Ratios and Proportions Study Guide
Introduction to Ratios and Proportions
A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?
—Martin Gardner (1914– )
You may have seen and have even worked with ratios and proportions in your daily life, whether in sports scores or maybe on a map of your state. This lesson will explain ratios and proportions and how to best work with them.
Ratios are a way of comparing numbers, and they can be expressed in three ways. Suppose your school has five people with blonde hair for every seven people with brown hair. This information, expressed as a ratio, would be as follows:
- as a sentence:
There is a five-to-seven ratio of blondes to brunettes.
- using a colon:
- using a bar:
Like fractions, ratios can be reduced to smaller terms. A person may count that your class has 10 blondes and 14 brunettes and state that there is a 10-to-14 ratio. This is accurate because reduces to .
Sometimes, you will encounter ratio questions that deal with scale drawings. Scale drawings are used to represent objects that are too large or too small to be drawn or built to actual size. The scale is determined by the ratio of a given length on the drawing or model to its corresponding length in real life.
How does this work? Suppose you are given a map with a scale of 1 inch = 90 miles. If the distance between Virginia Beach, Virginia, and New York City, New York, on the map is 4 inches, what is the actual distance?
The ratio is . Set that equal to and cross multiply: 1x = (90)(4) x = 360
So, the actual distance is 360 miles.
A proportion is an equation that shows that two ratios are equivalent. Usually, proportions are written using ratios with bars.
When written as a:b = c:d, a and d are the extremes (they are on the end), and b and c are the means (they are in the middle). The product of the means equals the product of the extremes.
When written as afraction, the cross products are equal. This is useful indetermining whether two fractions are a proportion. Try one. Are and a proportion? Multiply the cross products to see:
- 24 × 9 = 18 ×12?
- 216 = 216; these fractions represent a proportion.
Proportions can be used to convert units. Look at the following conversions:
- 1 foot = 12 inches
- 3 feet = 1 yard
- 1 mile = 5,280 feet
- 1 minute = 60 seconds
- 1 hour = 60 minutes
- 1 meter = 10 decimeters
- 1 cup = 8 ounces
- 1 pint = 2 cups
- 1 quart = 2 pints
- 1 gallon = 4 quarts
- 1 meter = 100 centimeters
- 1 meter = 1,000 millimeters
If you know that 1 minute equals 60 seconds, you can easily figure out how many minutes are in 300 seconds by setting up a proportion:
When you use the fraction form of proportions, you can cross multiply to solve for any unknown. Cross multiplying, you get 1 × 300 = 60 × ?, or 300 = 60 × ?. Dividing both sides by 60, you get ? = 5.
You should always convert your units before you set up a proportion. Suppose you knew that 12 inches of fabric costs $0.60, and you needed to order 20 feet of it to design an outfit. First, you would convert the 20 feet into inches. You know 1 foot equals 12 inches, so you need to multiply:
20 feet ×= 240 inches
To find the price for 240 inches, set up a proportion:
12 × ? = (0.60)(240)
12 ×? = 144
? = = $12
Direct and Inverse Proportions
If two proportions are directly proportional, then one increases by a certain factor as the other increases by the same factor. If one decreases by a certain factor, the other decreases by that same factor.
If you have an after-school or weekend job, the amount of money you earn may be directly proportional to the amount of hours that you work. May be if you work twice as long, you make twice as much. If you work half the amount you usually work in one week, you earn half the money you usually get.
Two proportions are inversely proportional if an increase by a certain factor for one is accompanied by a decrease by that same factor for the other.
Find practice problems and solutions for these concepts at Ratios and Proportions Practice Questions