# Solved Examples: Variations

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
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For SAT Students
Any question labeled SAT-C is a question that allows a calculator.
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For GCSE and Malta Students
All work is shown to satisfy (and actually exceed) the minimum for awarding method marks.
Calculators are allowed for some questions. Calculators are not allowed for some questions.

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
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For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Indicate each type of variation.
Write the proportion.
Convert to an equation.
Solve the equation.
Show all work.

(1.) (a.) The quantity, y varies directly with the square of x
If y = 24 when x = 3, determine y when x is 4.

(b.) The quantity, y varies inversely with the square of x
If y = 8 when x = 3, determine y when x is 4

(c.) The quantity, x varies directly with the square of y and inversely with z
If x = 40 when y = 4 and z = 2, determine x when y = 10 and z = 25

$(a.) \\[3ex] y \propto x^2 ...Direct\;\;Proportion \\[3ex] y = kx^2 ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[3ex] kx^2 = y \\[3ex] k = \dfrac{y}{x^2} \\[5ex] y = 24 \\[3ex] x = 3 \\[3ex] k = \dfrac{24}{3^2} \\[5ex] k = \dfrac{24}{9} \\[5ex] k = \dfrac{8}{3} \\[5ex] x = 4 \\[3ex] y = ? \\[3ex] y = kx^2 \\[3ex] y = \dfrac{8}{3} * 4^2 \\[5ex] y = \dfrac{8}{3} * 16 \\[5ex] y = \dfrac{128}{3} \\[5ex] (b.) \\[3ex] y \propto \dfrac{1}{x^2} ...Inverse\;\;Variation \\[5ex] y = \dfrac{k}{x^2} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] yx^2 = k \\[3ex] k = x^2y \\[3ex] y = 8 \\[3ex] x = 3 \\[3ex] k = 3^2 * 8 \\[3ex] k = 72 \\[3ex] x = 4 \\[3ex] y = ? \\[3ex] y = \dfrac{k}{x^2} \\[5ex] y = \dfrac{72}{4^2} \\[5ex] y = \dfrac{72}{16} \\[5ex] y = \dfrac{9}{2} \\[5ex] (c.) \\[3ex] x \propto \dfrac{y^2}{z} ...Combined\;\;Variation \\[5ex] x = \dfrac{ky^2}{z} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] \dfrac{ky^2}{z} = x \\[5ex] ky^2 = zx \\[3ex] k = \dfrac{zx}{y^2} \\[5ex] x = 40 \\[3ex] y = 4 \\[3ex] z = 2 \\[3ex] k = \dfrac{2(40)}{4^2} \\[5ex] k = \dfrac{80}{16} \\[5ex] k = 5 \\[3ex] y = 10 \\[3ex] z = 25 \\[3ex] x = ? \\[3ex] x = \dfrac{ky^2}{z} \\[5ex] x = \dfrac{5(10)^2}{25} \\[5ex] x = \dfrac{5(100)}{25} \\[5ex] x = 5(4) \\[3ex] x = 20$
(2.) ACT Given that y varies directly as the square of x, if y = 20 when x = 2, what is y when x = 3?

$F.\;\; 75 \\[3ex] G.\;\; 45 \\[3ex] H.\;\; 30 \\[3ex] J.\;\; 21 \\[3ex] K.\;\; 15 \\[3ex]$

$y \propto x^2 ...Direct\;\;Proportion \\[3ex] y = kx^2 ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[3ex] kx^2 = y \\[3ex] k = \dfrac{y}{x^2} \\[5ex] y = 20 \\[3ex] x = 2 \\[3ex] k = \dfrac{20}{2^2} \\[5ex] k = \dfrac{20}{4} \\[5ex] k = 5 \\[3ex] x = 3 \\[3ex] y = ? \\[3ex] y = kx^2 \\[3ex] y = 5(3)^2 \\[3ex] y = 5(9) \\[3ex] y = 45$
(3.) ACT The table below gives the experimental data values for variables x and y.
Theory predicts that y varies directly with x.
Based on the experimental data, which of the following values is closest to the constant of variation?
(Note: The variable y varies directly with the variable x provided that y = kx for some nonzero constant k, called the constant of variation.)
 x y 2.75 8.50 14.75 16.75 21.00 0.140 0.425 0.750 0.850 1.050

$F.\;\; -2.61 \\[3ex] G.\;\; 0.05 \\[3ex] H.\;\; 3.61 \\[3ex] J.\;\; 15.90 \\[3ex] K.\;\; 20.00 \\[3ex]$

$y = kx \\[3ex] kx = y \\[3ex] k = \dfrac{y}{x} \\[5ex] (2.75, 0.14) \\[3ex] k = \dfrac{0.14}{2.75} \approx 0.051 \\[5ex] (8.5, 0.425) \\[3ex] k = \dfrac{0.425}{8.5} = 0.05 \\[5ex] (14.75, 0.75) \\[3ex] k = \dfrac{0.75}{14.75} \approx 0.051 \\[5ex] (16.75, 0.85) \\[3ex] k = \dfrac{0.85}{16.75} \approx 0.051 \\[5ex] (21, 1.05) \\[3ex] k = \dfrac{1.05}{21} = 0.05 \\[5ex]$ 0.05 is closest to the constant of variation.
(4.) ACT Let k be a constant of proportionality and let w, x, y, and z be positive real number variables.
In which of the following equations does x vary directly with y, directly with the square of w, and inversely with z?

$A.\;\; x = \dfrac{kw^2}{yz} \\[5ex] B.\;\; x = \dfrac{kw^2y}{z} \\[5ex] C.\;\; x = \dfrac{ky}{w^2z} \\[5ex] D.\;\; x = \dfrac{kz}{w^2y} \\[5ex] E.\;\; x = kw^2yz \\[3ex]$

$x \propto \dfrac{yw^2}{z} ...Combined\;\;Variation \\[5ex] x = \dfrac{kyw^2}{z} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] x = \dfrac{kw^2y}{z}$
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(10.) JAMB If x varies inversely as the cube root of y and x = 1 when y = 8, find y when x = 3

$A.\;\; \dfrac{1}{3} \\[5ex] B.\;\; \dfrac{2}{3} \\[5ex] C.\;\; \dfrac{8}{27} \\[5ex] D.\;\; \dfrac{4}{9} \\[5ex]$

$x \propto \dfrac{1}{\sqrt[3]{y}} ...Inverse\;\;Variation \\[5ex] x = \dfrac{k}{\sqrt[3]{y}} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] k = x \cdot \sqrt[3]{y} \\[3ex] x = 1 \\[3ex] y = 8 \\[3ex] k = 1 \cdot \sqrt[3]{8} \\[3ex] k = 1 \cdot 2 \\[3ex] k = 2 \\[3ex] x = 3 \\[3ex] y = ? \\[3ex] x \cdot \sqrt[3]{y} = k \\[3ex] \sqrt[3]{y} = \dfrac{k}{x} \\[5ex] y = \left(\dfrac{k}{x}\right)^3 \\[5ex] y = \dfrac{k^3}{x^3} \\[5ex] y = \dfrac{2^3}{3^3} \\[5ex] y = \dfrac{8}{27}$
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