A ratio is a comparison of two quantities using division. The ratio of quantities a and b can be expressed as a to b, a : b, or, where b ≠ 0. Ratios are usually expressed in simplest form.

Example 1: a) The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is Find the athlete-to-student ratio to the nearest tenth. To find this ratio, divide the number of athletes by the total number of students.

b) The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) To find this ratio, divide the number of school days by the total number of days in a year.

Extended Ratios can be used to compare three or more quantities. The expression a : b : c means that the ratio of the first two quantities is a : b, the ratio of the last two quantities is b : c, and the ratio of the first and last quantities is a : c.

Example 2: a) In ΔEFG, the ratio of the measures of the angles is 5 : 12 : 13. Find the measures of the angles. Just as the ratio or 5 : 12 is equivalent to or 5x : 12x, the extended ratio 5 : 12 : 13 can be written as 5x : 12x : 13x. Write and solve an equation to find the value of x. __ x ___ 5x5x 5x + 12x + 13x= 180Triangle Sum Theorem 30x= 180Combine like terms. x= 6Divide each side by 30. So, the measures of the angles are 5(6) or 30°, 12(6) or 72°, and 13(6) or 78°.

b) The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles. Just as the ratio or 3 : 5 is equivalent to or 3x : 5x, the extended ratio 3 : 5 : 7 can be written as 3x : 5x : 7x. Write and solve an equation to find the value of x. __ 3 5 5x5x ___ 3x3x 3x + 5x + 7x= 180Triangle Sum Theorem 15x= 180Combine like terms. x= 12Divide each side by 30. So, the measures of the angles are 3(12) or 36°, 5(12) or 60°, and 7(12) or 84°.

An equation stating that two ratios are equal is called a proportion. In the proportion the numbers a and d are called the extremes of the proportion,,while the numbers b and c are called the means of the proportion. The product of the extremes ad and the means bc are called the cross products.

The converse of the Cross Products Property is also true. If ad = bc and b ≠ 0 and d ≠ 0, then. That is, and form a proportion. You can use the Cross Products Property to solve a proportion.

Example 3: Solve the proportion. a) 6y = 18.2(9)Cross Products Property 6y = 163.8Multiply y = 27.3Divide each side by 6

b) (4x – 5)6 = 3(–26)Cross Products Property 24x – 3 = –78Simplify 24x = –48Add 30 to each side x = –2Divide each side by 24

c) (7n – 1)2 = 8(15.5)Cross Products Property 14n – 2 = 124Simplify 14n = 126Add 2 to each side n = –2Divide each side by 14

Example 4: a) Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat. Students who have a pet Total number of students 18 ● 870= 30xCross Products Property 15,660= 30xSimplify. 522= xDivide each side by 30.

b) Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job. Students who have a part time job Total number of students 20 ● 810= 50xCross Products Property 16,200= 50xSimplify. 324= xDivide each side by 50.