The heart of middle school mathematics, and a key part of algebra readiness, is understanding ratios and rates. The overview and lessons below are tools to prepare students, usually in Grades 6 and up, who are ready to learn about these concepts. The lessons below will typically cover two days of instruction.
Ratios and Rates
A ratio is a comparison of two numbers or measurements. The numbers or measurements being compared are sometimes called the terms of the ratio. For example, if a store sells 6 red shirts and 8 green shirts, the ratio of red to green shirts is 6 to 8. You can write this ratio as 6 red/8 green, 6 red:8 green—or when writing fast or trying to make a point—simply 6/8 or 6:8. Both expressions mean that there are 6 red shirts “for every” 8 green shirts. Notice how you can rewrite 6/8 as 3/4, no different from any other time a math concept can appear as a fraction.
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. This is not a ratio of two like units, such as shirts. This is a ratio of two unlike units: cents and ounces. The first term of the ratio (69¢) is measured in cents, and the second term (12) 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, and similar to the shirt ratio, can enter calculations as the fraction 69/12. But notice that this time, a new unit is created: cents per hour.
Rates are used by people every day, such as when they work 40 hours per week or earn interest every year at a bank. When rates are expressed as a quantity of 1, such as 2 feet per second (that is, per 1 second) or 5 miles per hour (that is, per 1 hour), they can be defined as unit rates. You can write any rate as a unit rate by reducing the fraction so it has a 1 as the denominator or second term. As a unit rate example, you can show that the unit rate of 120 students for every 3 buses is 40 students per bus.
120/3 = 40/1
You could also find the unit rate by dividing the first term of the ratio by the second term.
120 ÷ 3 = 40
When a price is expressed as a quantity of 1, such as $25 per ticket or $0.89 per can, it is called a unit price. If you have a non-unit price, such as $5.50 for 5 pounds of potatoes, and want to find the unit price, divide the terms of the ratio.
$5.50 ÷ 5 pounds = $1.10 per pound
The unit price of potatoes that cost $5.50 for 5 pounds is $1.10 per pound.
Rates in the Real World
Rate and unit rate 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 = ?/12
60/3 = 240/12
Removing the units makes the calculation easier to see. However, it is important to remember the units when interpreting the new ratio.
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 × 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.
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 pens = $0.90 per pen
$0.90 × 10 pens = $9.00
Finding the cost of one unit enables you to find the cost of any number of units.
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