📏 Enter known values
Formula Reference
Worked example
We solve the formula for \(d\).
If we use the result to calculate Value A, we should get the value you entered:
Rule of three: the formula
The rule of three finds an unknown value when two quantities are directly proportional. If A corresponds to B, then C corresponds to D:
\[ \frac{A}{B} = \frac{C}{D} \quad\Rightarrow\quad D = \frac{B \times C}{A} \]
It is also called cross multiplication: the two products across the equals sign are equal (A × D = B × C). Solving for each value gives:
\[ A = \frac{B \times C}{D} \]
\[ B = \frac{A \times D}{C} \]
\[ C = \frac{A \times D}{B} \]
How to use the calculator
- Enter in A and B the two values you already know that go together (for example, 3 kg and $12).
- Enter in C the new value (5 kg).
- Leave D empty and press Calculate. You can leave any of the four fields empty instead, and the calculator finds that one.
Worked examples
1. Price: 3 kg cost $12. How much do 5 kg cost?
\[ D = \frac{12 \times 5}{3} = \frac{60}{3} = 20 \]
The 5 kg cost $20.
2. Recipe: 300 g of flour serve 4 people. How much for 6?
\( D = \frac{300 \times 6}{4} = 450 \) g of flour.
3. Distance: a car travels 150 km in 2 hours. How far does it go in 5 hours at the same speed?
\( D = \frac{150 \times 5}{2} = 375 \) km.
4. Percentage: if 80 is 100 %, what percentage is 20?
\( D = \frac{100 \times 20}{80} = 25 \) %. For this kind of problem there is also the percentage calculator.
5. Map scale: 2 cm on the map are 50 km in reality. How much are 7 cm?
\( D = \frac{50 \times 7}{2} = 175 \) km.
Direct or inverse rule of three?
Before calculating, check how the two quantities are related:
- Direct: when one goes up, the other goes up in the same proportion (more kilos, higher price). This is the case the calculator solves.
- Inverse: when one goes up, the other goes down (more workers, fewer days). Then A × B = C × D, and D = A × B ÷ C.
Inverse example: 4 workers finish a job in 6 days, so 8 workers need \( D = \frac{4 \times 6}{8} = 3 \) days. To solve it with this calculator, swap the second row: A = 8, B = 6 and C = 4.
Proportion table: flour by number of people
With 300 g for 4 people, each person takes 75 g:
| People | Flour |
|---|---|
| 1 | 75 g |
| 2 | 150 g |
| 3 | 225 g |
| 4 | 300 g |
| 5 | 375 g |
| 6 | 450 g |
| 8 | 600 g |
| 10 | 750 g |
| 12 | 900 g |
| 20 | 1,500 g |
When Do You Need the Rule of Three?
Knowing the price of one amount, the rule of three gives the price of any other: 3 kg for $12 means 5 kg for $20.
Find the better deal in secondsA recipe for 4 people can be adapted to 6 or 10 by scaling every ingredient in the same proportion.
Cook for any number of guestsWith the rate of one currency, the rule of three converts any amount: if 1 USD is 0.92 EUR, 250 USD is 230 EUR.
Quick conversions when travelling or shopping onlineAt a constant speed or consumption, distance, time and fuel are proportional: 150 km in 2 hours means 375 km in 5 hours.
Plan trips and fuel stopsThe scale of a map or a floor plan is a proportion: 2 cm for 50 km means 7 cm for 175 km.
Turn drawing measurements into real onesProportion problems appear from primary school to university entrance exams, and percentages are a rule of three with 100.
The most used tool in school mathsCommon Mistakes
⚠️ Using It When the Relation Is Inverse
⚠️ Dividing by the Wrong Value
⚠️ Mixing Up the Rows
⚠️ Mixing Units
⚠️ Applying It Without Proportionality
⚠️ Rounding Too Early
Applications by Industry
Retail & Shopping
- Unit prices: Price per kilo or per litre to compare packages
- Discounts: How much is saved on any amount
- Stock: Units needed for a given number of days
Cooking & Catering
- Recipes: Scaling ingredients to the number of servings
- Costs: Cost per portion from the cost of the whole batch
- Yield: Raw weight needed for a cooked weight
Construction
- Materials: Paint, tiles or cement for a given area
- Mixes: Keeping the proportions of concrete and mortar
- Plans: Real lengths from the scale of a plan
Finance
- Currency: Converting amounts with the exchange rate
- Taxes: Tax on any amount from a known rate
- Salaries: Pay for partial months or hours worked
Health
- Dosage training: Proportion exercises in nursing and pharmacy courses
- Nutrition: Calories or protein for a different serving size
- Hydration: Water intake by body weight
Education
- Proportions: The standard method for proportion problems
- Percentages: Every percentage is a rule of three with 100
- Science: Unit conversions and concentrations in chemistry