With the rule of three calculator, you can easily solve any rule of three problem online: simply enter three values and you’ll see the result, along with the step-by-step calculation and the formula.
The rule of three, also known as the three-term rule, is a method of calculation that allows you to determine a fourth value from three known values. You’ll need it when shopping, for recipes, when calculating wages, or when working with percentages. Here, you’ll find a simple explanation of the rule of three, complete with a formula, table and examples. And if you’re in a hurry, the rule of three calculator above will work out any problem with three values for you and help you work out any ratio.
The most important facts at a glance
- Three values known, one unknown: That is where the name ‘rule of three’ comes from. The calculation is carried out in three steps, the three ‘sätze’.
- Proportional: The more, the more. Formula: x = b ÷ a × c.
- Inversely proportional: The more, the less. Formula: x = a × b ÷ c.
- Tip: Always work out the figure for 1 piece, 1 hour or 1 % first. Then the rest is easy.
- Percentage: The rule of three is the basis of all percentage calculations. 100 % is the base value.
Quick guide to the rule of three calculator
Here’s how to use the online rule of three calculator to work out the rule of three. Simply enter your problem as a sentence, exactly as it appears in your workbook:
- Known ratio: For example, «If 12 apples cost CHF 3.60». Units such as ‘apples’ or ‘CHF’ are optional, but they make the calculation easier to follow. The same applies to the euro.
- Ratio to be calculated: «…then there are 8 apples…». The result appears straight away, without having to click «Calculate».
- Method of calculation: Choose ‘proportional’ (the more, the more) or ‘inversely proportional’ (the more, the less). More apples cost more, so it’s proportional.
- Calculation method: On the right, you can see the three equations as they appear in the exercise book, with the ÷ and × symbols and the formula. Under «Graph», you can see the problem represented as a straight line or a curve.
With «Copy working», you can copy the solution as text, for example for your homework. With «Share with classmates», you can send a link that opens your specific exercise. The templates ‘Shopping’, ‘Time & Work’, ‘Recipes’, ‘Percentages’, ‘Speed’ and ‘Salary’ show you typical examples.
Free of charge and without an app: The rule of three calculator runs directly in your browser, on your mobile and on your computer. No download, no registration, no adverts in the calculator.
What is a rule of three?
A rule of three is a mathematical calculation, in which you compare three known quantities to determine a fourth. The best-known application is calculating percentages.
There are two explanations for the name. Either it comes from the fact that three values are known, or from the fact that the calculation is carried out in three steps, as we learn at school:
- Known ratio: 12 apples cost CHF 3.60.
- Ratio to a unit: 1 apple costs CHF 0.30.
- Ratio to be calculated: 8 apples cost CHF 2.40.
How is the rule of three calculated?
The rule of three is always worked out in three steps. You set the known values in proportion, reduce the result to a single unit, and then multiply by the value you are looking for.
Example: If 12 apples cost CHF 3.60, how much do 8 apples cost?
- 12 apples cost CHF 3.60.
- Calculate the price per apple: CHF 3.60 ÷ 12 = CHF 0.30.
- Multiply by the quantity required: CHF 0.30 × 8 = CHF 2.40.
So 8 apples cost CHF 2.40.
Rule of three formula
If you summarise these three steps, you get the rule of three formula. a and b are the known ratio, c is the new value and x is the result:
Proportional: x = b ÷ a × c
Inversely proportional: x = a × b ÷ c
In this example, that means: x = 3.60 ÷ 12 × 8 = 2.40. The calculator shows you this formula with your figures in the calculation history.
Rule of three table
Many people learn the rule of three using a table. This allows you to see at a glance how both sides change. For the apples, it looks like this:
| Step | Apples | Price |
|---|---|---|
| 1st sentence | 12 | CHF 3.60 |
| 2nd sentence (÷ 12) | 1 | CHF 0.30 |
| 3rd set (× 8) | 8 | CHF 2.40 |
In a proportional problem, you carry out the same calculation on both the left and the right. In an inversely proportional problem, you do the opposite on the right: if you divide on the left, you multiply on the right.
Cross calculation (St Andrew’s Cross)
Instead of using three equations, you can also work out proportional problems using the cross-multiplication method. You write the values one below the other and multiply them in a cross-multiplication format: 3.60 × 8 = 28.80, then you divide by 12. Result: 2.40. This is the same calculation as the formula, just written differently.
The rule of three formulas and everyday examples
In theory, the rule of three seems more dry than it actually is. It quickly becomes clear when you look at some examples.
Formula proportional rule of three
‘Proportional’ means that the first value is in the same ratio to the second as the third is to the fourth. In mathematics, this is referred to as a proportional relationship. If one value doubles, the other also doubles.
Example: 3 kg of potatoes cost CHF 5. How much do 2 kg cost?
- 3 kg = CHF 5
- 1 kg = CHF 5 ÷ 3 = CHF 1.67
- 2 kg = CHF 5 ÷ 3 × 2 = CHF 3.33
When working on problems like these, carry on calculating using the unrounded intermediate value wherever possible, otherwise a rounding error will occur. The calculator does this automatically.
Formula antiproportional rule of three
Inverse proportionality is the opposite: the more of one, the less of the other. If one value doubles, the other is halved.
Example: Travelling at 50 km/h, you’ll reach your destination in 4 hours. How long will it take you at 100 km/h?
- 50 km/h = 4 hours
- 1 km/h = 4 × 50 = 200 hours
- 100 km/h = 200 ÷ 100 = 2 hours
A second example: 3 painters take 12 days to paint a house. 4 painters take 3 × 12 ÷ 4 = 9 days.
Example: Calculating a salary using the rule of three
Working 42 hours a week, you earn CHF 1,260 a week, and you’re reducing your hours to 35. Calculated proportionally: CHF 1,260 ÷ 42 × 35 = CHF 1,050. That’s CHF 30 per hour. You can see how much you’ll take home in the Gross-net calculator, your hourly wage of Hourly Wage Calculator.
Example: Converting a recipe
A recipe for 4 people requires 500 g of flour. For 6 people, you calculate 500 ÷ 4 × 6 = 750 g. This applies to all ingredients in the same proportions.
When do you need the rule of three?
You always need to use the rule of three when two quantities are in a fixed ratio and you want to convert one of the values. Typical examples:
- Shopping: How much do 350 g of strawberries cost if you know the price per kilo?
- Work and time: How much longer does it take for more people to do the same job?
- Wages: How much do you earn if you work more or fewer hours?
- Travel: How much petrol do you need for a longer journey? There’s also the Petrol cost calculator.
- Percentages: Discount, VAT or interest.
Important: The rule of three is only accurate if the ratio is truly fixed. If there is a volume discount or fixed costs such as a standing charge, it will give an incorrect result.
Rule of three in percentage calculation
The rule of three is particularly useful for calculating percentages. First, you work out 1 % and then multiply by the percentage you are looking for. 100 % is always the base value.
Example: A television costs CHF 800 and has been reduced by 30 %. How much does it cost now?
- 100 % = CHF 800
- 1 % = CHF 800 ÷ 100 = CHF 8
- 70 % = CHF 8 × 70 = CHF 560
Using the rule of three, you can work out all three figures in a percentage calculation:
| Wanted | Example | Result |
|---|---|---|
| Percentage | 30 % at CHF 800 | £240 |
| Percentage | CHF 240 out of CHF 800 | 30 % |
| Base value (reverse) | 70 % cost CHF 560 | CHF 800 |
In the rule of three calculator for percentages, select the «Percentages» template. For pure percentage problems, the Percentage calculator even faster, for the VAT on the VAT calculator.
Double rule of three
In a double or compound rule of three, two quantities change at the same time. You simply work it out as two rules of three in succession.
Example: 3 machines produce 120 parts in 4 hours. How many parts can 5 machines produce in 6 hours?
- Calculate for 1 machine and 1 hour: 120 ÷ 3 ÷ 4 = 10 parts.
- Extrapolate to 5 machines and 6 hours: 10 × 5 × 6 = 300 pieces.
In the rule of three calculator, you first enter the number of machines (3 machines = 120 parts, then 5 machines = 200 parts) and use the result to work out the hours (4 hours = 200 parts, then 6 hours = 300 parts).
The Rule of Three in Excel
If a, b and c are in cells A2, B2 and C2, the Excel formula for the proportional rule is =B2/A2*C2. For the inverse-proportional rule of three, you write =A2*B2/C2. This way, you can convert entire lists in one go.
You can find more calculators for everyday life and personal finances in the Computer Overview.
FAQ
How do you use the rule of three?
Solving a rule problem involves three steps: write down the known ratio, convert it to a single unit, and multiply by the value you are looking for. Example: 12 apples cost CHF 3.60, 1 apple costs CHF 0.30, 8 apples cost CHF 2.40.
What is the rule of three?
Proportional: x = b ÷ a × c. Inversely proportional: x = a × b ÷ c. Here, a and b are the known values and c is the new value.
How can I tell whether it is proportional or inversely proportional?
Ask yourself: Does one value increase when the other increases? If so, they are proportional (the more of one, the more of the other). If it decreases, they are inversely proportional (the more of one, the less of the other); for example, more workers, fewer days.
How do you calculate percentages using the rule of three?
100 % is the base value. First work out 1 % by dividing by 100, then multiply by the percentage you’re looking for. 30 % of CHF 800 is CHF 8 × 30 = CHF 240.
What is a double rule of three?
In the double rule of three, two quantities change at the same time, for example, the number of machines and the number of hours. You work it out by performing two rules of three in succession.
What is the English term for ‘rule of three’?
The rule of three is called «rule of three» in English. «Direct proportion» means proportional, and «inverse proportion» means anti-proportional.
Does the rule of three calculator cost anything?
No, the rule of three calculator is free of charge. It works in your browser on your mobile and computer, without the need for an app or registration.