Solving for the missing term

A proportion says two ratios are equal, with one value unknown:

a : b = c : x, so x = b × c ÷ a

The reliable way to do this without errors is cross-multiplication. If 3 : 8 = 12 : x, then 3x = 96 and x = 32. The trap is mixing up which side each number belongs to — always write the ratio with units attached, so 3 kg : 8 litres makes it obvious that the unknown is in litres.

Simplifying: divide by the greatest common divisor

To reduce a ratio to lowest form, divide every term by the GCD of all of them. The classic example is a screen resolution:

1920 : 1080. The GCD is 120, so dividing gives 16 : 9. Same shape, smallest whole numbers.

Note the third row. Ultrawide is sold as 21 : 9 but 2560 : 1080 simplifies to 64 : 27, which is 2.370 rather than 2.333. The marketing name is a rounded approximation, not the true ratio — worth knowing if you are calculating exact crop dimensions.

To find a missing dimension at a fixed aspect ratio, use the same proportion. A 16 : 9 image 2,560 pixels wide is 2,560 × 9 ÷ 16 = 1,440 pixels tall.

The mistake that ruins mixes: part-to-part versus part-to-whole

This is the single biggest source of ratio errors, and it costs real money on building sites.

A ratio of 2 : 3 does not mean two thirds and one third. It means 2 parts and 3 parts, so 5 parts total — two fifths and three fifths. Reading 2 : 3 as 2/3 rather than 2/5 is off by 67%.

Worked example with three terms. A standard concrete mix is 1 : 2 : 4 cement to sand to gravel, which is 7 parts total. For 0.7 cubic metres of mix, each part is 0.1 m³:

• Cement: 1 part = 0.1 m³
• Sand: 2 parts = 0.2 m³
• Gravel: 4 parts = 0.4 m³

Someone treating the 1 as one quarter — because there are three ingredients — would order 0.175 m³ of cement, 25% too much, and the mix would be wrong as well as wasteful.

Scaling recipes and quantities

Scaling is multiplication by a single factor applied to every term. A bread dough of 500 g flour to 325 g water is a 2 : 1.3 ratio, or 65% hydration. To make 800 g of flour instead, the factor is 800 ÷ 500 = 1.6, so water becomes 325 × 1.6 = 520 g. Every ingredient gets the same 1.6, or the ratio breaks.

Two honest cautions about scaling in the kitchen and the workshop:

• Not everything scales linearly. Cooking time, pan surface area and leavening do not. Doubling a cake batter does not double the baking time, and salt and spice often need less than a full doubling.
• Ratios by weight and by volume are different. A cup of flour and a cup of water do not weigh the same. Baking ratios are quoted by weight for exactly this reason.

Ratio, proportion, rate and percentage

Related ideas that are easy to conflate:

• A ratio compares quantities of the same kind: 3 : 2 boys to girls.
• A rate compares different kinds: 60 km per hour. Units survive in the answer.
• A proportion is a statement that two ratios are equal, which is what you solve.
• A percentage is a ratio with the whole fixed at 100. The ratio 3 : 2 is 60% to 40% of the total, or 150% as a comparison of first to second — which is why percentages need their base stated explicitly.

Questions people actually ask

Can a ratio have decimals?

Yes, and it often should. A 1 : 1.618 ratio is exact enough for design work, and forcing it to whole numbers as 1000 : 1618 gains nothing. Whole numbers matter when you are counting discrete items.

Does the order matter?

Always. 3 : 2 and 2 : 3 are different ratios, and a ratio written without labels is an invitation to error. Write cement : sand, not just 1 : 2.

How do I split an amount in a given ratio?

Add the parts, divide the amount by that total, then multiply by each part. Splitting 4,500 in the ratio 4 : 5 gives 9 parts of 500, so 2,000 and 2,500.

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