Rule of three (proportion) calculator
Solve simple and compound proportions, including direct and inverse proportions, with the formula.
Directly or inversely proportional
A compound proportion problem involves three or more quantities. Before setting up the calculation, you need to understand how each quantity relates to the one you’re trying to find:
- Directly proportional: When one quantity increases, the other increases in the same proportion. More machines mean more products made.
- Inversely proportional: When one quantity increases, the other decreases in the same proportion. More workers mean fewer days to finish the job.
The key is to compare each quantity with the unknown separately, assuming the other quantities stay constant.
The arrow method in 4 steps
- Create a table with one column for each quantity and two rows (the known situation and the new situation). Put x in the column for the unknown.
- Draw an arrow in the x column (by convention, point it down).
- For each other quantity, ask: “If this increases, does x increase or decrease?” If it increases, draw an arrow in the same direction (direct proportion); if it decreases, draw an arrow in the opposite direction (inverse proportion).
- Flip the fractions for quantities that are inversely proportional, then set the ratio for x equal to the product of the other ratios.
Example 1: workers, hours, and days
Six workers, working 8 hours a day, build a wall in 10 days. How many days will it take 4 workers, working 6 hours a day, to build the same wall?
| Workers | Hours per day | Days |
|---|---|---|
| 6 | 8 | 10 |
| 4 | 6 | x |
- More workers → fewer days: inverse proportion.
- More hours per day → fewer days: inverse proportion.
Flip both fractions: 10/x = (4/6) × (6/8) = 24/48 = 1/2. So, x = 20 days.
Check that the answer makes sense: the total work in the original situation is 6 × 8 × 10 = 480 worker-hours. In the new situation, 4 × 6 = 24 worker-hours per day; 480 ÷ 24 = 20 days.
Example 2: faucets and tanks
Three identical faucets fill 2 tanks in 6 hours. How long will it take 4 faucets to fill 5 tanks?
| Faucets | Tanks | Hours |
|---|---|---|
| 3 | 2 | 6 |
| 4 | 5 | x |
- More faucets → fewer hours: inverse proportion (flip to 4/3).
- More tanks → more hours: direct proportion (keep 2/5).
6/x = (4/3) × (2/5) = 8/15 → x = 6 × 15 ÷ 8 = 90 ÷ 8 = 11.25 hours, or 11 hours and 15 minutes (0.25 h × 60 = 15 min).
Example 3: machines, hours, and parts
Five machines produce 400 parts in 4 hours. How many parts will 8 identical machines produce in 6 hours?
- More machines → more parts: direct proportion.
- More hours → more parts: direct proportion.
400/x = (5/8) × (4/6) = 20/48 → x = 400 × 48 ÷ 20 = 960 parts.
Production increased from 400 to 960 parts, an increase of 560 ÷ 400 = 1.4, or 140%. To check percentage changes like this, use the Percentage Calculator.
During a test, set up the table and arrows on paper. To check your answer at home, the Rule of Three Calculator solves simple and compound problems involving direct or inverse proportion and shows the formula.
The same approach works with four quantities
A print shop with 2 printers running 5 hours a day prints 3,000 flyers in 3 days. How many days will it take 3 printers running 4 hours a day to print 7,200 flyers?
- More printers → fewer days: inverse proportion (use 3/2).
- More hours per day → fewer days: inverse proportion (use 4/5).
- More flyers → more days: direct proportion (use 3,000/7,200).
3/x = (3/2) × (4/5) × (3,000/7,200) = 1.2 × 0.41666... = 0.5 → x = 6 days.
Quick check: Calculate the output per unit of “work.” In the original situation, 2 × 5 × 3 = 30 printer-hours produced 3,000 flyers, or 100 flyers per printer-hour. In the new situation, 7,200 flyers require 72 printer-hours. With 3 × 4 = 12 printer-hours per day, that’s 72 ÷ 12 = 6 days. If both methods give the same answer, you’ve got it right.
Common mistakes
- Comparing quantities with each other instead of comparing each one with x.
- Forgetting to flip the fraction for an inversely proportional quantity.
- Mixing units: hours and minutes, meters and centimeters. Convert them before setting up the table.
- Not checking the answer: if fewer workers finish faster, you may have reversed an arrow.
Updated October 2026.
Frequently asked questions
How do you tell whether a proportion is direct or inverse?
Compare each quantity with the unknown, assuming the others stay constant: if both increase together, it’s direct proportion; if one increases while the other decreases, it’s inverse proportion.
What’s the difference between a simple and a compound proportion problem?
A simple proportion problem involves two quantities; a compound one involves three or more. The same direct-or-inverse reasoning applies to each quantity in relation to the unknown.
How do you convert 11.25 hours to hours and minutes?
The whole number is the number of hours, and the decimal part multiplied by 60 gives the minutes: 0.25 × 60 = 15. So, 11.25 hours is 11 hours and 15 minutes.