CHAPTER 5
Ratios, Rates, and Proportions
Ratios and Rates
A ratio is a way of showing how two or more things compare in terms of size, number, or amount. For example, if a club has 3 members who are men for every 2 members who are women, you would say that the ratio of men to women club members is 3 to 2.
There are three common ways to write ratios. Using the ratio of men to women as an example:
• As a statement: as in the preceding example, “the ratio of men to women is 3 to 2”
• In a fraction: 
• In ratio notation: 3:2
Ratios are always written in lowest terms. Therefore, the ratio 6:10 (said “six to ten”) must be simplified to 3:5.
A ratio that compares two quantities with different units is called a rate. Rates are common in everyday life. A very common example is the speed limit on major highways: 65 miles per hour. This is a ratio of 65 miles for each hour.
Word Problems with Ratios and Rates
On the GED® test, ratios and rates often come up in word problems. You may also be asked to simplify a given ratio. Let’s look at a couple of examples.
EXAMPLE 1
A craftsman uses 48 nails to attach 200 feet of siding. As a fraction, what is the ratio of nails to feet of siding?
The problem is asking for the ratio of nails to feet of siding, so the order matters. As a fraction, the ratio is 
.
EXAMPLE 2
The list shows the price of different quantities of ketchup at three different stores.
To the nearest hundredth, which store has the lowest price per ounce?
Each price per quantity can be written as a ratio. To find the price “per ounce,” you will need to simplify each ratio until it has a denominator of 1.
To make the denominator of each ratio 1, divide each by its denominator.
Now that you have found the rate per unit (the rate where the denominator is 1), you can see that Outlet Mart has the lowest price per ounce.
EXERCISE 1
Ratios and Rates
Directions: Match the ratio on the left with its equivalent ratio on the right. There is only one correct answer for each of the problems 1 through 3. Not all of the answers on the right will be used.
1. 3:4
2. 2:9
3. 8:6
A. 
B. 
C. 
D. 
E. 
Write each of the following ratios or rates as fractions in their lowest terms.
4. A writer types 500 words every 2 hours.
5. The new trail has 3 miles for every 2 miles of the old trail.
Fill in the blanks for questions 6 and 7.
6. Laura made a scale model of the Alamo. The front of the actual church building is 63 feet long by 33.5 feet high. If Laura’s model is 15.12 inches long, how tall is it? _____________
7. What is the magnitude of the scale change? 1 foot = __________ inches
Solve each of the following word problems.
8. A company’s workforce is composed of 10 men for every 12 women. What is the ratio of men to women at this company?
9. The Andersons and the Lamberts are on road trips. The Andersons traveled 300 miles in 6 hours, while the Lamberts traveled 200 miles in 5 hours. Which family traveled at a faster average rate?
10. Miguel’s garden is four times larger than Tama’s garden. Write the fraction that represents the ratio of the size of Miguel’s garden to the size of Tama’s garden.
Proportions
Proportions are equations involving equal ratios or rates. Often, problems with proportions will require you to find a missing value, usually shown as x or some other letter. You can do this through a process called cross multiplying.
EXAMPLE 3
What is the value of x if 
Cross multiplying means multiplying diagonally across the equals sign. When you cross multiply in this problem, you get:
(3 × 12) = (4 × x) or 36 = 4x
You can then find the unknown value by dividing both sides by 4: x = 9.
EXAMPLE 4
What is the value of b if 
?
Cross multiply: 10b = 10.
Divide both sides by the number in front of b: b = 1.
EXERCISE 2
Proportions
Directions: Find the unknown value that makes each of the statements below true.
1. 
2. 
3. 
4. 
5. 
Answers are on page 514.
Word Problems with Proportions
There are many real-life applications of proportions, and these can show up in word problems on the GED® test. Let’s look at some examples.
EXAMPLE 5
A car is traveling at a rate of 60 miles per hour. If this rate is maintained, how long will it take the car to travel 180 miles?
Set up the proportion 
where h represents the number of
hours. Note that both numerators are in miles and both denominators are in hours. In proportions like this one, the numerators and denominators must each be in the same units.
Cross multiply: 60h = 180.
Divide both sides by 60: h = 3.
The car will take 3 hours to travel 180 miles.
EXAMPLE 6
The ratio of links to images on a website is 3:10. If there are 500 images on the website, how many links does it contain?
The proportion is 
, where L is the number of links.
Cross multiply: 1500 = 10L.
Divide both sides by 10: 150 = L.
There are 150 links on the website.
EXERCISE 3
Word Problems with Proportions
1. There are 40 men in a sporting league and the ratio of men to women in the league is 5:3. How many women are in the league?
2. It took a bicyclist 4 hours to cover 30 miles of hilly terrain. At this rate, how many hours will it take her to cover 15 more miles?
3. A factory makes a party snack mix containing 20 pounds of peanuts for every 2 pounds of cashews. If a new batch of the mix contains 100 pounds of peanuts, how many pounds of cashews does it contain?
4. If a data entry specialist can enter 40 patient records in 3 hours, how long will it take him to enter 60 records?
5. A hiking map is drawn so that each inch represents 10 miles. If a popular trail is 16 miles long, how long will it be on the hiking map?
Answers are on page 514.