7.1 OBJ: Use ratios and proportions. EQ: How do we use ratios to set up proportions?

VOCABULARY Ratio: a comparison of a number a and a nonzero number b using division. All ratios must be in reduced form.

60 cm : 200 cm can be written as the fraction . 60 cm 200 cm Example 1 Simplify Ratios Simplify the ratio. a. 60 cm : 200 cm b. 3 ft 18 in. SOLUTION a. 60 cm : 200 cm can be written as the fraction . 60 cm 200 cm = 60 cm 200 cm 60 ÷ 20 200 ÷ 20 3 10 Simplify. is read as “3 to 10.” = 3

Example 1 3 ft 3 · 12 in. b. = 18 in. 36 ÷ 18 = 18 ÷ 18 = Simplify Ratios = 3 · 12 in. 18 in. 3 ft b. = 36 ÷ 18 18 ÷ 18 2 1 = 4

In the diagram, AB : BC is 4 : 1 and AC = 30. Find AB and BC. Example 2 Use Ratios In the diagram, AB : BC is 4 : 1 and AC = 30. Find AB and BC. SOLUTION Let x = BC. Because the ratio of AB to BC is 4 to 1, you know that AB = 4x. AB + BC = AC Segment Addition Postulate 4x + x = 30 Substitute 4x for AB, x for BC, and 30 for AC. 5x = 30 Add like terms. x = 6 Divide each side by 5. 5

To find AB and BC, substitute 6 for x. Example 2 Use Ratios To find AB and BC, substitute 6 for x. AB = 4x = 4 · 6 = 24 BC = x = 6 ANSWER So, AB = 24 and BC = 6. 6

Example 3 Use Ratios The perimeter of a rectangle is 80 feet. The ratio of the length to the width is 7 : 3. Find the length and the width of the rectangle. SOLUTION The ratio of length to width is 7 to 3. You can let the length l = 7x and the width w = 3x. 2l + 2w = P Formula for the perimeter of a rectangle 2(7x) + 2(3x) = 80 Substitute 7x for l, 3x for w, and 80 for P. 14x + 6x = 80 Multiply. 20x = 80 Add like terms. x = 4 Divide each side by 20. 7

To find the length and width of the rectangle, substitute 4 for x. Example 3 Use Ratios To find the length and width of the rectangle, substitute 4 for x. l = 7x = 7 · 4 = 28 w = 3x = 3 · 4 = 12 ANSWER The length is 28 feet, and the width is 12 feet. 9

In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG. Checkpoint Use Ratios 1. In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG.

In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG. Checkpoint Use Ratios 1. In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG. ANSWER EF = 16; FG = 8 2. The perimeter of a rectangle is 84 feet. The ratio of the length to the width is 4 : 3. Find the length and the width of the rectangle. ANSWER length, 24 ft; width, 18 ft

VOCABULARY Ratio: a comparison of a number a and a nonzero number b using division. Proportion: an equation that states that two ratios are equal.

Cross Product Property In a proportion, the product of the extremes is equal to the product of the means.

Ex: Solve the Proportion. 14 14 x = 1

Example 4 Solve the proportion . = y + 2 6 5 3 SOLUTION = y + 2 6 5 3 Solve a Proportion Solve the proportion . = y + 2 6 5 3 SOLUTION Write original proportion. = y + 2 6 5 3 5 · 6 = 3(y + 2) Cross product property 30 = 3y + 6 Multiply and use distributive property. 30 – 6 = 3y + 6 – 6 Subtract 6 from each side. 24 = 3y Simplify. Divide each side by 3. = 3y 3 24 8 = y Simplify. 17

Solve the proportion. We do…. Checkpoint Solve a Proportion Solve the proportion. We do…. 1. = 6 8 3 x ANSWER 4 2. = 15 y 5 3 ANSWER 9 3. = m + 2 5 14 10 ANSWER 5

From the scale on the map, you know that 1 inch represents 180 miles. Example 5 Write and Solve a Proportion Use the map of Texas to estimate the distance between Dallas and Houston. SOLUTION From the scale on the map, you know that 1 inch represents 180 miles. 19

On the map, the distance between Dallas and Houston Example 5 Write and Solve a Proportion On the map, the distance between Dallas and Houston is 1 inches, which can be written as 1.25 inches. 1 4 Let x represent the actual distance between Dallas and Houston. You can write the following proportion. Write a proportion. = 1.25 in. x mi 1 in. 180 mi. 1 · x = 180 · 1.25 Cross product property x = 225 Simplify. ANSWER The distance between Dallas and Houston is about 225 miles. 20

Homework Worksheet 7.1A