Lesson 2: Proportional Relationships
- What does proportional mean?
Two quantities are proportional when they always change in the same way.
This means: If one doubles, the other doubles too. If one is cut in half, the other is cut in half. - How do we check if two things are proportional?
We check the ratio between them.
- If the ratio is always the same, then they are proportional.
- Another way: divide one quantity by the other. If the result is always the same number, the relationship is proportional.
- Constant of Proportionality
This is the number that shows how the two quantities are related.
Example: If cost = 2 × weight, then “2” is the constant of proportionality.
Part B.
1. Isabelle’s family buys frozen yogurt. The cost is based on the weight.
Here are the results:
- 12.5 ounces costs $5
- 10 ounces costs $4
- 5 ounces costs $2
- 8 ounces costs $3.20
Question: Is the cost proportional to the weight?
Step 1: Find the cost per ounce for each example.
12.5 oz → 5 ÷ 12.5 = 0.40 per ounce
10 oz → 4 ÷ 10 = 0.40 per ounce
5 oz → 2 ÷ 5 = 0.40 per ounce
8 oz → 3.20 ÷ 8 = 0.40 per ounce
Step 2: All the ratios are equal (0.40 per ounce).
Answer: Yes, cost is proportional to weight. The constant of proportionality is $0.40 per ounce.
2. A recipe book gives this conversion: 8 ounces = 1 cup, 16 ounces = 2 cups, 4 ounces = 0.5 cup.
Question: Are ounces proportional to cups?
Step 1: Compare ounces to cups.
8 ÷ 1 = 8
16 ÷ 2 = 8
4 ÷ 0.5 = 8
Step 2: Each time, the ratio is 8. That means 8 ounces = 1 cup always.
Answer: Yes, ounces are proportional to cups. The constant is 8 ounces per cup.
3. Jose burns calories by jumping rope.
Here is the chart:
- 1 minute → 11 calories
- 2 minutes → 22 calories
- 3 minutes → 33 calories
- 4 minutes → 44 calories
a) Is the number of calories proportional to the time?
Check ratios:
11 ÷ 1 = 11
22 ÷ 2 = 11
33 ÷ 3 = 11
44 ÷ 4 = 11
All equal → Yes, it is proportional.
b) If Jose jumps for 6.5 minutes, how many calories will he burn?
6.5 × 11 = 71.5 calories.
Answer: Yes, calories are proportional to time. He will burn about 71.5 calories in 6.5 minutes.
4. Alex works in the summer. In 4 weeks, he earns $112.
a) If his earnings are proportional to the number of weeks, how much will he earn in 8 weeks?
Step 1: Find unit rate → 112 ÷ 4 = 28.
Step 2: Each week he earns $28.
Step 3: In 8 weeks → 28 × 8 = 224.
Answer: He will earn $224 in 8 weeks.
b) Are his total earnings proportional to the number of weeks worked?
Yes, because each week he earns the same amount, $28. The ratio of money to weeks is always 28:1.
Part C. Extra Practice Problems
5. A juice blend is mixed in a ratio of 3 cups cranberry to 5 cups apple. Complete the table.
Cranberry: 3, 6, 9, 12
Apple: 5, 10, 15, 20
Answer: Yes, these are proportional because the ratio cranberry ÷ apple is always 3 ÷ 5 = 0.6.
6. John fills a bathtub that is 18 inches deep. In 2 minutes, the water is 3 inches deep. Will it take 10 more minutes to reach 18 inches?
Step 1: Find rate → 3 inches ÷ 2 minutes = 1.5 inches per minute.
Step 2: To reach 18 inches: 18 ÷ 1.5 = 12 minutes.
Step 3: He already used 2 minutes, so needs 10 more minutes.
Answer: Yes, he is correct.
7. A car travels 50 km in 2 hours. Is distance proportional to time?
50 ÷ 2 = 25 km per hour.
If the car keeps the same speed, every time doubled gives distance doubled.
Answer: Yes, proportional with constant 25 km/hour.
8. A printer prints 60 pages in 4 minutes. How many pages in 10 minutes? Is this proportional?
60 ÷ 4 = 15 pages per minute.
In 10 minutes: 15 × 10 = 150 pages.
Answer: Yes, proportional.
9. A worker earns $72 for 6 hours. How much in 9 hours?
72 ÷ 6 = 12 dollars per hour.
9 × 12 = 108.
Answer: $108, proportional.
10. A school has 120 students and 4 teachers. If they keep the same ratio, how many teachers for 300 students?
Ratio = 120:4 = 30:1.
For 300 students → 300 ÷ 30 = 10.
Answer: 10 teachers.
11. A car uses 12 liters of gas for 96 km. How many km per liter?
96 ÷ 12 = 8.
Answer: 8 km per liter.
12. A recipe uses 2 cups flour for 3 cups sugar. How much flour for 12 cups sugar?
Ratio = 2:3.
Multiply both by 4 → 8:12.
Answer: 8 cups flour.
13. A bike rental costs $24 for 3 hours. What is the cost for 7 hours?
24 ÷ 3 = 8 per hour.
7 × 8 = 56.
Answer: $56.
14. A shop sells 15 pens for $4.50. How many pens for $9?
4.50 → 15 pens.
9 → double, so 30 pens.
Answer: 30 pens.
15. 5 sandwiches cost $20. What is the cost per sandwich?
20 ÷ 5 = 4.
Answer: $4 each.
16. A runner runs 18 km in 2 hours. How far in 5 hours?
18 ÷ 2 = 9 km/hour.
5 × 9 = 45.
Answer: 45 km.
17. A factory makes 300 toys in 5 hours. How many toys in 8 hours?
300 ÷ 5 = 60 toys/hour.
60 × 8 = 480.
Answer: 480 toys.
18. A person saves $150 in 3 months. How much in 12 months?
150 ÷ 3 = 50 per month.
12 × 50 = 600.
Answer: $600.
19. A recipe calls for 4 eggs for 6 cups of flour. How many eggs for 15 cups of flour?
Ratio = 4:6.
Simplify to 2:3.
For 15 cups → 15 ÷ 3 = 5.
2 × 5 = 10 eggs.
Answer: 10 eggs.
20. A car travels 270 miles in 6 hours. What is the constant of proportionality (unit rate)?
270 ÷ 6 = 45.
Answer: 45 miles per hour.