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How to use this calculator#
- Choose the mode that matches the questionSimplify reduces a ratio to lowest terms. Proportion finds a missing fourth term from three known ones. Share divides a total into ratio parts. They are three different questions with three different inputs.
- Keep the terms in a consistent orderIf you write cement:sand, keep cement first everywhere. Swapping the order halfway through a proportion inverts the answer, and a 2:3 mix silently becomes 3:2 — a 50% error in the first ingredient.
- Convert both terms to the same unit firstA ratio of 500 g to 2 kg is not 500:2. Convert to 500:2000, which simplifies to 1:4. Mixed units are the single biggest source of wrong ratios in recipes, mixes and map scales.
- Check that the shares add back to the totalAfter splitting a total, sum the shares. If they do not return the original figure you have divided by one term instead of the sum of all terms — the most common ratio error there is.
The formula#
Simplify, solve, and share
Simplify: a:b → (a ÷ g) : (b ÷ g) where g = GCD(a, b) • Proportion: a:b = c:x ⟹ x = (b × c) ÷ a • Share: each share = Total × (its term ÷ sum of all terms)
- a
- First term of the ratio (the antecedent)
- b
- Second term of the ratio (the consequent)
- c
- The known term on the other side of a proportion
- x
- The unknown term being solved for
- g
- Greatest common divisor of all the terms
- Total
- The amount being divided up in Share mode
A ratio with decimals is still valid but scales to whole numbers by multiplying every term by the same factor: 1.5:2 becomes 3:4 when both are doubled. Multiplying or dividing all terms by the same number leaves a ratio unchanged; adding or subtracting the same number does not.
Simplifying a ratio#
Divide both sides of a ratio by their greatest common divisor. For 24:36 the GCD is 12, so the ratio reduces to 2:3 — the same relationship written in the smallest whole numbers. You can also express it as 1:1.5 by dividing both terms by the first, which is handy for comparing mixes. The mistake to avoid is subtracting the same amount from both sides, which changes the ratio entirely.
Solving a proportion for a missing term#
Two equal ratios form a proportion, A:B = C:D, and cross-multiplying gives A×D = B×C. So if 3:5 = 12:x, then x = (5 × 12) ÷ 3 = 20. This single rule drives recipe scaling, map scales, unit pricing and model dimensions. Keep the units in matching positions on both sides — mixing cups-to-grams on one side with grams-to-cups on the other silently inverts your answer.
Splitting a total by a ratio#
To share a total in a given ratio, add the ratio terms to find the number of parts, divide the total by that number, then multiply each term by the part value. Splitting $500 in the ratio 3:2 gives 5 parts worth $100 each, so the shares are $300 and $200. People often divide by one term instead of the sum, producing amounts that do not add back to the total.
Worked examples#
Splitting profit three ways
£4,200 of profit shared between three partners in the ratio 5:3:2.
- Total parts = 5 + 3 + 2 = 10
- One part = 4,200 ÷ 10 = 420
- Shares = 5 × 420 = 2,100; 3 × 420 = 1,260; 2 × 420 = 840
- Check: 2,100 + 1,260 + 840 = 4,200 ✓
£2,100, £1,260 and £840 — which is 50%, 30% and 20% of the total.
Scaling a recipe with a proportion
A recipe uses 250 g of flour for 4 servings. You need 7 servings.
- Set up the proportion 4:250 = 7:x
- Cross-multiply: 4 × x = 250 × 7 = 1,750
- x = 1,750 ÷ 4 = 437.5
- The scale factor is 7 ÷ 4 = 1.75, so apply it to everything else too
437.5 g of flour. A 300 ml liquid becomes 300 × 1.75 = 525 ml, and a 20-minute bake does not scale at all.
Reference tables#
| Ratio | First share | Second share | As percentages |
|---|---|---|---|
| 1:1 | 1/2 | 1/2 | 50% / 50% |
| 1:2 | 1/3 | 2/3 | 33.33% / 66.67% |
| 1:3 | 1/4 | 3/4 | 25% / 75% |
| 1:4 | 1/5 | 4/5 | 20% / 80% |
| 2:3 | 2/5 | 3/5 | 40% / 60% |
| 3:2 | 3/5 | 2/5 | 60% / 40% |
| 3:4 | 3/7 | 4/7 | 42.86% / 57.14% |
| 2:5 | 2/7 | 5/7 | 28.57% / 71.43% |
| 5:3 | 5/8 | 3/8 | 62.5% / 37.5% |
| 1:9 | 1/10 | 9/10 | 10% / 90% |
| 1:19 | 1/20 | 19/20 | 5% / 95% |
Read 1:3 as one part to three parts — a quarter of the total, not a third. Misreading a part-to-part ratio as a fraction is the commonest ratio mistake in print.
| Dilution | Concentrate as % of mix | ml per litre of mix | fl oz per US gallon of mix |
|---|---|---|---|
| 1:4 | 20% | 200 | 25.6 |
| 1:8 | 11.11% | 111 | 14.2 |
| 1:10 | 9.09% | 91 | 11.6 |
| 1:16 | 5.88% | 59 | 7.5 |
| 1:20 | 4.76% | 48 | 6.1 |
| 1:32 | 3.03% | 30 | 3.9 |
| 1:64 | 1.54% | 15 | 2.0 |
| 1:128 | 0.78% | 8 | 1.0 |
Cleaning-product labels often use a looser convention where 1:32 means one ounce of product into a full 32 ounces of water, giving 4 fl oz per gallon rather than 3.9. The difference is small at high dilutions and significant at low ones — check which convention the label means before mixing.
Common mistakes#
- Adding or subtracting the same amount from both termsTaking 1 off each side of 3:2 gives 2:1, a completely different relationship — 60:40 becomes 67:33. Only multiplication and division by the same number preserve a ratio, which is why simplifying uses the GCD.
- Reading a part-to-part ratio as a fractionA 1:3 mix is one part in four, or 25% — not one third. A 40:60 split of a class means 40% and 60%, but a 40:60 ratio written as 2:3 is the same thing. Decide which frame you are in before quoting a number.
- Dividing the total by one term instead of the sumSplitting $500 in 3:2 by dividing 500 by 3 gives $166.67 a part, and the two shares then come to $833.33 — more than you started with. Always divide by the sum of the terms.
- Inverting the units in a proportionIf the left side is servings:grams, the right side must be servings:grams too. Writing 4:250 = x:7 instead of 4:250 = 7:x solves for the wrong quantity and returns 0.112 instead of 437.5.
Frequently asked questions#
How do I simplify a ratio?
Divide every term by the greatest common divisor of all the terms. 24:36 both divide by 12, giving 2:3.
How do I convert a ratio to a percentage?
Divide one term by the sum of all terms and multiply by 100. In 3:2, the first share is 3 ÷ 5 × 100 = 60%.
Can a ratio contain decimals?
Yes. A ratio like 1.5:2 is valid, and it scales up to the equivalent whole-number ratio 3:4 by multiplying both terms by two.
What is the difference between a ratio and a fraction?
A ratio compares two parts to each other (3:2), while a fraction compares one part to the whole (3/5). The same data, framed differently.
Key terms#
- Antecedent
- The first term of a ratio — the 3 in 3:2.
- Consequent
- The second term of a ratio — the 2 in 3:2.
- Proportion
- A statement that two ratios are equal, such as 3:5 = 12:20. Solved by cross-multiplication.
- Part-to-whole
- A comparison of one share against the total rather than against the other share. 3:2 is part-to-part; 3/5 is the same information part-to-whole.
- Scale factor
- The number every term is multiplied by to move between two equivalent ratios — 1.75 when scaling a 4-serving recipe to 7.
- Unit rate
- A ratio rewritten with 1 as its second term, such as 62.5 g per serving. The form that makes prices and speeds comparable.
- Dilution ratio
- Concentrate to solvent, usually written 1:N. The concentrate is 1/(N + 1) of the finished mixture, not 1/N.
Sources#
- Prealgebra 2e — ratios and rate — OpenStax, Rice University
- Pre-algebra: ratios and rates — Khan Academy
- Elementary Algebra 2e — solving proportions — OpenStax, Rice University
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