Lesson 1: An Experience in Relationships as Measuring Rate

What is a Ratio?
A ratio is a way to compare two numbers.
Example: A class has 8 boys and 12 girls.
The ratio of boys to girls is written as 8:12.
This means: “For every 8 boys, there are 12 girls.”

What is a Rate?
A rate is a ratio that compares two different types of things.
Example: A car goes 100 miles in 2 hours.
The rate is “100 miles for 2 hours.”

What is a Unit Rate?
A unit rate tells us how much happens in just 1 unit of the other quantity.
From the car example: 100 miles ÷ 2 hours = 50.
This means the car goes 50 miles in 1 hour.

Practice questions:

1. A car travels 420 miles in 7 hours. What is the speed in miles per hour?
Step 1: Write the comparison → 420 miles to 7 hours = ratio 420:7.
Step 2: We want to know how many miles happen in 1 hour.
Step 3: Divide both numbers by 7 (because the second number is 7, and we want it to become 1).
420 ÷ 7 = 60
7 ÷ 7 = 1
Step 4: New ratio = 60:1.
Answer: The car goes 60 miles in 1 hour (60 mph).

2. A store served 360 customers in 30 days. How many customers per day?
Step 1: Ratio = 360 customers to 30 days → 360:30.
Step 2: We want “per day,” so the second number (denominator, 30) should become 1.
Step 3: Divide both numbers by 30.
360 ÷ 30 = 12
30 ÷ 30 = 1
Step 4: New ratio = 12:1.
Answer: The store serves 12 customers per day.

3. A runner completes 40 meters in 16 seconds. What is the speed per second?
Step 1: Ratio = 40 meters to 16 seconds → 40:16.
Step 2: We want “per second,” so the second number should be 1.
Step 3: Divide both numbers by 16.
40 ÷ 16 = 2.5
16 ÷ 16 = 1
Step 4: New ratio = 2.5:1.
Answer: The runner goes 2.5 meters per second.

4. A shopper buys 5 pounds of fruit for $7.96. What is the cost per pound?
Step 1: Ratio = $7.96 to 5 pounds → 7.96:5.
Step 2: We want “per pound,” so the second number should be 1.
Step 3: Divide both numbers by 5.
7.96 ÷ 5 = 1.59
5 ÷ 5 = 1
Step 4: New ratio = 1.59:1.
Answer: The fruit costs $1.59 per pound.

5. The ratio of right-handed to left-handed students is 18:4. Write three equivalent ratios.
Start with 18:4. Equivalent ratios are made by multiplying or dividing both numbers by the same value.

First equivalent ratio (18:4, multiply by 2):
18 × 2 = 36
4 × 2 = 8
Ratio = 36:8

Second equivalent ratio (18:4, multiply by 3):
18 × 3 = 54
4 × 3 = 12
Ratio = 54:12

Third equivalent ratio (18:4, divide by 2):
18 ÷ 2 = 9
4 ÷ 2 = 2
Ratio = 9:2

Answer: Equivalent ratios are 36:8, 54:12, and 9:2.

6. Mr. Rowley has 16 homework papers and 14 exit tickets. Ms. Rivera has 64 homework papers and 60 exit tickets. Are their ratios the same?
Mr. Rowley: Ratio = 16:14.
Both numbers can be divided by 2.
16 ÷ 2 = 8
14 ÷ 2 = 7
Simplified ratio = 8:7

Ms. Rivera: Ratio = 64:60.
Both numbers can be divided by 4.
64 ÷ 4 = 16
60 ÷ 4 = 15
Simplified ratio = 16:15

Compare: 8:7 is not the same as 16:15.
Answer: The ratios are not equivalent.

7. Jonathan can play 3 hours of video games for every 5 hours of homework. Lucas can play 30 minutes for every 1 hour of homework. Who gets more?
Jonathan: Ratio = 3:5.
Divide both by 5.
3 ÷ 5 = 0.6
5 ÷ 5 = 1
Ratio = 0.6:1 → 0.6 hours = 36 minutes.

Lucas: Ratio = 30 minutes:1 hour.
That is 0.5 hours per 1 hour homework = 30 minutes.

Compare: Jonathan gets 36 minutes, Lucas gets 30 minutes.
Answer: Jonathan gets more game time.

8. At Euclid Middle School, 30 girls tried out for lacrosse and 12 were selected. 40 boys tried out and 16 were selected. Do boys and girls have the same ratio of selection?
Girls: Ratio = 12:30.
Both numbers can be divided by 6.
12 ÷ 6 = 2
30 ÷ 6 = 5
Simplified = 2:5

Boys: Ratio = 16:40.
Both numbers can be divided by 8.
16 ÷ 8 = 2
40 ÷ 8 = 5
Simplified = 2:5

Compare: Both are 2:5.
Answer: Ratios are the same.

9. Devon buys a 6-pack of drinks for $2.99. His sister says each drink costs “just over $2.” Is she correct?
Ratio = 2.99:6.
Divide to find per 1 drink.
2.99 ÷ 6 ≈ 0.50
Each drink is 50 cents.
Answer: His sister was wrong. Each drink costs $0.50.

10. Lizzie’s school buys 350 agenda books for $1,137.50. The school wants $1,500 profit. What is the least they can charge per book?
Step 1: Cost per book = 1137.50 ÷ 350 = 3.25.
Step 2: Profit per book = 1500 ÷ 350 = 4.29.
Step 3: Total = 3.25 + 4.29 = 7.54.
Answer: At least $7.54 per book.

Part C. Extra Practice Problems

11. A runner jogs 12 miles in 2 hours. What is the speed?
12 ÷ 2 = 6
Answer: 6 miles per hour

12. A factory makes 240 toys in 8 hours. How many toys per hour?
240 ÷ 8 = 30
Answer: 30 toys per hour

13. A car travels 180 miles on 6 gallons of gas. How many miles per gallon?
180 ÷ 6 = 30
Answer: 30 miles per gallon

14. 9 apples cost $4.50. What is the cost of 1 apple?
4.50 ÷ 9 = 0.50
Answer: $0.50 per apple

15. Simplify the ratio 20:25.
Both numbers can be divided by 5.
20 ÷ 5 = 4
25 ÷ 5 = 5
Answer: 4:5

16. Write two equivalent ratios to 15:10.
Multiply both by 2 → 30:20
Multiply both by 3 → 45:30
Answer: 30:20 and 45:30

17. There are 50 students, 20 boys and 30 girls. What is the ratio of boys to girls?
20:30 → divide both by 10 → 2:3
Answer: 2:3

18. Six pens cost $9. What is the price of one pen?
9 ÷ 6 = 1.50
Answer: $1.50 per pen

19. A bike goes 24 miles in 2 hours. What is the speed?
24 ÷ 2 = 12
Answer: 12 miles per hour

20. Which is better: 30 pencils for $9 or 20 pencils for $6?
30 ÷ 9 = 3.33 pencils per dollar
20 ÷ 6 = 3.33 pencils per dollar
Answer: Both are the same value

21. A baker makes 90 cupcakes in 3 hours. How many cupcakes per hour?
90 ÷ 3 = 30
Answer: 30 cupcakes per hour

22. A swimmer swims 100 meters in 50 seconds. What is the speed?
100 ÷ 50 = 2
Answer: 2 meters per second

23. Simplify the ratio 45:60.
Both numbers can be divided by 15.
45 ÷ 15 = 3
60 ÷ 15 = 4
Answer: 3:4

24. Class A has 12 boys and 8 girls. Class B has 18 boys and 12 girls. Are the ratios the same?
Class A: 12:8 = divide by 4 → 3:2
Class B: 18:12 = divide by 6 → 3:2
Answer: Yes, both are 3:2

25. A movie ticket costs $48 for 4 people. What is the price per person?
48 ÷ 4 = 12
Answer: $12 per person

26. Four pizzas feed 12 people. How many people per pizza?
12 ÷ 4 = 3
Answer: 3 people per pizza

27. Write three equivalent ratios to 9:6.
Multiply both by 2 → 18:12
Multiply both by 3 → 27:18
Divide both by 3 → 3:2
Answer: 18:12, 27:18, 3:2

28. A store sells 15 bottles of juice for $18. What is the cost per bottle?
18 ÷ 15 = 1.20
Answer: $1.20 per bottle

29. A plane flies 600 miles in 3 hours. What is the speed?
600 ÷ 3 = 200
Answer: 200 miles per hour

30. A marathon is 26 miles. If someone runs it in 4 hours, what is the speed per hour?
26 ÷ 4 = 6.5
Answer: 6.5 miles per hour