## Ratio and Proportion

Ratios
A ratio compares two numbers or two quantities. The two numbers being compared are called terms. A ratio can be written in three different ways.

a to b
a : b

Each is read as “the ratio of a to b.” Ratios can be simplified by canceling common factors. Two ratios are equivalent if they represent the same number.

A rate is a special ratio that compares two quantities that have different units of measure. If the second term in a rate is 1, the ratio is called a unit rate. Unit rates are often described as a quantity per unit of time or per item. Sometimes rates are written with a slash rather than the word per, such as mi/h for miles per hour or \$2/dozen for \$2 per dozen. Students will frequently encounter rates, such as being in a car traveling 30 miles per hour, making a long-distance telephone call that costs 20¢ per minute, skating at an ice rink that costs \$10 for 2 hours, and so on.

Proportions
A proportion is a statement that shows two ratios are equivalent. A quick way to tell whether two ratios form a proportion is to use cross multiplication. To cross multiply, find the product of the numerator on one side with the denominator on the other. If the ratios are equivalent, the cross products are equal and if the cross products are equal, the ratios are equivalent. Here is the algebraic explanation of why this statement is true.

a, b, c, and d form any proportion, where b ≠ 0 and b ≠ 0.
If you multiply both sides of the equation by , which is the multiplicative inverse of , the equation is still true because you have performed the same operation on both sides of the equals sign.
Because is equal to 1, the numerator and denominator must be equal.

The cross products a × d and b × c are equal.

A proportion is often used when one ratio is known and only part of a second ratio is known, such as “The ratio of girls to boys in a class is 6:8 and there are 12 boys in the class.” A proportion can be set up and solved to find how many girls there are in the class. There are various ways of solving a proportion. One way is to reduce the known ratio to simplest form and then find an equivalent fraction that matches the corresponding known term for the other ratio.

Set up the proportion.
Reduce to
Find a common denominator.
There are 9 girls in the class.
x = 9

Another method of solving a proportion is using cross multiplication.

Set up the proportion.
Set the two cross products equal.
8 × x = 6 × 12
Simplify 6 × 12.
8 × x = 72
Solve the equation by dividing both sides by 8.
There are 9 girls in the class.
x = 9

Two important uses of proportions are speed and scale drawings. Speed is a rate—the ratio of distance to time. This leads to the formula relating speed, distance, and time: s = , where s is the speed, d is the distance, and t is the time. Speed is usually given as a unit rate, such as miles per hour or meters per second. In a scale drawing, the scale is the ratio of the measurements in the drawing of an object to the corresponding measurements of the actual object. By using proportions, students can find lengths needed to make a scale drawing or can find the actual lengths of an object based on a given scale drawing.

Teaching Model 16.4: Write and Solve Proportions