Inverse Variation

Introduction

If one worker needs 12 hours to complete a job the above table shows the relationship between number of workers and the amount of hours to complete the job. From this table, we can see that the amount of hours necesary to complete the job is equal to 12 divided by the number of worker.

Conclusion: As the number of workers increases, the number of hours to complete the job decreases.

If Sarah goes from her house to Jane's house, which have a distance of 300 miles between them the above table shows the relationship between the speed of the car and the number of hours requaired to drive to Jane's house. ( We assume that the speed is constant). From this table, we can see that the number of hours is equal to 300 divided by the speed of the car.

Conclusion: As she increases the speed of her car, the number of hours driving to Jane's house decreases.

In a birthday party there are 120 cookies to distribute them equitably to the children, the above table shows the relationship between the total number of children and the number of cookies corresponding to each child. From this table, we can see that the number of cookies corresponding to each child is equal to 120 divided by the total number of children in the party.

Conclusion: As the total number of children increases, the number of cookies corresponding to each child decreases.

In all of these relations, one variable is equal to a non-zero constantan divided by other variable. Such relations are referred to as inverse proportions or inverse variations.

Definition

Given two variables x and y, the statements

y varies inversely as x,

y is inversely proportional to x.

Finding the Formula for Inverse Variation Relations

Inverse variation relations have the form $y=\frac{k}{x}$ where x and y are variables and k is a non-zero constant. This is a relationship between two variables in which the product is a constant. The formula for an inverse variation relationship can be found with the following steps.

Examples

Example 1

r is inversely proportional to s. If r = 15 when s = 3, then write the formula for the relation between r and s.

Conclusion: the formula to express the relationship between r and s is $r=\frac{\mathrm{45}}{s}$

Example 2

c varies inversely as d. If c = 100 when d = 0.2, then write the formula for the relation between c and d.

Conclusion: the formula to express the relationship between c and d is

$c=\frac{\mathrm{20}}{d}$

Steps to Solve Simple Inverse Variation Problems

Simple inverse variation problems consist in find a formula and verify what happen with a quantity when we change the other quantity of the problem.

Examples

Example 1

f is inversely proportional to g. If f = 5 when g = 25, then find the value of g when

f = 15.

Step 1: Find formula	1: Translate the statement into a inverse variation formula. f is inversely proportional to g means $f=\frac{k}{g}$
	2: Substitute known values to find k. $5=\frac{k}{\mathrm{25}}$ k =5 · 25 = 125
	3: Substitute k and write the formula. $f=\frac{\mathrm{125}}{g}$
Step 2: Identify known variables and substitute into formula. $\mathrm{10}=\frac{\mathrm{125}}{g}$
Step 3: Solve for Unknown. g = 12.5

Conclusion: g = 12.5 when f = 10.

Example 2

t varies inversely as q. If t = 240 when q = 0.01, then find the value of t when q = 8

Conclusion: t = 0.3 when q = 8

Steps to Solve Word problems of Inverse Variation

Word problems of inverse variation are problems where the involved quantities have names, but the way to solve them is not different from the last section. The steps to solve then are the same

Examples

Example 1

Let a rectangle which its area does not change. The length of a rectangle varies inversely as the width. If the length of a rectangle is 20 ft when the width is 8 ft, find the length of the rectangle when the width is 10 ft.?

Step 1: Find formula	1: Translate the statement into a inverse variation formula. The length of a rectangle varies inversely as the width means $\mathrm{length}=\frac{k}{\mathrm{width}}$
	2: Substitute known values to find k. $\mathrm{20}=\frac{k}{8}$ k =20 · 8 = 160
	3: Substitute k and write the formula. $\mathrm{length}=\frac{\mathrm{160}}{\mathrm{width}}$
Step 2: Identify known variables and substitute into formula. $\mathrm{lenght}=\frac{\mathrm{160}}{\mathrm{10}}$
Step 3: Solve for Unknown. length = 16

Conclusion: The length of a rectangle which area is 160 feet square is 16 when its width is 10.

Example 2

The time to complete a project is inversely proportional to number of people who is doing the project. A certain project can be completed by 5 workers in 24 days. In order to finish the project sooner, the company plans to hire additional workers. How many workers are needed to finish the project in 15 days?

Step 1: Find formula	1: Translate the statement into a inverse variation formula. The time to do a project is inversely proportional to number of worker doing the project means $\mathrm{time}=\frac{k}{\mathrm{\#\; workers}}$
	2: Substitute known values to find k. $\mathrm{24}=\frac{k}{5}$ k =24 · 5 = 120
	3: Substitute k and write the formula. $\mathrm{time}=\frac{\mathrm{120}}{\mathrm{\#\; workers}}$
Step 2: Identify known variables and substitute into formula. $\mathrm{15}=\frac{\mathrm{120}}{\mathrm{\#\; workers}}$
Step 3: Solve for Unknown. # workers = 8

Conclusion: The company needs 8 workers to finish the project in 15 days.