In-depth
Proportions Calculator: How to Solve Any Proportion
Proportion is one of the most widely used mathematical tools in everyday life: from cooking recipes to reading maps, from scales in technical drawings to currency conversion problems. This calculator allows you to find the unknown term of any proportion instantly.
What is a proportion
A proportion is an equality between two ratios. It is written in the form:
a : b = c : d
and reads "a is to b as c is to d." The terms have specific names:
- a and d are the extremes
- b and c are the means
The fundamental property
The golden rule of proportions is:
The product of the means equals the product of the extremes.
In formula: a × d = b × c
This property allows you to find any unknown term when the other three are known.
How to find the unknown term
Given the proportion a : b = c : x, the formulas for finding the missing term are:
| Unknown term | Formula | Example (2:5 = 6:x) |
|---|---|---|
| x = d (fourth term) | x = (b × c) ÷ a | x = (5 × 6) ÷ 2 = 15 |
| x = c (third term) | x = (a × d) ÷ b | — |
| x = b (second term) | x = (a × d) ÷ c | — |
| x = a (first term) | x = (b × c) ÷ d | — |
Practical examples
Cooking recipes
A recipe for 4 people requires 300 g of flour. How much flour is needed for 6 people?
Proportion: 4 : 300 = 6 : x
Solution: x = (300 × 6) ÷ 4 = 450 g
Scales and maps
A map has a scale of 1:50,000. If two cities are 3 cm apart on the map, what is the real distance?
Proportion: 1 : 50,000 = 3 : x
Solution: x = 3 × 50,000 = 150,000 cm = 1.5 km
Currency conversion
If 1 euro equals 1.08 dollars, how much are 250 euros worth?
Proportion: 1 : 1.08 = 250 : x
Solution: x = 1.08 × 250 = $270
Percentages as proportions
35% of 200 can be expressed as a proportion: 35 : 100 = x : 200
Solution: x = (35 × 200) ÷ 100 = 70
Other properties of proportions
Besides the fundamental property, proportions have other useful properties:
- Inversion: if a : b = c : d, then b : a = d : c
- Alternation: if a : b = c : d, then a : c = b : d
- Composition: if a : b = c : d, then (a + b) : b = (c + d) : d
- Division: if a : b = c : d, then (a − b) : b = (c − d) : d
Direct and inverse proportion
In a direct proportion, the quantities grow together: the more hours I work, the more I earn. In an inverse proportion, one grows while the other decreases: the more workers there are, the less time is needed. In the latter case, the formula changes:
a × b = c × d (instead of a × d = b × c)
Example: 3 workers complete a job in 12 days. How many days would it take 4 workers?
3 × 12 = 4 × x → x = 36 ÷ 4 = 9 days
How to use the calculator
Enter the three known terms of the proportion and leave the unknown term field empty. Press "Calculate" and the calculator will automatically find the missing value by applying the fundamental property of proportions. You can enter whole numbers, decimals, or fractions.
