Ratios

What a ratio is

A ratio compares two quantities by division: how much of one there is for each amount of the other. The ratio of $2$ cups of flour to $3$ cups of sugar is written

read “two to three.” The same comparison can also be written as the fraction $\frac{2}{3}$. That link to fractions is what makes ratios easy: scaling, simplifying and comparing work here exactly as they do for fractions.

Two cautions come with the definition. First, order matters. The ratio of flour to sugar is $2 : 3$, but the ratio of sugar to flour is $3 : 2$. Whatever quantity you name first is written first. Second, a ratio is not a total. Saying flour to sugar is $2 : 3$ does not claim there are exactly $2$ and $3$ cups. It says that for every $2$ cups of flour there are $3$ cups of sugar. You might really have $4$ and $6$ cups, or $20$ and $30$, and the comparison is still $2 : 3$.

Equivalent ratios

Doubling a recipe keeps the flavor, and doubling both parts of a ratio keeps the comparison. Use twice as much of the $2 : 3$ mixture: the flour goes from $2$ cups to $2 \times 2 = 4$, and the sugar from $3$ cups to $3 \times 2 = 6$. Those $4$ cups of flour form two groups of $2$, and the $6$ cups of sugar form two groups of $3$. Each group of flour still faces a group of sugar, so $4 : 6$ makes the same comparison as $2 : 3$.

Because a ratio is a comparison and not a count, multiplying or dividing both quantities by the same nonzero number leaves it unchanged. Ratios related this way are called equivalent ratios:

Each one describes the same relationship, just measured in larger or smaller batches. The picture below shows the first two. For every $2$ flour squares there are $3$ sugar squares; doubling every square to $4$ and $6$ keeps the two colors in step.

This is exactly the rule for equivalent fractions. Writing the ratio $2 : 3$ as $\frac{2}{3}$, scaling both parts is the same step that turns $\frac{2}{3}$ into $\frac{4}{6}$.

Take the ratio $7 : 11$ and scale both parts by $3$, which gives $21 : 33$. As a fraction that is $\frac{21}{33}$, and the $3$ that scaled both parts is still sitting in both the top and the bottom: $21 = 3 \times 7$ and $33 = 3 \times 11$. Cancel that shared $3$ and the fraction becomes $\frac{7}{11}$ again, so $21 : 33$ is the comparison $7 : 11$ over again, which is why scaling both parts by the same number never changes what a ratio compares.

You can scale the recipe ratio yourself. The top bar below is one batch of the $2 : 3$ mixture, cut into $5$ equal parts, one for each part of that ratio. In that bar, $2$ parts are shaded for flour and the other $3$ are left for sugar. The bar under it is the same batch, and you choose how finely it is cut.

Cut each part into three and count what you get: $6$ shaded parts against $9$ unshaded, which is the ratio $6 : 9$ from the list above. Cut into four instead and it is $8 : 12$. The edge of the shaded region never moves while you do this, because the batch itself never changes size, only how finely it is sliced. The steady edge is the claim in the proof made visible: $2 : 3$, $6 : 9$ and $8 : 12$ are one relationship counted in smaller and smaller pieces, not different mixtures. The readout below states this two ways: first as a fraction of the whole batch, flour and sugar together (you will meet that comparison properly later in this lesson), and then as the flour-to-sugar ratio itself, the part-to-part comparison this section is about. Watch that second number: it is the one that keeps matching $2 : 3$ no matter how finely the batch is cut.

Simplifying a ratio

A ratio is in lowest terms (or simplest form) when its two parts share no common factor larger than $1$. To get there, divide both parts by their greatest common factor, the same move you use to reduce a fraction. Simplest form is handy because it is the smallest whole-number version of the comparison. So two ratios of whole numbers are equivalent exactly when they reduce to the same lowest terms.

Part-to-part and part-to-whole

When a whole is split into groups, there are two different comparisons you can make, and confusing them is the single most common ratio mistake. A part-to-part ratio compares one group to another group. A part-to-whole ratio compares one group to the total.

Suppose a class has only boys and girls, no other kind of student, with $2$ boys for every $3$ girls. The part-to-part ratio of boys to girls is $2 : 3$. Adding the parts only gives the whole class because there is nothing left out: $2 + 3 = 5$ students in every batch of $2$ boys and $3$ girls. So the part-to-whole ratio of boys to all students is $2 : 5$, and as a fraction boys make up $\frac{2}{5}$ of the class, not $\frac{2}{3}$. The denominator of a part-to-whole fraction is the sum of the parts being compared, and that sum is the true whole only when those parts are the only ones there are.

Using a ratio to find an amount

A ratio lets you find one quantity from the other once you know a single actual amount. The key is the scale factor: the number you multiply a part of the ratio by to reach the real amount. Because both parts scale by the same factor, finding it for one part hands you the other.

Comparing two ratios

Two paint mixes can use the same two colors and still not match. One mix is $3$ parts blue to $4$ parts white, and another is $5$ parts blue to $7$ parts white. Which one packs in more blue? Call one ratio stronger than another when it has more of the first quantity for the same amount of the second. To decide which of two ratios is stronger, write each ratio as a fraction and compare the fractions, using a common denominator exactly as you would to compare any two fractions.

Common mistakes