Quick Answer

The ratio 12:8 simplifies to 3:2. In the proportion 3:4 = 9:x, the missing value x is 12.

Simplify a Ratio

First Value

Second Value

Solve Proportion (a:b = c:?)

a

b

c

Scale a Ratio

First Value

Second Value

Scale Factor

Common Examples

Input	Result
Simplify 24:36	2:3
Simplify 100:75	4:3
Solve 2:5 = 6:?	15
Scale 3:4 by 5	15:20
Simplify 48:16	3:1

How It Works

Simplifying a Ratio

To simplify a ratio a:b, find the greatest common divisor (GCD) of both terms and divide each term by it.

Simplified Ratio = (a / GCD(a, b)) : (b / GCD(a, b))

The GCD is the largest positive integer that divides both numbers evenly. The Euclidean algorithm is the standard method for computing GCD: repeatedly divide the larger number by the smaller and take the remainder until the remainder is zero.

Solving a Proportion

A proportion states that two ratios are equal: a:b = c:d. To find the missing fourth term d, use the cross-multiplication property.

d = (b x c) / a

This follows directly from the definition of equal ratios. If a/b = c/d, then cross-multiplying gives a x d = b x c, so d = (b x c) / a.

Scaling a Ratio

Scaling multiplies both terms of a ratio by the same factor, preserving the relationship between the two quantities.

Scaled Ratio = (a x factor) : (b x factor)

This is useful for converting a ratio into actual quantities. For instance, a recipe with a 2:3 ratio of flour to sugar scaled by 4 produces 8:12 (or 8 cups flour to 12 cups sugar).

Worked Example

To simplify 24:36: GCD(24, 36) = 12. Divide both terms: 24/12 = 2, 36/12 = 3. The simplified ratio is 2:3.

To solve 2:5 = 6:d: d = (5 x 6) / 2 = 30 / 2 = 15. So 2:5 = 6:15.

To scale 3:4 by a factor of 5: 3 x 5 = 15, 4 x 5 = 20. The scaled ratio is 15:20.

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Frequently Asked Questions

What is a ratio?

A ratio is a comparison of two quantities that shows how many times one value contains or is contained within the other. Ratios can be written as a:b, a/b, or 'a to b.' For example, a 3:2 ratio means the first quantity is 1.5 times the second.

How do I simplify a ratio?

Find the greatest common divisor (GCD) of both numbers, then divide each number by the GCD. For example, 18:12 has a GCD of 6, so it simplifies to 3:2.

What is a proportion?

A proportion is an equation stating that two ratios are equal. For example, 1:2 = 3:6 is a proportion. If one of the four values is unknown, you can solve for it using cross-multiplication.

Can ratios include decimals?

Yes. Ratios can use decimal values, though they are often expressed as whole numbers for clarity. A ratio like 2.5:5 is valid and simplifies to 1:2.

What is the difference between a ratio and a fraction?

A ratio compares two separate quantities (e.g., boys to girls in a class = 3:2), while a fraction represents a part of a whole (e.g., 3/5 of the class are boys). They use similar notation but serve different purposes.

Learn More

Percentages, Fractions, and Ratios Explained

How percentages, fractions, and ratios relate to each other, with conversion methods and worked examples for each. A reference guide for everyday math.