Section 6.4 Ratio, 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. The most meaningful way to compare the sizes of two numbers is to divide them. For two nonzero numbers, a and b, the ratio of a to b is written as a:b (read a to b) or ab.

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

EXAMPLE 1 Writing Ratios SOLUTION

EXAMPLE 2 Writing a Ratio Involving Units Find the ratio of 18 inches to 2 feet. It’s tempting to simply write 18/2, but this is deceiving—it makes it seem like 18 inches is 9 times as much as 2 feet, which is of course silly. Instead, to make the ratio meaningful, we want the units to be the same. Since 1 foot is 12 inches, 2 feet is 24 inches. So the ratio is Now we can reduce. The unit inches divides out. The ratio of 18 inches to 2 feet is 3:4. SOLUTION

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.

EXAMPLE 3 Deciding if a Proportion is True Decide if each proportion is true or false. In each case, we will cross multiply and see if the two products are equal. (a) 3 x 15 = 45; 5 x 9 = 45 The proportion is true. (b) 5 x 2 = 10; 3 x 7 = 21 The proportion is false. (c) 14 x 8 = 112; 16 x 7 = 112 The proportion is true. SOLUTION

EXAMPLE 4 Solving a Proportion Solve the proportion for x. SOLUTION

EXAMPLE 5 Solving a Proportion Solve the proportion: SOLUTION

EXAMPLE 5 Solving a Proportion SOLUTION

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 6 Applying Proportion to Fuel Consumption

SOLUTION Step 1 Identify the ratio statement. The ratio the problem gives us is 12 gallons of gas to drive 228 miles. Step 2 Write the ratio as a fraction. Step 3 Set up the proportion. We need to find the number of gallons of gas needed to drive 380 miles, so we’ll call that x. The ratio we already have is gallons compared to miles, so the second ratio in our proportion should be as well. We have x gallons, and 380 miles, so the proportion is

EXAMPLE 6 Applying Proportion to Fuel Consumption SOLUTION Step 4 Solve the proportion. Step 5 Answer the question. The SUV will burn 20 gallons of gas to make the last 380 miles, so they can make it without stopping.

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? EXAMPLE 7 Applying Proportion to Wildlife Population

SOLUTION Step 1 Identify the ratio statement. Five of 80 fish caught were tagged. Step 2 Write the ratio as a fraction. Step 3 Set up the proportion. We want to know the number of fish in the lake, so call that x. The comparison in the lake overall is 35 tagged : x total, so the proportion is

EXAMPLE 7 Applying Proportion to Wildlife Population SOLUTION Step 4 Solve the proportion. Step 5 Answer the question. There are approximately 560 fish 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.

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 8 Using Direct Variation to Find Wages SOLUTION Let y = the total amount earned x = the number of days you work k = $95 per day (as we saw above) Then y = 95x is the variation equation. For x = 6 days: y = 95 x 6 = $570. For x = 15 days: y = 95 x 15 = $1,425.

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. EXAMPLE 9 Using Direct Variation to Find a Weight SOLUTION Step 1 Write the equation of variation, y = kx. y = the weight x = length of cable in feet k = the constant4 lbs = k 20 ft

EXAMPLE 9 Using Direct Variation to Find a Weight SOLUTION Step 2 Solve for k. Now we know that the equation of variation can be written as y = 0.2x. Step 3 Solve the problem for the new values of x and y. So 75 feet of cable will weigh 15 pounds.

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

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 10 Using Inverse Variation to Find Driving Time

EXAMPLE 10 Using Direct Variation to Find Driving Time SOLUTION Let y = the time it takes to drive the distance x = the average speed k = 378 miles (the distance) Then the variation equation is If they average 60 miles per hour: If they average 35 miles per hour:

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? EXAMPLE 11 Using Inverse Variation to Construction SOLUTION Step 1 Write the equation of variation. Let y = strength of the beam in pounds it can support x = length of the beam k = the constant of proportionality The variation equation is y = kx3, since y varies inversely with the cube of x.

EXAMPLE 11 Using Direct Variation to Construction SOLUTION Step 2 Find k. Step 3 Substitute in the given value for x. In this case, it is 15. A 15-foot beam can support pounds.