1. Ratio and Proportion
7-1

2. EXAMPLE 1
Simplify ratios
Simplify the ratio. b.
5 ft
20 in.
64 m : 6 m a.
SOLUTION a.
Write
64 m : 6 m as
64 m
6 m .
Then divide out the units and simplify.
64 m
6 m = 32 3 =
32 : 3 b.
To simplify a ratio with unlike units, multiply by a conversion factor.
5 ft
20 in. =
5 ft
20 in.
12 in.
1 ft = 60 20 = 3 1

3. GUIDED PRACTICE
for Example 1
Simplify the ratio.
yards to 3 yards
SOLUTION
Write
24 yards : 3 yards as 24 3
Then divide out the units and simplify. 24 3 = 8 1 =
8 : 1

4. GUIDED PRACTICE
for Example 1
Simplify the ratio.
cm : 6 m
SOLUTION
To simplify a ratio with unlike units, multiply by a conversion factors.
150cm 6m =
150 6 1m
100cm = 1 4
= 1 : 4

5. EXAMPLE 4 Solve proportions Solve the proportion. ALGEBRA a. 5 10 x 16=
SOLUTION a. 5 10 x 16 =
Write original proportion. =
10 x
5 16
Cross Products Property =
10 x 80
Multiply. = x 8
Divide each side by 10.

6. EXAMPLE 4 Solve proportions b. 1 y + 1 = 2 3y SOLUTION 1 y + 1 2 3y b.
Write original proportion. =
2 (y + 1)
1 3y
Cross Products Property =
2y + 2 3y
Distributive Property = y 2
Subtract 2y from each side.

7. GUIDED PRACTICE for Example 4 5. 2 x 5 8 = SOLUTION 2 x 5 8 = = 5 x
Write original proportion. =
5 x
2 8
Cross Products Property =
5 x 16
Multiply. = x 16 5
Divide each side by 5 .

8. GUIDED PRACTICE for Example 4 6. 1 x – 3 4 3x = SOLUTION 1 x – 3 4 3x
Write original proportion. 3x
4(x – 3) =
Cross Products Property 3x
4x – 12 =
Multiply.
3x – 4x
– 12 =
Subtract 4x from each side.
– x =
– 12 x = 12

9. GUIDED PRACTICE for Example 4 7. y – 3 7 y 14 = SOLUTION y – 3 7 y 14
Write original proportion. =
14(y – 3)
7 y
Cross Products Property
14y – 42 7y =
Multiply.
Subtract 7y from each side and add 42 to each side.
14y – 7y 42 = y = 6
Subtract , then divide

10. EXAMPLE 2
Use a ratio to find a dimension
Painting
You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484 feet and that the ratio of its length to its width is 9 : 2. Find the area of the wall.
SOLUTION
Write expressions for the length and width. Because the ratio of length to width is 9 : 2, you can represent the length by 9x and the width by 2x.
STEP 1

11. Use a ratio to find a dimension
EXAMPLE 2
Use a ratio to find a dimension
STEP 2
Solve an equation to find x. =
2l + 2w P
Formula for perimeter of rectangle =
2(9x) + 2(2x)
484
Substitute for l, w, and P. =
484
22x
Multiply and combine like terms. = 22 x
Divide each side by 22.
Evaluate the expressions for the length and width. Substitute the value of x into each expression.
STEP 3
Length = 9x = 9(22) = 198
Width = 2x = 2(22) = 44
The wall is 198 feet long and 44 feet wide, so its area is
198 ft
44 ft =
8712 ft . 2

12. The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90. ANSWER
EXAMPLE 3
Use extended ratios
ALGEBRA The measures of the angles in CDE are in the extended ratio of 1 : 2 : 3. Find the measures of the angles.
SOLUTION
Begin by sketching the triangle. Then use the extended ratio of 1 : 2 : 3 to label the measures as x° , 2x° , and 3x° . =
180 o
x + 2x + 3x
Triangle Sum Theorem = 6x
180
Combine like terms. = 30 x
Divide each side by 6.
The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90. o
ANSWER

13. GUIDED PRACTICE
for Examples 2 and 3
3. The perimeter of a room is 48 feet and the ratio of its length to its width is 7 : 5. Find the length and width of the room.
SOLUTION
Write expressions for the length and width. Because the ratio of length is 7 : 5, you can represent the length by 7x and the width by 5x.
STEP 1

14. Solve an equation to find x.
GUIDED PRACTICE
for Examples 2 and 3
STEP 2
Solve an equation to find x. =
2l + 2w P
Formula for perimeter of rectangle =
2(7x) + 2(5x) 48
Substitute for l, w, and P. = 48
24x
Multiply and combine like terms. = 2 x
Evaluate the expressions for the length and width. Substitute the value of x into each expression.
STEP 3
Length = 7x + 7(2) = 14 ft
Width = 5x + 5(2) = 10 ft

15. The angle measures are 20 , 3(20 ) = 60 , and 5(20 ) = 100. ANSWER
GUIDED PRACTICE
for Examples 2 and 3
4. A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles. x
3x 5x
SOLUTION
Begin by sketching the triangle. Then use the extended ratio of 1 : 3 : 5 to label the measures as x° , 2x° , and 3x° . =
x + 3x + 5x
180 o
Triangle Sum Theorem = 9x
180
Combine like terms. = 20 x
Divide each side by 9.
The angle measures are 20 , 3(20 ) = 60 , and 5(20 ) = 100. o
ANSWER