Hong Life CalculatorsUnderstanding the proportion solver — concept and principle
A proportion is an equation stating that two ratios are equal, a : b = c : x (or a/b = c/x). If three of the four terms are known, the fourth is fixed. This is the classic "rule of three" used to scale recipes, convert map distances, convert currencies at a fixed rate and estimate costs at a constant unit price.
How it is calculated
The calculator solves for the fourth term with cross-multiplication:
x = b × c ÷ a
- a and b form the known ratio (for example 3 cups of flour for 4 people).
- c is the new quantity that corresponds to a (9 cups).
- x is the unknown that corresponds to b (how many people).
Keep each pair in matching units: a and c in the same unit, b and x in the same unit. The value a must not be zero.
The principle
If a/b = c/x, multiplying both sides by b·x gives a·x = b·c: in a true proportion, the product of the outer terms (extremes) equals the product of the inner terms (means). Dividing by a isolates x. Another way to see it: c is c/a times as large as a, so the matching term must be the same multiple of b, giving x = b × (c/a). The method works only when the two quantities are directly proportional, meaning that doubling one always doubles the other and both are zero together.
Worked example
- Inputs: a = 3, b = 4, c = 9. The proportion is 3 : 4 = 9 : x.
- Cross-multiply: 3 × x = 4 × 9 = 36.
- Divide by a: x = 36 ÷ 3.
- Result: x = 12.
- Check: 3/4 = 0.75 and 9/12 = 0.75, so the two ratios are equal.
| a | x |
|---|---|
| 0.75 | 48 |
| 1.5 | 24 |
| 2.25 | 16 |
| 3 | 12 |
| 3.75 | 9.6 |
| 4.5 | 8 |
| 6 | 6 |
How to read the result
x is the value that keeps the ratio unchanged. If c is larger than a, x will be larger than b by the same factor (here both grow three times). A quick sanity check is to compare the two quotients a/b and c/x; they must be equal. If the answer seems far off, the terms were probably entered in the wrong positions.
Common mistakes
- Placing the terms in the wrong order, for example writing 3 : 4 = x : 9, which answers a different question.
- Using the rule for inverse relationships. If more workers finish a job in less time, time is inversely proportional and x = a × b ÷ c, not b × c ÷ a.
- Mixing units within a pair, such as hours with minutes.
Frequently asked questions
What is the rule of three?
It is the traditional name for solving a proportion when three terms are known: multiply the two terms that sit diagonally and divide by the remaining one.
How can I tell whether a relationship is direct or inverse?
Ask what happens when one quantity doubles. If the other also doubles, it is direct. If the other halves, such as speed and travel time for a fixed distance, it is inverse.
How is a proportion used with percentages?
A percentage is a proportion with 100 as one term. To find 15% of 80, solve 15 : 100 = x : 80, which gives x = 12.
Sources: Euclid, Elements, Book V (definitions of ratio and proportion) and Book VI · Euclid, Elements, Book VII, Proposition 19 (product of extremes equals product of means for numbers in proportion)