Ratio and Proportion Calculator

What this calculator answers

Four ratio questions, each with exact answers and the steps written out with your numbers:

• Simplify a ratio: the smallest whole numbers in the same proportion, even when the terms are decimals, fractions or mixed numbers, with up to six terms. You also get the total number of parts, each term’s fraction and percentage of the whole, and the 1 : n and n : 1 forms.
• Solve a proportion: the missing term of A : B = C : D. Fill in all four terms instead and it checks whether the two ratios are equal.
• Divide a total in a ratio: each share of an amount, such as $1,000 split 2 : 3 : 4, rounded to the cent if you like, with the rounded shares still adding up to the total.
• Scale a ratio: every term multiplied by a factor, or scaled so one term reaches a value you choose.

How to use it

• Calculation: pick one of the four. Each opens on its own example.
• Ratio (simplify, divide and scale): type the terms with colons between them, like 3:4, 2:3:5 or 1.5 : 2.25 : 0.75. Decimals, fractions (1/2 : 3/4), mixed numbers (1 1/2 : 2) and repeating decimals written with parentheses (0.(3) for 0.333…) all work, and so does 3 to 4. You can paste a ratio copied from elsewhere. When a term is written another way, such as 1 1/2 or 3 to 4, the line under the box shows how it was read. If you prefer a box for each term, choose One box per term and use Add term; the terms move across when you switch.
• A, B, C and D (solve): the proportion A : B = C : D. Leave the term you want empty, or type x or ? in it. To solve for a term without clearing its box, pick it in Solve for. Fill in all four to check the proportion.
• Total to divide and Round the shares (divide): the amount to split, and whether to keep the exact shares, round them to the cent or round them to whole numbers.
• Scale by (scale): a factor such as 3, 2.5 or 1/4, or a new value for one term, counted from the left.

Results update as you type. The Try buttons simplify 1/2 : 3/4, solve a 1 : 25,000 map scale, check whether 4 : 6 = 10 : 15, split $1,000 in the ratio 2 : 3 : 4, and scale 16 : 9 to a height of 1,080. To compare two ratios, press Save for comparison, change the inputs and save again.

How to simplify a ratio

Divide every term by their greatest common divisor (GCD), the largest whole number that divides all of them. The result is the same ratio in the smallest whole numbers.

$$a \mathbin{:} b \mathbin{:} c = \dfrac{a}{g} \mathbin{:} \dfrac{b}{g} \mathbin{:} \dfrac{c}{g}, \qquad g = \gcd(a, b, c)$$

• $a$, $b$ and $c$ are the terms as whole numbers (the same rule works for two terms or six).
• $g$ is their greatest common divisor. When $g = 1$, the ratio is already in its simplest form.

Terms that are decimals or fractions are first turned into whole numbers by multiplying every term by the same number, as the next sections show.

Worked example: a potting mix of 1.5 : 2.25 : 0.75

A potting mix uses 1.5 buckets of compost, 2.25 buckets of garden soil and 0.75 bucket of perlite. What is the ratio in whole numbers, and how much of each goes into a 30-liter planter?

1. Clear the decimals: the most decimal places any term needs is 2 (in 2.25 and 0.75), so multiply every term by 100: 150 : 225 : 75.
2. Find the greatest common divisor: GCD(150, 225, 75) = 75.
3. Divide every term by 75: 150 ÷ 75 = 2, 225 ÷ 75 = 3 and 75 ÷ 75 = 1. The mix is 2 : 3 : 1.
4. Total parts: 2 + 3 + 1 = 6. Compost is 2/6 = 1/3 of the mix (33.33%), soil 3/6 = 1/2 (50%) and perlite 1/6 (16.67%).

For the planter, one part is 30 ÷ 6 = 5 liters, so it takes 10 liters of compost, 15 of soil and 5 of perlite. If you have 6 buckets of compost instead, the scale factor is 6 ÷ 1.5 = 4, and the mix becomes 6 : 9 : 3 buckets. Those two answers are what the calculator shows when you switch to dividing or scaling.

Simplifying ratios with decimals or fractions

Multiply every term by the same number to make them all whole, then divide by the GCD. Multiplying every term by the same number keeps the ratio the same.

• Decimals: multiply by 10, 100 or 1,000, whichever clears the term with the most decimal places. 0.4 : 1.25 becomes 40 : 125, then 8 : 25 after dividing by 5.
• Fractions: multiply by the least common denominator. For 1/2 : 3/4 the denominators are 2 and 4, so multiply by 4: 2 : 3.
• Mixed numbers: write them as improper fractions first. 1 1/2 : 2 is 3/2 : 2, and multiplying by 2 gives 3 : 4.
• Repeating decimals: 0.(3) is exactly 1/3, so 0.(3) : 1 is 1/3 : 1, or 1 : 3. Rounding it to 0.33 first would give 33 : 100, a different ratio.

How to solve a proportion (find the missing value)

Cross-multiply: in a true proportion, the product of the outer terms equals the product of the inner terms. Then divide to get the unknown term.

$$\frac{A}{B} = \frac{C}{D} \quad\Longleftrightarrow\quad A \times D = B \times C$$

So $D = B \times C \div A$, $C = A \times D \div B$, $B = A \times D \div C$ and $A = B \times C \div D$.

Map distance. A map has a scale of 1 : 25,000, and two points are 3.2 cm apart on it. How far apart are they on the ground? Set up map : ground on both sides, 1 : 25,000 = 3.2 : D. Then 1 × D = 25,000 × 3.2 = 80,000, so the distance is 80,000 cm, which is 800 meters.

Checking two ratios. Is 4 : 6 the same as 10 : 15? The cross products are 4 × 15 = 60 and 6 × 10 = 60. They are equal, so yes: both ratios simplify to 2 : 3.

A proportion divides by its second terms, so B and D can’t be 0. If A is 0 when you solve for D, there is no single answer: 0 : 5 = 3 : D has no solution, and 0 : 5 = 0 : D holds for any D above 0. The calculator says which case you have instead of showing an error.

How to divide an amount in a given ratio

Add the terms to get the total number of parts, divide the amount by it to get one part, then multiply one part by each term.

Splitting $1,000 in the ratio 2 : 3 : 4. There are 2 + 3 + 4 = 9 parts, so one part is $1,000 ÷ 9 = $111.11… The exact shares are $222.22…, $333.33… and $444.44… Rounded down to the cent they come to $222.22 + $333.33 + $444.44 = $999.99, one cent short. The missing cent goes to the share with the largest remainder, the third (0.44 of a cent), so the shares are $222.22, $333.33 and $444.45, which add up to exactly $1,000.00. This is the largest-remainder rule, known as Hamilton’s method. The same rule rounds to whole numbers, such as 25 items shared 1 : 1 : 1 (9, 8 and 8).

How to scale a ratio up or down

Multiply every term by the same factor: above 1 to enlarge, below 1 to shrink. To make one term reach a set value, the factor is that value ÷ the term’s current value.

A 16 : 9 screen 1,080 pixels tall. The height is the second term, so the factor is 1,080 ÷ 9 = 120. The width is 16 × 120 = 1,920, and the scaled ratio is 1,920 : 1,080. It still simplifies to 16 : 9, because scaling never changes a ratio’s proportions.

Writing a ratio in 1:n form

Divide every term by the first term. 2 : 3 : 1 becomes 1 : 1.5 : 0.5, and a map scale of 2 cm : 500 m (2 cm : 50,000 cm) becomes 1 : 25,000. The n : 1 form divides by the last term instead: 3 : 4 becomes 0.75 : 1. Both forms make ratios easy to compare at a glance. The 1 : n form isn’t defined when the first term is 0, and the n : 1 form when the last term is 0.

Ratios, fractions and percentages: how they relate

A two-term ratio a : b can be written as the fraction a/b, which compares the first amount with the second: in 2 : 3, the first amount is 2/3 of the second. A share of the whole is different: it divides a term by the total number of parts. In 2 : 3 there are 5 parts, so the first term is 2/5 of the whole, which is 40%, and the second is 3/5, or 60%. A percentage is a ratio with 100 as its second term, so 40% is 40 : 100, which simplifies to 2 : 5.

Reading the result

• The headline is the answer to the calculation you chose: the simplified ratio, the missing term (or whether the proportion is true), the shares or the scaled ratio. A decimal marked ≈ is rounded for display; the exact fraction is shown next to it.
• The tiles add the total number of parts, the 1 : n and n : 1 forms, the cross products, one part of the total or the scale factor, depending on the calculation.
• How this was calculated lists each step with your numbers.
• Parts of the whole draws each term’s share as a bar, with its number of parts and percentage.
• The tables list each term as entered, in smallest whole numbers, as a fraction and a percentage of the whole, and its share or scaled value. Equivalent ratios lists the simplified ratio times 1 to 10, the ratio table used in class. Each table has a CSV download.

Assumptions and limitations

• Every term of a ratio is in the same unit. The calculator reads numbers only, so it can’t convert 2 kg and 500 g for you.
• Terms are 0 or more, at most 1,000,000,000,000 each, with up to 15 decimal places. At least one term must be above 0.
• Everything is worked out exactly with fractions, so simplifying 1.5 : 2.25 : 0.75 gives exactly 2 : 3 : 1, not 1.99999 : 3 : 1. Decimals marked ≈ are rounded to 6 significant figures for display only.
• Rounded shares follow the largest-remainder rule. Some agreements split rounding differently, for example giving every leftover cent to the first person, so check yours if the cents matter.

Common mistakes

• Mixing units. 2 kg : 500 g is not 2 : 500. Convert first: 2,000 g : 500 g is 4 : 1.
• Swapping the order. 2 : 3 and 3 : 2 are different ratios. Keep the terms in the order the question names them.
• Treating a term as a share of the whole. In 2 : 3 the first amount is 2/5 of the total, not 2/3 and not 2%.
• Setting up a proportion with mismatched units. Put the same kind of quantity in the same place on both sides: map : ground = map : ground, never map : ground = ground : map.
• Rounding each share on its own. $1,000 in 2 : 3 : 4 rounded share by share gives $999.99. Round with the largest-remainder rule so the shares add up.
• Typing a ratio as a fraction. In the ratio box, 3/4 is one number, three quarters. Type 3:4 for the ratio of 3 to 4.