What Is a Ratio?

A ratio is a way of comparing two or more quantities to show how large one is relative to another. If you have 3 apples and 2 oranges, the ratio of apples to oranges is "3 to 2." Ratios show up everywhere in daily life, from mixing paint colors to reading a recipe to comparing prices, which is why understanding ratios is one of the most useful skills in basic math.

Unlike a simple subtraction (which tells you the difference between two amounts), a ratio tells you the relationship between them, specifically how many times one quantity fits into or compares with another.

A ratio between two quantities \(a\) and \(b\) can be written in three equivalent ways:

• Using a colon: \(a : b\)
• Using the word "to": \(a\) to \(b\)
• As a fraction: \(\frac{a}{b}\)

So the ratio of 3 apples to 2 oranges can be written as \(3 : 2\), "3 to 2," or \(\frac{3}{2}\). All three mean the exact same comparison. This fraction form is why ratios are closely linked to fractions, and it can be helpful to review converting among ratios, fractions and decimals once you are comfortable with the basics here.

Ratios can compare a part to another part, or a part to the whole. In a classroom with 12 girls and 8 boys:

• The part-to-part ratio of girls to boys is \(12 : 8\), which simplifies to \(3 : 2\).
• The part-to-whole ratio of girls to all students is \(12 : 20\), which simplifies to \(3 : 5\).

Notice both ratios above were simplified. Just like a fraction, a ratio is simplified by dividing every term by their greatest common factor. \(12 : 8\) has a greatest common factor of \(4\), so dividing both terms by \(4\) gives \(3 : 2\).

Two ratios are called equivalent ratios when they express the same relationship, even though the numbers look different. You can create an equivalent ratio by multiplying or dividing every term of a ratio by the same nonzero number.

For example, starting with \(3 : 2\):

• Multiply both terms by \(2\): \(6 : 4\)
• Multiply both terms by \(3\): \(9 : 6\)

So \(3:2\), \(6:4\), and \(9:6\) are all equivalent ratios. Checking whether two ratios are equivalent is often done by writing them as fractions and seeing if they simplify to the same value, or by cross-multiplying: \(a:b\) and \(c:d\) are equivalent if \(a \times d = b \times c\).

Example: A recipe calls for 4 cups of flour and 6 cups of sugar. Write the ratio of flour to sugar in simplest form, and give one equivalent ratio.

Step 1: Write the ratio as given: \(4 : 6\).

Step 2: Find the greatest common factor of \(4\) and \(6\), which is \(2\), and divide both terms: \(4 \div 2 : 6 \div 2 = 2 : 3\).

Step 3: To find an equivalent ratio, multiply both terms of \(2:3\) by the same number, say \(5\): \(10 : 15\).

So \(4:6\), \(2:3\), and \(10:15\) all describe the exact same comparison between flour and sugar.

A ratio compares two quantities, while a proportion is a statement that two ratios are equal, such as \(\frac{2}{3} = \frac{10}{15}\). Proportions are used to solve for an unknown quantity when you know a ratio must stay consistent, for example scaling a recipe up or down. Once you feel confident with what a ratio is, it is worth moving on to proportions to see how ratios are used to solve real problems.

This lesson covers the core definition of a ratio, but there is much more to explore, including comparing ratios with different units (called rates), and applying ratios to word problems involving mixtures, maps, and scale drawings. Visit the full lesson on ratios for more practice problems and detailed examples.

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