Ratio Calculator

A ratio compares quantities. Written 3:2, it says that for every three of the first thing there are two of the second — and nothing about how many there are in total.

That last point is the source of most confusion with ratios. 3:2 describes a relationship that holds at 3 and 2, at 30 and 20, and at 300 and 200. Only the proportion is fixed.

Ratios against fractions

A ratio of 3:1 is usually a part-to-part comparison: three of one thing for each one of another. As a part-to-whole fraction, the first thing is 3/4 of the total, because the total is four parts.

Reading 3:1 as "three quarters" or as "three times" both work, and they are different statements. Confusing them is the most common ratio error, and it appears wherever mixes are involved: a 3:1 paint-to-thinner mix is 75% paint, not 300%.

A ratio can also be part-to-whole, and it is worth checking which is meant. "One in four" is a part-to-whole statement equal to 1:3 as a part-to-part ratio.

Dividing an amount in a ratio

The standard procedure, in three steps.

Divide ₹60,000 between three partners in the ratio 3:2:1.

Add the parts. 3 + 2 + 1 = 6.

Find one part. 60,000 ÷ 6 = 10,000.

Multiply out. 30,000, 20,000 and 10,000.

Check by adding: ₹60,000. The check is worth doing every time, because it catches both arithmetic slips and a misread ratio.

The same method handles any number of terms, and it is what sits behind profit sharing in a partnership, dividing an inheritance in stated shares, and splitting a bill by consumption rather than equally.

Scaling: multiply, never add

To scale a ratio, multiply every term by the same number. 2:3 scaled by 4 is 8:12. It is not 6:7, which is what adding 4 to each term would give — and which is a completely different proportion.

This matters in the kitchen and on site. A recipe for four scaled to six is multiplied by 1.5, every ingredient. A concrete mix specified as 1:2:4 — cement, sand, aggregate — holds at any batch size as long as every part is multiplied by the same factor.

The one place addition does appear is in comparing before and after. Adding one part of water to a 1:3 mix changes it to 1:4, and that is a deliberate change to the ratio rather than a scaling of it.

Reducing to simplest form

24:36 and 2:3 are the same ratio. Divide both terms by their greatest common divisor, 12.

Simplest form makes ratios comparable at a glance, which is the point of writing them down. It also reveals when two apparently different ratios are the same: 15:25 and 21:35 are both 3:5.

Where three or more terms are involved, the GCD is taken across all of them. 12:18:30 reduces by 6 to 2:3:5.

Proportions and the cross-multiplication rule

A proportion says two ratios are equal, and it is the workhorse of everyday arithmetic:

a/b = c/d, therefore a × d = b × c.

If 5 kg of rice costs ₹350, what do 12 kg cost?

5/350 = 12/x, so x = (350 × 12) ÷ 5 = ₹840.

This is direct proportion: one goes up, the other goes up. Inverse proportion is the other pattern, where the product rather than the quotient is constant. If 6 workers take 12 days, then 8 workers take (6 × 12) ÷ 8 = 9 days, because the total work is fixed.

Distinguishing the two is the whole difficulty. Ask whether doubling the first quantity should double the second or halve it.

Ratios in the world

Aspect ratios. A 16:9 screen is 16 units wide for every 9 tall. A 1920-pixel width therefore implies 1920 × 9 ÷ 16 = 1080. Older televisions were 4:3; cinema uses 2.39:1.

Gear ratios. A 3:1 gear turns the output once for every three input turns, trading speed for torque.

Map scales. 1:50,000 means one unit on the map is 50,000 in reality, so 1 cm is 500 m.

Concrete. M20 grade is nominally 1:1.5:3 by volume — cement, sand, coarse aggregate. Nominal mixes are specified this way in IS codes for small works, with design mixes used above that.

Shareholding. A company owned 60:30:10 divides both control and dividends in those proportions, which is why the ratio matters more than the number of shares.

Financial ratios. A current ratio of 2:1 means current assets are twice current liabilities. A debt-to-equity of 1:1 means the company is financed half by borrowing.

Where ratios mislead

No absolute size. A ratio of 3:1 says nothing about whether the quantities are three and one or three million and one million. "Twice as many complaints" is not informative without the counts.

Order matters. 3:2 and 2:3 are different ratios, and labelling which term is which prevents a whole class of error. A cement-to-sand ratio written backwards produces very weak concrete.

Units must match. Comparing 500 grams to 2 kilograms is 1:4, not 500:2. Convert to a common unit before writing the ratio down.

Percentages of different bases. Two ratios of percentages cannot be combined without knowing the bases, in exactly the way that averages of percentages cannot.

The golden ratio, briefly

The ratio 1:1.618, usually written φ, has the property that the ratio of the whole to the larger part equals the ratio of the larger part to the smaller.

It appears genuinely in mathematics — it is the limit of the ratio between consecutive Fibonacci numbers — and its claimed appearances in art, architecture and the human body are mostly retrofitted. The Parthenon and the Mona Lisa are cited constantly and fit only if you choose the measuring points generously.

It is a real and interesting number. It is not a design rule, and a layout is not improved by conforming to it.

Three-term ratios and continued proportion

Ratios extend to any number of terms, and the arithmetic is the same: every term scales together.

Combining two-term ratios into one is the operation that takes care. If A:B is 2:3 and B:C is 4:5, the two cannot simply be written as 2:3:5 — the value of B differs between them. Scale each until B matches: multiply the first by 4 and the second by 3, giving 8:12 and 12:15, so A:B:C is 8:12:15.

This is the standard method for combining shareholdings, mixing three-component recipes, and any chain of comparisons where the middle term is shared.

A continued proportion is the special case where the ratio between successive terms is constant — 2:4:8:16, each term double the last. That is a geometric sequence, and it is the same structure that compound interest produces.

Rates are ratios with units

A rate is a ratio between quantities measured in different units, and it behaves identically.

Speed is distance to time. Fuel efficiency is distance to volume. Price per kilogram, interest per year, rupees per dollar — all ratios, all scalable in the same way.

The reason this matters is that unit conversion is ratio arithmetic. To convert 60 km/h to metres per second, multiply by the ratio 1000 m per km and by 1 hour per 3600 s: 60 × 1000 ÷ 3600 = 16.67 m/s. Writing the units alongside the numbers and cancelling them is the reliable way to check any conversion, because a conversion done the wrong way round leaves the wrong units behind.

Unit rates — the value at 1 — are what make comparison possible. Two packets at ₹85 for 400 g and ₹120 for 600 g are ₹212.50 and ₹200 per kilogram, and only the unit rate makes that visible.

Equivalent ratios

Two ratios are equivalent when one is the other with both sides multiplied, or both divided, by the same number. They describe the same proportion at a different size.

a:b is equivalent to ka:kb for any k other than zero. Reducing instead divides both sides by their greatest common divisor.

Working backwards is the same operation. 16:40 divided through by 8 is 2:5, so 16:40 is equivalent to every ratio in the first three rows.

To test whether two ratios are equivalent, cross-multiply. 2:5 and 6:15 are equivalent because 2 × 15 and 5 × 6 are both 30; 2:5 and 3:6 are not, because 2 × 6 is 12 and 5 × 3 is 15. Reducing both ratios to lowest terms reaches the same verdict from the other direction.

The simplify mode above does both halves: it reduces the ratio you enter, then scales the reduced form by any factor, which is the quickest way to turn a ratio into the quantities a recipe, a mix or a dilution actually needs.

What this calculator assumes

• Ratios are part-to-part unless you are dividing a total, in which case the parts are summed first.
• Terms are reduced to simplest form using the greatest common divisor across all of them.
• Division of an amount distributes any rounding remainder to the largest share, so the parts always sum to the original total.
• Terms must be positive; a zero term makes the ratio undefined rather than infinite.
• Units are yours to keep consistent — the calculator sees numbers, not kilograms and grams.