Section 6.4Ratio, Proportion and Variation

Math in Our World

Learning Objectives

Write ratios in fraction form. Solve proportions. Solve real-world problems using proportions. Solve real-world problems using direct variation. Solve real-world problems using inverse

variation.

Ratios

A ratio is a comparison of two quantities using division.

For two nonzero numbers, a and b, the ratio of a to b is written as a:b (read a to b) or ab .a

b

EXAMPLE 1 Writing Ratios

According to the Sporting Goods Manufacturers’ Association, 95.1 million Americans participate in recreational swimming, 56.2 million Americans participate in recreational biking, 52.6 million Americans participate in bowling, and 44.5 million Americans participate in freshwater fishing. Find each:

(a) The ratio of recreational swimmers to recreational bikers

(b) The ratio of people who fish to people who bowl

Proportions

A proportion is a statement of equality of two ratios.

Two ratios form a proportion if the cross products of their numerators and denominators are equal.

For example, the ratio of 4:7 and 8:14 can be written as a proportion as shown. 4 8

7 14

, if .a c

ad bcb d

EXAMPLE 3 Deciding if a Proportion is True

Decide if each proportion is true or false.

EXAMPLE 4 Solving a Proportion

Solve the proportion for x.

EXAMPLE 5 Solving a Proportion

Solve the proportion:

EXAMPLE 6 Applying Proportion toFuel Consumption

While on a spring break trip, a group of friends burns 12 gallons of gas in the first 228 miles, then stops to refuel. If they have 380 miles yet to drive, and the SUV has a 21-gallon tank, can they make it without refueling again?

EXAMPLE 7 Applying Proportion toWildlife Population

As part of a research project, a biology class plans to estimate the number of fish living in a lake thought to be polluted. They catch a sample of 35 fish, tag them, and release them back into the lake. A week later, they catch 80 fish and find that 5 of them are tagged. About how many fish live in the lake?

Variation

A quantity y is said to vary directly with x if there is some nonzero constant k so that y = kx. The constant k is called the constant of proportionality.

Two quantities are often related in such a way that if one goes up, the other does too, and if one goes down, the other goes down as well. This would show direct variation.

EXAMPLE 8 Using Direct Variation to Find Wages

Suppose you earn $95 per day. Write a variation equation that describes total pay in terms of days worked, and use it to find your total pay if you work 6 days and if you work 15 days.

EXAMPLE 9 Using Direct Variation to Find a Weight

The weight of a certain type of cable varies directly with its length. If 20 feet of cable weighs 4 pounds, find k and determine the weight of 75 feet of cable.

Variation

A quantity y is said to vary inversely with x if there is some nonzero constant k so that y = kx. k

x

EXAMPLE 10 Using Inverse Variation to Find Driving Time

The time it takes to drive a certain distance varies inversely with the speed, and the constant of proportionality is the distance. A family has a vacation cabin that is 378 miles from their residence. Write a variation equation describing driving time in terms of speed. Then use it to find the time it takes to drive that distance if they take the freeway and average 60 miles per hour, and if they take the scenic route and average 35 miles per hour.

EXAMPLE 11 Using Inverse Variation to Construction

In construction, the strength of a support beam varies inversely with the cube of its length. If a 12-foot beam can support 1,800 pounds, how many pounds can a 15-foot beam support?