Math Background

Fractions, Ratios, Rates, and Percents: Overview

Earlier in the year students studied basic operations with fractions and decimals and were introduced to statistics and probability. This background gave them a foundation for learning about ratios and percents. Students need to understand the concepts of ratios, rates, and percents. They will encounter many forms of numerical comparisons in the media. Television, magazines, and newspapers often quote statistics in their presentations. Numerical data, graphs, and charts frequently contain rates, percents, and ratios. Medical information, business statistics, census data, and sports data are often reported using ratios and percents.

A ratio compares two numbers. The numbers are called the terms of the ratio. For example, if you have 2 dimes, 3 nickels, and 4 quarters, you can write the ratio of nickels to quarters in several ways: word form—3 to 4; ratio form—3:4; or fraction form—three-fourths . The ratio of nickels to all nine coins would be three-ninths , or in simplest form one-third . Ratios can be used to describe many different circumstances. Suppose a fifth-grade spelling bee is in progress, and there are 11 students remaining in the competition. Four boys are still trying to win the championship. The ratio of remaining boys to remaining girls is 4 to 7. The ratio of remaining girls to remaining boys is 7:4. The ratio of remaining boys to remaining students is 4:11, and so on.

Ratios that are equal to each other are called equivalent ratios. You can find an equivalent ratio by multiplying or dividing each term of a ratio by the same number.
For example, one-fourth = one-fourth x two-halves = two-eighths
one-fourth and two-eighths are equivalent ratios. Other ratios equivalent to one-fourth are three-twelfths , four-sixteenths , five-twentieths , and twenty-five-hundredths . Any ratio has an infinite number of equivalent ratios.

A rate is a ratio that compares two quantities using different units. A common comparison is distance and time. If you drive 90 miles in 2 hours, your rate can be written as ninety-halves = forty over one , or 45 miles per hour. Your car's speedometer is a gauge that measures a rate—miles per hour. Another common rate is cost per unit. Ramona earns $5.00 for washing her parents' van. If this job takes her 2 hours to complete, her rate is five dollars over two = two dollars and fifty cents over one , or $2.50 per hour. A bag of 9 oranges costs $1.98. The rate, or cost per unit, is one dollar and ninety-eight cents over nine = twenty-two cents over one , or $0.22 per orange.

Percent means “per hundred” and is a ratio comparing a number to 100. Students' weekly quizzes and tests are often graded with a percent that compares the number of correct answers to the total number of questions. Seventy percent is written as 70% and can be written as the ratio seventy-hundredths . A percent can be written as a ratio and as a decimal. For example, 35% = thirty-five-hundredths = 0.35. The ratio thirty-five-hundredths can be simplified to seven-twentieths by dividing both the numerator and the denominator by 5. The percent 35%, the ratio thirty-five-hundredths , the decimal 0.35, and the fraction seven-twentieths are equivalent. They all name the same number.

You can also write a fraction, such as four-fifths , as a percent. four-fifths = four-fifths x twenty-twentieths = eighty-hundredths = 0.80 = 80%

If the fraction doesn't have a denominator of 100, you multiply the numerator and denominator by the same number to create an equivalent fraction with a denominator of 100. Then write the equivalent percent.

You can show students how to use a number line like the one shown below to make comparisons among fractions, decimals, and percents. For example, compare three-twentieths , 40%, and 0.55.


When comparing these three numbers on a number line, you can see that 0.55 is the farthest to the right, the farthest from 0, and is therefore the greatest number. The fraction three-twentieths is closest to 0 and is the least number. Forty percent (40%) is between three-twentieths and 0.55. So three-twentieths < 40% < 0.55. The number line also shows us that one-fourth and 0.25, one-half and 0.5, and three-fourths and 0.75 occupy the same points on the line. They name the same number.

Another way to compare these numbers is to rewrite the ratio and percent as a decimal and then compare them to the decimal 0.55.

three-twentieths = fifteen-hundredths = 0.15
40% = forty-hundredths = 0.4

0.15 is less than 0.4, which is less than 0.55. So three-twentieths < 40% < 0.55.

You could also rewrite the three numbers as percents or ratios and compare them.

Following are some problems involving percents and ratios.

Zack's collection of 80 sports cards includes players in major-league football, basketball, baseball, and hockey. His favorite sport is baseball, and 60% of his cards are baseball players. How many cards are baseball players?
60% of 80 cards = 0.60 x 80 = 48 cards
So 48 cards are baseball players.

Dwayne's geography quiz had 10 questions, and he got 8 correct. The ratio of correct to incorrect answers is 8 to 2. His percent score or ratio of correct answers to total number of questions is eight-tenths = eighty-hundredths = 0.80 = 80%.

Ella's book bag contains 10 markers—6 red, 3 blue, and 1 orange. The ratio of red markers to orange markers is 6 to 1, and the ratio of orange markers to blue markers is 1 to 3. To determine what percent of Ella's markers is red, blue, or orange, compare each color to the total number of markers.

Red = 6 to 10 or six-tenths = sixty-hundredths = 0.60 = 60% Blue = 3 to 10 or three-tenths = thirty-hundredths = 0.30 = 30% Orange = 1 to 10 or one-tenth = ten-hundredths = 0.10 = 10%

Houghton Mifflin Math Grade 5