The missing number in a proportion
If 4 tickets cost $10, how much do 6 cost? If a recipe for 4 people needs 300 g of flour, how much for 6? These are proportions: three numbers are known, and the fourth follows from them. The calculator finds it. Type the three numbers in the sentence and the answer appears straight away. Many countries call this method the rule of three; in English it is usually taught as solving a proportion, or cross multiplication.
Direct proportion: more gives more
When one quantity doubles and the other doubles too, the proportion is direct. Write it as a fraction: 4 is to 10 as 6 is to x, so 4/10 = 6/x. Cross multiply: x = 6 × 10 ÷ 4 = 15. In plain words: find the value for one (10 ÷ 4 = 2.50 per ticket), then multiply by the number you want (2.50 × 6 = 15).
- Shopping: 4 bagels for $5, so 6 bagels cost 6 × 5 ÷ 4 = $7.50.
- Recipes: 300 g of flour for 4 people, so 300 × 6 ÷ 4 = 450 g for 6 people.
- Travel: 120 miles on 4 gallons, so 300 miles need 300 × 4 ÷ 120 = 10 gallons.
- Maps and scale: 1.5 inches on the map for 6 miles, so 4 inches stand for 4 × 6 ÷ 1.5 = 16 miles.
Inverse proportion: more gives less
Sometimes, when one quantity goes up, the other goes down. Three painters take 12 days to paint a house; how long would 4 painters take? More painters, fewer days. The product stays the same: 3 × 12 = 36 painter-days, so 4 painters need 36 ÷ 4 = 9 days. Tap “direct (more gives more)” in the sentence to switch to “inverse (more gives less)”, and the formula becomes first value × its result ÷ new value.
- Speed and time: a trip takes 3 hours at 40 mph; at 60 mph it takes 3 × 40 ÷ 60 = 2 hours.
- Sharing: a bag of snacks lasts 6 children 4 days; for 8 children, 6 × 4 ÷ 8 = 3 days.
- Machines: 5 printers finish a job in 8 hours; 4 printers need 5 × 8 ÷ 4 = 10 hours.
Which one is it?
Ask yourself: if the first quantity doubles, does the second double (direct) or halve (inverse)? Price and quantity, ingredients and servings, distance and fuel are direct. Workers and time, speed and time, people and how long supplies last are inverse. Some situations are neither: a bus ticket that costs the same for 1 or 5 stops, or a bulk price that drops per item. A proportion only works when the ratio really stays the same.
Where proportions come up
- Percentages. A percentage is a proportion with 100: if 40 out of 50 is x out of 100, x = 80. The percentage calculator handles all the usual cases.
- Unit prices. Comparing two pack sizes is a pair of proportions. The unit price calculator does both and tells you which is cheaper.
- Grades. Scaling a score to 100 (18 out of 24 is 75 out of 100) is a direct proportion; for an average of several marks, use the average calculator.
- Cooking and DIY. Scaling a recipe, mixing paint or concrete, converting a pattern size.
A tip for checking your answer: the result of a direct proportion should move in the same direction as the change you made (6 is more than 4, so the answer should be more than 10). In an inverse proportion, it should move the other way. If it does not, tap the kind of proportion to switch.
Decimals are fine in every box. The only thing the calculator refuses is a division by zero: the first value can’t be 0 in a direct proportion, nor the new value in an inverse one. You can also tap the microphone and say “4 gives 10, 6 gives what”. Your last numbers are kept in your browser, and Share gives a link that reopens the same proportion.