1. Exercise Use the LCM to rename these ratios with a common denominator.
1 2
2 3
and
3 6
4 6 ,

2. Exercise Use the LCM to rename these ratios with a common denominator.
2 3
3 5
and
10 15
9 15 ,

3. Exercise Use the LCM to rename these ratios with a common denominator.
1 3
3 4
2 5 , ,
and
20 60
45 60 ,
24 60

4. Exercise
Change these ratios to equivalent ratios by multiplying by 1 in the form of 2 over 2 and 3 over 3.
2 5
4 10
6 15 ,

5. Exercise
Change these ratios to equivalent ratios by multiplying by 1 in the form of 2 over 2 and 3 over 3.
4 3
8 6
12 9 ,

6. Proportion
A proportion is a statement of equality between two ratios.

7. 6 8 9 12 = 1st 3rd 2nd 4th The 1st and 4th are called the extremes.
The 2nd and 3rd are called the means.

8. Property of Proportions
The product of the extremes is equal to the product of the means.

9. 6 8
9 12 =
6 8
9 12 =
8 6
12 9 =
also

10. 6 8
9 12 =
6 8
9 12 =
9 6
12 8 =
also

11. 6 8
9 12 =
6 8
9 12 =
6 9
8 12 =
also

12. Example 1
3 5
12 20
Given = , write a proportion by inversion of ratios and a proportion by alternation of terms.
5 3
2012 =
3 12
5 20 =

13. Example 2 6 15 8 n Solve = for n. 8 n 6 15 = 6n 6 1206 = 6n = 15 • 8

14. Example 3 32 x 108 27 Solve = for x. 108 27 32 x = 108x 108 864108 =

15. Example
4 x
7 2
Solve = .
x ≈ 1.14

16. Example
x 8
9 50
Solve =
x = 1.44

17. Example 4
The ratio of adults to students on a bus is 2 to 7. If there are 8 adults, how many students are on the bus?
8 n
2 7 =
2n 2
56 2 =
2n = 56
n = 28 students

18. Example 5
Sam can paint 200 ft. of privacy fence in 3 hr. To the nearest foot, how many feet can he paint in 5 hr.?
3 hr. 5 hr.
200 ft. x ft. =
3x 3
1,000 3 =
3x = 200(5)
x ≈ 333 ft.
3x = 1,000

19. Example
A rectangle whose width is 5 ft. and whose length is 12 ft. is similar to a rectangle whose width is 8 ft. What is the length of the larger rectangle?
19.2 ft.

20. Example
What is the height of a tree if a 6 ft. man standing next to the tree makes an 8 ft. shadow and the tree makes a 50 ft. shadow?
37.5 ft.

21. Example
If a car can go 250 mi. on 8 gal., how many gallons will it take to go on a 600 mi. trip?
19.2 gal.

22. Example
If a 50 ft. fence requires 84 2” x 4”s, how many 2” x 4”s are needed for a 160 ft. fence?
269

23. Example
The product of ratios is also a ratio. For example, suppose a production line can assemble 130 cars per hour. How many days, at 8 hr./day, will it take to produce 100,000 cars?
97 days