Part 5: Proportional Reasoning
Two quantities have a proportional relationship if two pairs of the quantities have equivalent ratios. In other words, if one quantity doubles, the other quantity doubles (and their ratio remains constant). For example, a car with a fuel efficiency of 100 kilometres per 8 litres of gas (ratio 100 : 8) has the same fuel efficiency as a car that gets 200 kilometres per 16 litres of gas (ratio 200 : 16). The equivalent ratios can also be expressed as equivalent unit rates using fractions: 100/8 = 200/16 = 12.5.
Here are a couple of proportional relationship problems where we have to find how much one quantity changes in relation to a change in another quantity.
- Word problem: Jacob’s car goes 100 kilometres for every 8 litres of gas. If Jacob puts 44 litres of gas in his car, how far can he drive?
- 100 kilometres are proportional to 8 litres.
- Divide by 8: 100/8 = 12.5 kilometres are proportional to 8/8 = 1 litre.
- Multiply by 44: 44 x 12.5 = 550 kilometres are proportional to 44 x 1 = 44 litres.
- Jacob can drive 550 kilometres on 44 litres of gas.
- Check: unit rates are 100/8 = (100/8) / (8/8) = (44 x 100/8) / (44 x 8/8) = 550/44 = 12.5.
- Word problem: Karita runs 1 lap of the track in 1.5 minutes. If Karita runs at this pace for 30 minutes, how many laps of the track could she complete?
- 1 track lap is proportional to 1.5 minutes.
- Divide by 1.5 (equivalent to dividing by 3/2 or multiplying by 2/3): 2/3 track laps are proportional to 1.5/1.5 = 1 minute.
- Multiply by 30: 30 x 2/3 = 20 track laps are proportional to 30 minutes.
- Karita could complete 20 laps of the track in 30 minutes.
- Check: unit rates are 1/1.5 = (1/1.5) / (1.5/1.5) = (30 x 1/1.5) / (30 x 1.5/1.5) = 20/30 = 2/3.
Inversely proportional relationships
Two quantities have an inversely proportional relationship if two pairs of the quantities have equivalent products. In other words, if one quantity doubles, the other quantity halves (and their product remains constant). For example, if 1 person takes 6 hours to complete a task, then 2 people working together at the same rate take 6/2 = 3 hours to complete the task. The number of person-hours remains constant: 1 person working for 6 hours = 6 person-hours, while 2 people working for 3 hours = 6 person-hours.
Here are a couple of inversely proportional relationship problems where we have to find how much one quantity changes in relation to a change in another quantity.
- Word problem: Siblings Ruth, Theo, and Anil share a bedroom. If one sibling takes 4.5 hours to clean their room by themselves, how long does it take the three siblings to clean their room if they work together at the same rate?
- 1 person times 4.5 hours = 4.5 person-hours.
- Divide by 3: 4.5 person-hours divided by 3 people = 1.5 hours.
- It takes the three siblings 1.5 hours to clean their room if they work together.
- Check: 1 person times 4.5 hours = 4.5 person-hours, 3 people times 1.5 hours = 4.5 person hours.
- Word problem: Patrick, Amanda, and Rachel eat pizza at the same rate and can eat a whole pizza between the three of them in 20 minutes. If Patrick is taking a break from eating pizza, how long does it take Amanda and Rachel to eat a whole pizza between the two of them?
- 3 people times 20 minutes = 60 person-minutes.
- Divide by 2: 60 person-minutes divided by 2 people = 30 minutes.
- It takes Amanda and Rachel 30 minutes to eat a whole pizza between the two of them.
- Check: 3 people times 20 minutes = 60 person-minutes, 2 people times 30 minutes = 60 person-minutes.
The video below works through some examples of applying proportions.
Do the following exercises to practice applying proportions.