Ratios and proportions, without the confusion
How to read a ratio, simplify it, compare part-to-part and part-to-whole, and use proportions to scale recipes, maps and prices.
A ratio compares two quantities. A proportion says two ratios are equal. That is the whole framework. The tricky part is knowing what is being compared and what the numbers represent.
Reading a ratio
A ratio of 3:2 can mean:
- for every 3 of the first thing, there are 2 of the second;
- the first quantity is 3/2 times the second;
- the quantities are in a 3-to-2 relationship.
It does not necessarily mean there are exactly 3 and 2 items. The actual amounts could be 30 and 20, or 300 and 200.
Simplify a ratio by dividing both sides by a common factor:
- 12:8 = 3:2
- 45:15 = 3:1
Part-to-part vs part-to-whole
This distinction causes many mistakes.
If a class has 12 boys and 8 girls:
- The ratio of boys to girls is 12:8 = 3:2 (part-to-part).
- The fraction of students who are boys is 12/20 = 3/5 (part-to-whole).
- The fraction who are girls is 8/20 = 2/5.
The ratio 3:2 does not mean 3/2 of the class is boys. It means for every 3 boys there are 2 girls.
Rates and unit rates
A rate is a ratio with different units, such as miles per hour, dollars per pound or words per minute.
A unit rate expresses the rate for one unit of the second quantity. If 5 pounds of apples cost $12.50:
- Unit price = 12.50 ÷ 5 = $2.50 per pound.
Unit rates make comparisons easy. Which is cheaper: 12 oz for $3.60 or 18 oz for $5.40?
- $3.60 ÷ 12 = $0.30 per oz.
- $5.40 ÷ 18 = $0.30 per oz.
They are the same unit price. The package size and convenience may still matter.
Solving a proportion
A proportion has the form:
a / b = c / d
If you know three values, you can solve for the fourth. Cross-multiply:
a × d = b × c
Example: if 3 tickets cost $27, what do 7 tickets cost?
- 3 / 27 = 7 / x
- 3x = 189
- x = 63
Seven tickets cost $63, assuming the same price per ticket.
Cross-multiplication is a shortcut for multiplying both sides by the denominators. It works, but understand the setup first.
Scaling a recipe
A recipe for 4 servings calls for 2 1/2 cups of flour. You need 10 servings.
The scale factor is 10 ÷ 4 = 2.5.
- Flour = 2.5 × 2.5 = 6.25 cups.
Not every ingredient scales linearly. Salt, leavening and strong spices may need adjustment. Cooking times and pan sizes do not simply double. But the arithmetic gives a starting point.
Maps and drawings
A map scale of 1:50,000 means 1 unit on the map represents 50,000 of the same units on the ground.
- 4 cm on the map = 4 × 50,000 = 200,000 cm = 2 km.
Keep units consistent. Converting to the same unit before multiplying prevents errors.
Ratios in finance and mixtures
Ratios appear in:
- debt-to-income ratios;
- price-to-earnings ratios;
- mixing paint, concrete or fuel;
- odds and probabilities;
- aspect ratios for screens and photos.
An aspect ratio of 16:9 means for every 16 units of width there are 9 units of height. A 1920 × 1080 image has that ratio because 1920 ÷ 1080 ≈ 16 ÷ 9.
Common mistakes
- Treating a part-to-part ratio as a fraction. Use the total when you need part-to-whole.
- Adding ratios directly. 1:2 plus 1:3 is not 2:5. Convert to common terms or solve the underlying quantities.
- Mixing units. Convert before comparing.
- Assuming proportionality. Not every relationship is linear. Bulk discounts, taxes and fixed fees break simple proportions.
- Forgetting the scale factor. Write it down before multiplying every ingredient.
The short version
A ratio compares. A proportion sets two ratios equal. Simplify, keep units consistent and identify whether you have part-to-part or part-to-whole. Then the arithmetic is usually straightforward.
For the fraction mechanics, read fractions explained.