Rule of Three and Cross Multiplication Calculator
Fill in three boxes of the table and the fourth is calculated instantly, with the cross multiplication shown in full. Any box can be the unknown, and inverse proportion is supported too.
- Instant result
- Step-by-step working
- Direct or inverse
- Free, no sign-up
100% private — Everything is processed directly in your browser: your files, voice, and image are never sent to Chatzam's servers.
How to solve a rule of three
-
Name your two quantities
For example "People" and "Flour (g)". It's optional, but it makes the result easier to read.
-
Fill in three boxes
Each row pairs two values that go together. Leave the box you're looking for empty: it's calculated right away.
-
Read the result and the steps
The result appears in the empty box and below the table, with the cross multiplication written out in full so you can redo it by hand.
A proportion calculator that shows its work
Any box can be the unknown
The empty box is the one that gets calculated. The × button on a box clears it so it can be calculated instead.
Detailed working
Equal cross products, isolating x, then the final calculation: every step is written out with your numbers.
Inverse proportion
For situations where one quantity goes down when the other goes up, like the number of workers and the length of a job.
Ready-made examples
Recipe, price per kilo, currency conversion, speed and distance, percentage, workers and days: one click fills in the table.
How to do a rule of three, step by step
Here's an example: 3 notebooks cost $7.50, so how much do 5 notebooks cost? Put the values in a table with one row per situation: 3 and 7.50 on the first row, 5 and x on the second. In a proportion table, the cross products are equal: 3 × x = 7.50 × 5. Isolate x by dividing by the number next to it: x = 7.50 × 5 ÷ 3 = 37.50 ÷ 3 = $12.50. You can also use the unit rate: one notebook costs 7.50 ÷ 3 = $2.50, so 5 notebooks cost 5 × 2.50 = $12.50. Both methods give the same answer; cross multiplication has the advantage of working wherever the unknown is placed.
How to tell if two quantities are proportional
The rule of three only works when the two quantities are directly proportional, meaning you always go from one to the other by multiplying by the same number, called the constant of proportionality. That's the case for the price of goods sold by weight, the quantities in a recipe depending on the number of guests, a currency conversion at a fixed rate, or the distance traveled at a constant speed. It's not the case when there's a fixed charge: with a $4 base fare and $1.50 per mile, a 10-mile ride costs $19, but a 20-mile ride costs $34, not $38. The same goes for age and height, or for tiered pricing.
Inverse proportion: when one goes up, the other goes down
Some quantities move in opposite directions: if 6 workers finish a job in 10 days, 4 workers will take longer. What stays constant is the product of the two values on the same row: 6 × 10 = 60 worker-days. With 4 workers, it therefore takes 60 ÷ 4 = 15 days. Another example: a trip takes 2 h at 90 km/h; at 120 km/h, it takes 90 × 2 ÷ 120 = 1.5 h, or 1 h 30 min. In the calculator, choose "Inverse": the steps then show the equal products of each row instead of the cross multiplication. This model assumes everything else stays the same, for example that every worker works at the same pace.
Frequently Asked Questions
What is the rule of three?
It's the method for finding a fourth value from three others when two quantities are proportional. If 2 lb cost $8, it lets you find the price of 5 lb: 8 × 5 ÷ 2 = $20.
What's the difference between the rule of three and cross multiplication?
They're two names for the same calculation. It's called cross multiplication because you multiply the values placed diagonally in the table, then divide by the third one.
How do I calculate a percentage with the rule of three?
Put the total opposite 100, then the part opposite x. For 45 points out of 60: x = 45 × 100 ÷ 60 = 75%. The "Percentage" example fills in this table for you.
How do I know if I should choose inverse proportion?
Ask yourself what happens if one of the values doubles. If the other doubles too, the proportion is direct. If it's cut in half, like the length of a trip when the speed doubles, it's inverse.
Can I use decimal numbers?
Yes, type them with a period or a comma: 6.30 and 6,30 are both accepted. Set the number of decimal places in the result with the "Decimals" list. Nothing is sent over the internet: the calculation runs in your browser.