Rule of Three Calculator & Ratio Calculator
Use the rule of three calculator for direct or inverse proportions, solve proportions such as 3 : 4 = 12 : x by cross-multiplying, and simplify or scale ratios. See the calculation steps for each. Free, with no sign-up, right in your browser.
Built by Luca Stanek
- 3 calculation types
- With step-by-step calculations
- Share the result via a link
- Runs in the browser
- Proportion
- A · D = B · C
- Direct rule of three
- y₂ = y₁ ÷ x₁ · x₂
- Inverse rule of three
- y₂ = y₁ · x₁ ÷ x₂
Planning more than calculations? Launch your own website, app or learning environment with a VPS.
Proportion
Leave the unknown field blank. If all four fields are filled, the calculator checks whether the equation holds.
Find
Result
Missing value D
16
- Complete equation
- 3 : 4 = 12 : 16
- Factor C ÷ A
- × 4
- First ratio simplified
- 3 : 4
- As a decimal (A ÷ B)
- 0.75
The missing term is 16: 3 : 4 = 12 : 16.
Calculation steps (4)
- 1Set up: 3 : 4 = 12 : x, so 3 ÷ 4 = 12 ÷ x
- 2Cross-multiply: 3 · x = 4 · 12 = 48
- 3Solve for x: x = 48 ÷ 3 = 16
- 4Check: 3 · 16 = 4 · 12 = 48
Examples with step-by-step solutions
The rule of three and ratios, explained with examples
Four common problems, solved using the same calculator as above. Enter your own numbers above.
Direct rule of three
3 items cost 6 euros. How much do 5 items cost?
10 Euros
From 3 Pieces → 6 Euros, it follows that 5 Pieces → 10 Euros.
y₂ = y₁ ÷ x₁ · x₂
- 1.Known: 3 Pieces → 6 Euros
- 2.Find the value for 1 (divide): 1 Pieces → 6 ÷ 3 = 2 Euros
- 3.Find the value for the new quantity (multiply): 5 Pieces → 2 · 5 = 10 Euros
Inverse rule of three
3 workers need 12 days. How long do 4 workers need?
9 Days
From 3 Workers → 12 Days, it follows that 4 Workers → 9 Days.
y₂ = y₁ · x₁ ÷ x₂
- 1.Known: 3 Workers → 12 Days
- 2.Find the value for 1 (multiply): 1 Workers → 12 · 3 = 36 Days
- 3.Find the value for the new quantity (divide): 4 Workers → 36 ÷ 4 = 9 Days
Proportion
If a 16 : 9 image is 1920 pixels wide, how tall is it?
1,080
The missing term is 1,080: 16 : 9 = 1,920 : 1,080.
A · D = B · C
- 1.Set up: 16 : 9 = 1,920 : x, so 16 ÷ 9 = 1,920 ÷ x
- 2.Cross-multiply: 16 · x = 9 · 1,920 = 17,280
- 3.Solve for x: x = 17,280 ÷ 16 = 1,080
- 4.Check: 16 · 1,080 = 9 · 1,920 = 17,280
Simplify a ratio
36 boys, 48 girls: What is the ratio in simplest form?
3 : 4
36 : 48 simplifies to 3 : 4.
A : B = (A ÷ GCD) : (B ÷ GCD)
- 1.Greatest common divisor: gcd(36, 48) = 12
- 2.Divide both values by 12: 36 ÷ 12 = 3 and 48 ÷ 12 = 4
Quick overview
Proportions and ratios explained simply
When to use direct or inverse proportion, how to cross-multiply, and how to calculate a ratio.
Direct proportion: the more, the more
Double the quantity, double the value. First divide by the known quantity, then multiply by the new quantity: if 4 people need 500 g, 1 person needs 125 g and 6 people need 750 g.
Inverse proportion: the more, the less
Inverse proportion works the other way around: multiply first, then divide. If 3 workers need 12 days, 1 worker needs 36 days and 4 workers need 9 days.
Cross-multiply
In a proportion A : B = C : D, the product of the outer terms equals the product of the inner terms: A · D = B · C. This lets you find the missing term, whether it is A, B, C, or D.
Calculate and simplify a ratio
Divide both terms by their greatest common divisor (GCD). The calculator also shows the ratio as 1 : n, as n : 1, and as percentages of the whole.
Everyday examples
The calculator includes examples for a 16:9 aspect ratio, a 1:25,000 map scale, a 1:4 mixture, recipes, consumption and travel time. Click an example to fill in the fields.
What the calculator doesn't do
It handles simple rules of three. For a compound rule of three, solve two simple ones in sequence. Three-term ratios and splitting an amount according to a ratio are not supported.
Features
What the Rule of Three Calculator can do
A tool for ratios, proportions and the rule of three, for school, training, cooking and everyday life.
Three calculators in one
Solve or check proportions, simplify and scale ratios, and calculate direct and inverse proportions using the rule of three.
Step-by-step solutions, just like in class
Every step is shown: cross-multiplication, GCD and calculating the unit value. Handy for checking your work and learning.
Check proportion
When all four fields are filled in, the calculator compares A · D with B · C and tells you whether the equation holds.
Parts, 1 : n, and n : 1
When you simplify a ratio, you'll also see it as 1 : n, as n : 1, and as percentages of the whole.
Decimal comma or point, copy, and share
Both 12,5 and 12.5 work. Use “Copy result” and “Share link” to share the result and calculation steps.
Free and browser-based
No sign-up or installation needed: calculations run in your browser. Your numbers appear in the URL so a shared link opens your exact problem.
How it works
How to use the Rule of Three Calculator
No sign-up or installation needed. Everything runs in your browser.
Step 1
Choose calculation type
Choose Solve a proportion, Simplify & scale, or Rule of three. For the rule of three, choose direct or inverse proportion.
Step 2
Enter numbers
For a proportion, leave the unknown field blank. With four numbers, the calculator checks the equation. Click an example to fill in the fields.
Step 3
Read result
The result appears instantly with a step-by-step solution. Copy it or share the link.
Discover more
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FAQ
Frequently asked questions about Rule of Three Calculator
Quick answers about the rule of three, ratios and this calculator.
The rule of three solves problems with three known values in three steps: write down what you know (3 notebooks cost 6 euros), calculate the cost of one unit (1 notebook costs 2 euros), then calculate the cost of the quantity you need (5 notebooks cost 10 euros). The calculator shows these exact steps.
For direct proportion, “the more, the more” applies: y₂ = y₁ ÷ x₁ · x₂; twice the quantity means twice the price. For inverse proportion, “the more, the less” applies: y₂ = y₁ · x₁ ÷ x₂; twice as many workers take half the time. Here, x₁ is the known quantity, y₁ its value, and x₂ the new quantity. In “Rule of three” mode, choose the type and the calculator will show the corresponding formula.
Cross-multiply: the product of the outer terms equals the product of the inner terms, so 3 · x = 4 · 12 = 48. Then divide by 3 to get x = 16. In the calculator, simply leave the unknown field blank, whether you're looking for A, B, C, or D.
Yes. Fill in all four fields or select “Check only” under “Find”. The calculator multiplies the outer terms A · D and the inner terms B · C and shows whether the proportion holds.
Divide both terms by their greatest common divisor (GCD): for 1920 : 1080, the GCD is 120, giving 16 : 9. In “Simplify & scale” mode, you'll also see each term as a percentage of the whole—25 % and 75 % for a 1 : 3 mixture—as well as the ratio as 1 : n and n : 1.
Divide the target value by the corresponding term to get the factor, then multiply the other term by it. To scale 16 : 9 to a width of 1280 pixels, the factor is 1280 ÷ 16 = 80, and the height is 9 · 80 = 720. Enter a target value in “Simplify & scale” mode to do this.
Percentage calculations are a special case of the rule of three, with a base value of 100%. A percentage calculator is quicker for finding a percentage value, rate or base value. Use the rule of three when any two quantities are related, such as quantity and price or workers and days. Go to percentage calculator
Decimals, yes: the calculator accepts both 12,5 and 12.5 and recognizes thousands separators. When simplifying, it calculates exactly with up to 10 decimal places; beyond that, it rounds and flags the result as approximate. Enter fractions as decimals, such as 0,75 for 3/4. Negative numbers are allowed, but no term in a ratio can be 0 because you cannot divide by 0.
Any more questions?
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