Thinking in proportion
A recipe for 12 scaled up for 30, the cheaper tub of yogurt, a price after a 20% rise, a map where a centimetre is half a kilometre: each is two quantities that grow together, so that doubling one doubles the other. Ratio, rate, percent and scale are four names for that one idea, and the same few moves solve all of them.
Why think in proportion?
Many comparisons are about multiplying, not adding. A 2 kg bag of rice costs $6 and a 5 kg bag costs $14. Subtracting the prices tells you nothing, because the bags are different sizes. Dividing each price by its mass does: $3.00 a kilogram against $2.80.
What proportional thinking really does is keep a relationship fixed while the amounts change. Equivalent ratios, unit rates, percents and scales are all the same move: find the number one quantity is multiplied by, and multiply the other by it too.
It is one of the oldest tools in science. Around 240 BCE the Greek scholar Eratosthenes learned that at noon on midsummer's day the Sun shone straight down a well at Syene, while at Alexandria, to the north, shadows fell at about 1/50 of a full circle. So the distance between the cities was about 1/50 of the way round the Earth, and 50 times that distance gave the size of the whole planet.
Where it turns up
- Maps — Canada's topographic maps are printed at scales such as 1 : 50 000, where 1 cm on the paper is 500 m on the ground
- Fuel use — Canadian vehicle fuel-consumption labels give a rate, litres per 100 km, so a car using 8 L/100 km needs about 24 L for a 300 km trip
- Model trains — HO-scale models are built at 1 : 87, so a real rail car 20 m long becomes a model about 23 cm long
- Grocery shelves — shelf tags often show a unit price, such as the cost per 100 g, so packages of different sizes can be compared at a glance
- 1. The words, first
- 2. Two quantities that grow together
- 3. Equivalent ratios and proportions
- 4. Unit rates
- 5. Percent is a ratio to 100
- 6. Scale drawings and maps
The words, first
The idea: Proportional is the idea under all the others: two quantities where one is always the same multiple of the other.
| Word | What it means |
|---|---|
| Ratio | A comparison of two amounts of the same kind, written 3 : 2 and read “3 to 2”. |
| Equivalent ratios, proportion | Ratios that make the same comparison, like 3 : 2, 6 : 4 and 30 : 20. A proportion says two ratios are equal: 3 : 4 = 21 : 28. |
| Proportional | Two quantities are proportional when one is always the same multiple of the other. Double one and the other doubles, and zero goes with zero. |
| Rate | A comparison of two different kinds of quantity, such as 45 km in 2.5 hours, or $3.25 for 650 g. |
| Unit rate | A rate for one of something: 18 km in one hour. Prices are often given per 100 g instead, which is easier to read. |
| Percent | A ratio to 100. 35% means 35 : 100, or 35/100. |
| Scale | The ratio of a length on a drawing or map to the real length it stands for. |
Two quantities that grow together
The idea: In a proportional relationship, multiplying one quantity by any number multiplies the other by the same number. A ratio table shows it at a glance.
Worked example. A pancake recipe uses 2 cups of flour for 12 pancakes. How much flour for 30 pancakes?
| Flour (cups) | 2 | 1 | 5 |
|---|---|---|---|
| Pancakes | 12 | 6 | 30 |
- Halve both: 1 cup makes 6 pancakes. That is the unit rate.
- Multiply both by 5: 5 cups make 30 pancakes.
So 5 cups. Adding does not work: 12 + 18 = 30 pancakes, but 2 + 18 = 20 cups of flour is absurd. Add the same amount to both numbers and the ratio changes; multiply both by the same number and it stays.
Not everything is proportional. Sam is 3 and his sister is 6, twice his age. When Sam is 30 she will be 33, not 60. A taxi charging $4 to start plus $2 a kilometre is not proportional either: 5 km costs 4 + 10 = $14, but 10 km costs 4 + 20 = $24, not double. The test: does zero go with zero, and does doubling one double the other?
Equivalent ratios and proportions
The idea: Multiply or divide both terms by the same number. To fill a gap in a proportion, find that number from the pair you know completely.
3 : 4 = x : 28 → 4 × 7 = 28 → x = 3 × 7 = 21
Worked example: three parts. A concrete mix is 1 part cement to 2 parts sand to 3 parts gravel. How much of each goes into 30 buckets of mix?
- Parts in one batch: 1 + 2 + 3 = 6.
- One part: 30 ÷ 6 = 5 buckets.
- Each: cement 1 × 5 = 5, sand 2 × 5 = 10, gravel 3 × 5 = 15 buckets. Check: 5 + 10 + 15 = 30, and 5, 10 and 15 are 1, 2 and 3 each multiplied by 5 ✓.
Part to part, or part to whole? Cement to gravel is 1 : 3, part to part. Cement to the whole mix is 1 : 6, so cement is 1/6 of it — not one third.
When the multiplier is not whole. In 4 : 6 = 10 : x, going from 4 to 10 multiplies by 2.5, so x = 6 × 2.5 = 15.
Unit rates
The idea: Divide to find how much for one. Once every rate is “per one”, comparing is just reading which is bigger.
Worked example: the better buy. A 650 g tub of yogurt costs $3.25. A 1.5 kg tub costs $6.90.
- Same units: 1.5 kg is 1500 g.
- Price per 100 g. The small tub is 6.5 lots of 100 g, and 3.25 ÷ 6.5 = 0.50, so $0.50 per 100 g. The large tub is 15 lots, and 6.90 ÷ 15 = 0.46, so $0.46 per 100 g.
- Compare: the large tub is 4 cents cheaper per 100 g.
Worked example: speed. A cyclist rides 45 km in 2.5 hours. Her unit rate is 45 ÷ 2.5 = 18 km/h. At that rate, 63 km takes 63 ÷ 18 = 3.5 hours.
Rates flip. 18 km in one hour is also 60 minutes for 18 km, about 3.3 minutes a kilometre. “How far?” wants kilometres per hour; “how long?” wants minutes per kilometre.
Percent is a ratio to 100
The idea: A percent fixes the second term of a ratio at 100. So every percent question is a proportion with one gap in it.
18 out of 24 → 18/24 = 75/100 = 75% · 130% of 60 = 1.3 × 60 = 78
Worked example: finding the whole. 35% of the students in a school walk to school, and that is 63 students. How many students are in the school?
- As a proportion: 35 : 100 = 63 : x.
- The multiplier: 63 ÷ 35 = 1.8.
- The whole: x = 100 × 1.8 = 180 students. Check: 0.35 × 180 = 63 ✓.
A percent change is a multiplier. A 20% increase multiplies by 1.2, and a 20% decrease by 0.8. Up 20% and then down 20% ends at 1.2 × 0.8 = 0.96 of the start: 4% lower, not back where it began.
Scale drawings and maps
The idea: A scale is a ratio of drawing length to real length. Put both in the same unit, and it is a proportion like any other.
Worked example: a floor plan. A plan is drawn at 1 : 50. A bedroom on it measures 8 cm by 6.4 cm. How big is the real room?
- Multiply each length by 50: 8 × 50 = 400 cm and 6.4 × 50 = 320 cm.
- In metres: the room is 4 m by 3.2 m.
Worked example: a map. On a 1 : 50 000 map, a lake is 7.5 cm long. On the ground that is 7.5 × 50 000 = 375 000 cm. There are 100 cm in a metre, so that is 3750 m, or 3.75 km.
Choosing a scale. A gym 36 m long, which is 3600 cm, has to fit on a page 25 cm wide. At 1 : 100 it would be 3600 ÷ 100 = 36 cm, too long. At 1 : 200 it is 3600 ÷ 200 = 18 cm, which fits. A scale can enlarge, too: an ant drawn at 10 : 1 shows 6 mm as 60 mm.