What is a Proportion? – Proportion Definition
A proportion is a relationship between two quantities. It displays what portion of one part is contained in the whole. The result is typically expressed as a fraction, a colon ratio, a decimal, or a percent.
Just as a fraction is made up of two parts — a numerator and a denominator — so is a proportion. Suppose a cake is cut into 12 slices and 5 have been eaten. You have 12 − 5 = 7 slices remaining, so the proportion is 7/12, or equivalently 7:12.
The proportion calculator solves equations of the form A / B = C / D (direct proportion) or A × B = C × D (inverse proportion). Enter any three values and leave the unknown field blank — the calculator finds the missing value instantly.
Constant of Proportionality – Directly and Inversely Proportional
The constant of proportionality connects the ratios or products of two quantities. It is also called the coefficient of proportionality.
- Direct proportion: if y increases as x increases at a constant rate, the constant is k = y / x. The relationship is written y = kx.
- Inverse proportion: if y decreases as x increases so that their product is always the same, the constant is k = x × y. The relationship is written y = k / x.
Example — speed: speed (v) is the ratio of distance (d) to time (t): v = d / t. Here v is the constant of proportionality between d and t. If you travel 120 km in 2 hours, then v = 120 / 2 = 60 km/h. This is a direct proportional relationship.
How to Solve Proportions – Proportion Examples
To solve a proportion A / B = C / D, use cross-multiplication:
A × D = B × C
Then isolate the unknown value:
- Solve for A: A = (B × C) / D
- Solve for B: B = (A × D) / C
- Solve for C: C = (A × D) / B
- Solve for D: D = (B × C) / A
Example: A recipe calls for 3 cups of flour for every 4 servings. How much flour is needed for 9 servings?
3 / 4 = C / 9 → C = (3 × 9) / 4 = 6.75 cups
The Golden Ratio
The golden ratio (φ ≈ 1.618) is perhaps the most famous proportion in nature and art. It is defined by the proportion:
(A + B) / A = A / B = φ ≈ 1.618
This proportion appears in the spiral of nautilus shells, the arrangement of sunflower seeds, Renaissance paintings, and modern logo design. You can verify the golden ratio with this calculator by entering A and B such that A / B = (A + B) / A.
Law of Multiple Proportions
In chemistry, the law of multiple proportions (Dalton's law) states: when two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other are in simple whole-number proportions.
Example: Carbon and oxygen form two compounds — CO and CO₂. The oxygen masses that combine with 12 g of carbon are 16 g and 32 g, giving the ratio 16:32 = 1:2. This simple whole-number proportion confirms the atomic theory.
Proportional Tax (Flat Tax) Definition
A proportional tax (or flat tax) applies the same percentage rate to all taxpayers, regardless of income level. If the tax rate is 15%, a person earning $40,000 pays $6,000 and a person earning $200,000 pays $30,000. Their tax burdens grow in direct proportion to their incomes.
Using this calculator: A / B = C / D where A = tax paid by person 1, B = income of person 1, C = tax paid by person 2, D = income of person 2. In a true flat tax, A/B = C/D at all income levels.
Proportion in US Customary and Metric Measurements
Proportions are essential for unit conversion. This calculator supports:
- US Imperial / Customary: inches, feet, yards, miles; ounces, pounds, short tons; fluid ounces, cups, pints, quarts, gallons.
- Metric (SI): millimeters, centimeters, meters, kilometers; grams, kilograms, metric tons; milliliters, liters.
Example — recipe scaling: A cookie recipe uses 250 g of butter for 24 cookies. How much butter for 36 cookies? 250 / 24 = B / 36 → B = 375 g.
Currency Proportions
Proportions appear every time you convert between currencies or calculate a split cost. This calculator supports 20 world currencies including USD, RUB (Russian Ruble), EUR, GBP, JPY, CAD, AUD, CHF, CNY, INR and more.
Example: If 100 USD equals 9,200 RUB, how many RUB equal 250 USD? 100 / 9,200 = 250 / D → D = (9,200 × 250) / 100 = 23,000 RUB.