Lesson 2-F ~ Rates and Ratios with Complex Fractions 23

Materials Needed

None

Warm-Up Problems

1) Donna walked 18 miles in 4 hours at the same rate. What was her speed in miles per hour? 4.5 miles per hour

2) Simplify.

a) 4 __ 6 ∙ 9 __ 8 3 __ 4

b) 2 1 _ 2 ÷ 1 _ 2 5

3) Convert the rate 12 miles1 hour

to miles per minute. 0.2 mile per minute

Teaching Tips

Emphasize with students that it is best to write mixed numbers as improper fractions before they are used in a complex fraction.

Lesson 4 (Unit Rates) in the Oregon Focus on Proportionality book has rates with decimals. Students may want to convert the fractions in the rates and ratios used in this lesson to decimals first and follow the examples in Lesson 9 (Proportions and Similiar Figures). This may be a good way to check solutions, but focus on students using fractions throughout the lesson.

When multiplying fractions, remind students they can reduce the factors while multiplying instead of waiting to simplify the product at the end of the computation.

At this level, most students will choose Method 1 for simplifying complex fractions. However, some may choose Method 2 which is one that will be used in algebra to simplify equations with complex fractions involving variables.

24 Lesson 2-F ~ Rates and Ratios with Complex Fractions

Extra Example 1

Simplify each complex fraction.

a) 8

154

2 __ 15

b) 623

9

Extra Example 2

Sasha has many soccer fields to mow. She spent 3 _ 4 hour mowing 1 1 _ 8 soccer fields. Find the unit rate of hours per soccer field to find how long it takes Sasha to mow each soccer field.Sasha mows at a rate of 2 _ 3 hour per soccer field.

Teaching Tips (Continued)

Students may struggle with understanding conversions, especially using fractions. If so, you may want to ask students some conversion problems (e.g., change 30 feet per minute to miles per hour).

Consider discussing the meaning of fractions in word problems. For example, at the beginning of the lesson, explain why 1 _ 4 hour = 15 minutes or in Example 2 that 1 _ 3 hour means 20 minutes. Ask students what it means to fill 2 _ 3 of an aquarium.

Place the complex fractions on a number line to show students the values they represent.

Lesson 2-F ~ Rates and Ratios with Complex Fractions 25

Extra Example 3

Find the scale factor of the similar triangles.

58 inch

1 14

inch

12

or 1 : 2

Explore!

Step 1: a. 13200 feet 1 mile 60 min 792000 miles

30 min feet hour 158400 hours

= 5280 1

b. 5 miles per hour

Step 2: a. 2.5 b. 0.5 c. 5 miles per hour

Step 3: a. 2 12

or 52

b. 12

c. 5 miles per hour Step 4: Answers may vary.

26 Lesson 2-F ~ Rates and Ratios with Complex Fractions

Focused Assignment

Exercises 1, 3, 5–8, 11, 13–15, 19–20

Exercise Answers

1. 10

2. 16

3. 25

4. 47

5. 45

6. 16

7. Pedro is correct.

1628

= 64 and 1628

= 1

8. 3 inches per minute

9. 23

foot per second

10. 6 miles per hour

11. 58

page per minute

12. 2 18

cookies per hour

13. 1 25

innings per game

14. 9 entries per hour

15. 34

foot per hour

16. 4 29

miles per hour

17. 16

book per day

18. 330 miles

19. 1 12

hours

Lesson 2-F ~ Rates and Ratios with Complex Fractions 27

20. Corin made a mistake. 3 miles 60 min4 min 1 hour

= 45 miles per hour

21. 718

or 7:18

22. 120

or 1:20

23. 25

or 2:5

Communication Prompt

Explain how finding a unit rate that involves whole numbers is similar to finding a unit rate that involves fractions. For example, how is it similar to find the rate in feet per minute if one bug travels at a rate of 20 feet in 2 minutes and a second bug travels at a rate of 3 3 _ 4 feet in 1 _ 4 minute?

Exit Problems

1) Simplify 254

15

. 3

2 or 1 1

2

2) Find the unit rate:

132 yards

7 minutes

13

yard per minute

3) Rita swam 10 12

laps in 1 12

minutes. How many laps per minute did Rita swim? 7 laps per minute