Rates

A ratio is a comparison of two numbers or measurements. The numbers or measurements being compared are called the terms of the ratio. A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces. You can write this rate as 69¢/12 ounces or 69¢:12 ounces. Both expressions mean that you pay 69¢ "for every" 12 ounces of corn.

Rates are used by people every day, such as when they work 40 hours a week or earn interest every year at a bank. When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. If you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.

120/3 = 40/1

The unit rate of 120 students for every 3 buses is 40 students per bus. You could also find the unit rate by dividing the first term of the ratio by the second term. When prices are expressed as a quantity of 1, such as $25 per ticket or $0.89 per can, they are called unit prices. If you have a multiple-unit price, such as $5.50 for 5 pounds of potatoes, and want to find the single-unit price, divide the multiple-unit price by the number of units.

$5.50 ÷ 5 = $1.10

The unit price of potatoes that cost $5.50 for 5 pounds is $1.10 per pound.

Rates and unit rates are used to solve many real-world problems. Look at the following problem. "Tonya works 60 hours every 3 weeks. At that rate, how many hours will she work in 12 weeks?" The problem tells you that Tonya works at the rate of 60 hours every 3 weeks. To find the number of hours she will work in 12 weeks, write a ratio equal to 60/3 that has a second term of 12.

60/3 = 240/12

Tonya will work 240 hours in 12 weeks.

You could also solve this problem by first finding the unit rate and multiplying it by 12.

60/3 = 20/1
20 x 12 = 240

When you find equal ratios, it is important to remember that if you multiply or divide one term of a ratio by a number, then you need to multiply or divide the other term by that same number. Now let's take a look at a problem that involves unit price. "A sign in a store says 3 Pens for $2.70. How much would 10 pens cost?" To solve the problem, find the unit price of the pens, then multiply by 10.

$2.70 ÷ 3 = $0.90
$0.90 10 = $9.00

Finding the cost of one unit first makes it easy to find the cost of multiple units.

• Introducing the Concept
• Developing the Concept