Ratios

A ratio is a way of comparing two quantities of the same kind by division. If a class has $20$ girls and $25$ boys, the ratio of girls to boys is $20 : 25$ or, simplified, $4 : 5$.

Concept

The ratio $a : b$ (read "$a$ to $b$") means $\dfrac{a}{b}$ , how many "$b$-sized units" fit into $a$. It carries no unit of its own; the units of $a$ and $b$ must be the same.

If you compare $200$ g to $1$ kg, first convert: $1$ kg $= 1000$ g. So the ratio is $200 : 1000 = 1 : 5$.

Simplifying. Just like a fraction, divide both sides by their greatest common factor.

$24 : 36 \to 2 : 3 \quad (\div 12).$

Equivalent ratios. Multiplying or dividing both parts by the same number gives an equivalent ratio. $2 : 3 = 4 : 6 = 6 : 9 = 20 : 30$.

Comparing two ratios. Convert each to a fraction or to the same denominator, then compare. Is $3 : 5$ greater than $5 : 8$? $\tfrac{3}{5} = 0.6$ and $\tfrac{5}{8} = 0.625$. So $5 : 8 > 3 : 5$.

Ratios with three or more parts. Sometimes we say "divide $\mathbb{?}\,300$ between $A, B, C$ in the ratio $3 : 4 : 5$". Total parts $= 3+4+5 = 12$. Each part is worth $\mathbb{?}\,25$. So $A$ gets $\mathbb{?}\,75$, $B$ gets $\mathbb{?}\,100$, $C$ gets $\mathbb{?}\,125$.

Percentage as a ratio. Saying "$30\%$ marks" is saying "$30 : 100$" or $\tfrac{3}{10}$. Percentages, decimals, and ratios are three faces of the same underlying number.

Where ratios appear daily.

• Maps: a scale of $1 : 50{,}000$ means $1$ cm on the map represents $50{,}000$ cm $= 500$ m on the ground.
• Recipes: "$2$ cups flour, $1$ cup sugar" is a ratio of $2 : 1$.
• Mixing: paint mixing, cement-sand-gravel mixtures.
• Sport stats: a batting average is a ratio.
• Aspect ratio of a screen: $16 : 9$.

Worked examples

Example 1. Simplify the ratio $48 : 72$.

• $\gcd(48, 72) = 24$.
• $48 : 72 = 2 : 3$.

Example 2. Express the ratio of $750$ mL to $2$ L in lowest terms.

• Convert: $2$ L $= 2000$ mL.
• Ratio: $750 : 2000 = 3 : 8$.

Example 3. Divide $\mathbb{?}\,4500$ between Anu and Bina in the ratio $5 : 4$.

• Total parts $= 9$.
• Each part $= 4500 / 9 = 500$.
• Anu: $5 \cdot 500 = \mathbb{?}\,2500$. Bina: $4 \cdot 500 = \mathbb{?}\,2000$.

Example 4. Which is the larger ratio: $7 : 10$ or $4 : 5$?

• $\tfrac{7}{10} = 0.7$, $\tfrac{4}{5} = 0.8$.
• So $4 : 5$ is the larger ratio.

Try it yourself

1. Simplify $36 : 48$.
2. Express $400$ paise to $\mathbb{?}\,5$ as a ratio in lowest terms.
3. Divide $\mathbb{?}\,1800$ between $A$ and $B$ in the ratio $2 : 7$.
4. Three boys $A, B, C$ get marbles in the ratio $3 : 4 : 5$. If the total is $60$ marbles, how many does each get?
5. Compare ratios $9 : 13$ and $7 : 10$.
6. The ratio of length to width of a rectangle is $5 : 3$. If the length is $35$ cm, find the width.
7. A class has $32$ students of whom $12$ are boys. Find the ratio of (i) boys to girls (ii) girls to total.
8. On a map of scale $1 : 1{,}000{,}000$, two cities are $7.5$ cm apart. Find the real distance in km.

Activity

Recipe scaling. Pick a recipe from home that serves $4$ people. Convert every ingredient's quantity to its ratio with one ingredient (say, with flour). Then scale the whole recipe to serve $10$ , keeping the ratios identical. The result is mathematically guaranteed to taste just like the original, only more of it.