Chapter Twenty-One — Rational Equations and Inequalities

The Humongous Book of Algebra Problems

475

Direct and Indirect Variation

Turn a word problem into a rational equation

21.21 Assume that the value of x varies directly with the value of y according to

the constant of proportionality k. Identify two equations that describe the

relationship between x and y.

If x and y vary proportionally, then y = kx and .

Note: Problems 21.22–21.23 refer to the direct variation relationship described below.

21.22 A professional sports team notes that the ambient crowd noise at home

games is directly proportional to the attendance at those games. During one

game, the cheering of a maximum capacity crowd of 75,000 fans averages 90

decibels. Identify the constant of proportionality k.

Let a represent attendance and n represent the noise (in decibels) generated by

that population. If n varies directly with a, then (according to Problem 21.21)

n = ka. Substitute n = 90 and a = 75,000 into the equation and solve for k.

Note: Problems 21.22–21.23 refer to the direct variation relationship described in Problem

21.22.

21.23 Approximately how loud is a crowd of 62,000 fans?

Substitute a = 62,000 and k = 0.0012 into the variation equation n = ka to

calculate n.

Varying

proportionally means

the same thing as

varying directly.

Let’s say

k = 2. According

to the equation

y = kx, y is always twice

as big as x. Because

y is always two times

as large as x, y ÷ x

always equals 2.

This value

of k (and the

equation n = ka

that you plug it into)

come from Problem

21.22.

Chapter Twenty-One — Rational Equations and Inequalities

The Humongous Book of Algebra Problems

476

21.24 Assume x and y vary proportionally. If x = 16 when y = –3, calculate the value of

y when x = 6.

If x and y vary proportionally, then x = ky. Substitute x = 16 and y = –3 into the

equation and solve for k.

To determine the value of y when x = 6, substitute x = 6 and into the

proportionality equation.

21.25 Assume that the growth of a vine is directly proportional to the time it is

exposed to light. If the vine is exposed to 72 hours of light and grows 1.5

inches, how long will the vine grow when exposed to 200 hours of light?

Round the answer to the hundredths place.

Let l represent the length the vine grows when exposed to h hours of light. The

values vary proportionally, so l = kh, where k is a constant of proportionality.

Substitute l = 1.5 and h = 72 into the equation to calculate k.

To determine how long the vine grows after 200 hours of light, substitute

h = 200 and into the equation l = kh.

The vine grows approximately 4.17 inches when exposed to 200 hours of light.

To cancel out

the coefcient of

y, multiply both sides

of the equation by

its reciprocal.

Rounding

4.166666… to

the hundredths

place gives you

4.17.

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