pressure. If
the volume of a mass of gas at given temperature is 56 cu. in. when the pressure is 18 lb. Find the
volume when the pressure is 16 lb.
a. 36 cu. in b. 50 cu. in c. 63 cu. in d. 16128 cu. in
47. The volume of a right circular cylinder varies jointly as the height and the square of its radius. if
the volume of a right circular cylinder of radius 4 inches and a height of 7 inches is 352 cubic
inches, find the volume of another of radius 8 inches and height of 14 inches.
a. 2168 cu. in. b. 2186 cu. in. c. 2816 cu. in. d. 2861 cu. in.
48. The weight of a rectangular block of metal varies jointly as its length, width, and thickness. If the
weight of a 13 dm. by 8 dm. by 6 dm. block of aluminum is 18.2 kg., find the weight of a 16 dm. by
10 dm. by 4 dm. block of aluminum.
a. 17 kg. b. 18kg. c. 19 kg. d. 20kg.
49. The safe load for a horizontal beam supported at both ends varies jointly as the breadth and
the square of the depth, and inversely as the distance between the supports. Let L, w, d and l be
the load, breadth, depth and distance between supports. If a 4-by-6-in. beam 15 ft. long supports
1470 lb. when standing on edge, what is the safe load if the beam is turned on its side?
a. 800 b. 890 c. 980 d. 900
50. The electrical resistance of a uniform wire varies directly as the length and inversely as the
area of the cross section. Compare the resistance of 100 ft. wire of diameter 1/16 in. with that of
a 50-ft. wire of diameter 1/32 in.
a. The resistance of the longer wire is twice of that of the shorter
b. The resistance of the longer wire is ½ of that of the shorter
c. The resistance of the shorter wire is ½ of that of the longer
d. Their resistance is the same