Fractions, ratios and proportions

Fractions by hand

Without a calculator, fractions are usually faster than decimals, because they let you cancel before you multiply.

• Add or subtract over a common denominator: $\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.
• Multiply tops and bottoms, cancelling first: $\frac{14}{15} \times \frac{25}{28} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}$, after cancelling 14 with 28 and 25 with 15.
• Divide by multiplying by the reciprocal: $\frac{3}{4} \div \frac{9}{10} = \frac{3}{4} \times \frac{10}{9} = \frac{30}{36} = \frac{5}{6}$.

To compare two fractions, cross-multiply: $\frac{a}{b} > \frac{c}{d}$ (with positive denominators) exactly when $ad > bc$. Is $\frac{7}{12}$ or $\frac{11}{19}$ larger? $7 \times 19 = 133$ and $11 \times 12 = 132$, so $\frac{7}{12}$ is larger, by a hair. Benchmarks help too: a fraction whose top is more than half its bottom is above $\frac{1}{2}$.

A few decimal equivalents are worth knowing by heart: $\frac{1}{8} = 0.125$, $\frac{1}{6} \approx 0.167$, $\frac{1}{5} = 0.2$, $\frac{1}{4} = 0.25$, $\frac{1}{3} \approx 0.333$, $\frac{3}{8} = 0.375$.

A ratio is a recipe

A ratio $3 : 5$ of orange to apple juice says nothing about amounts; it says that for every 3 parts of orange there are 5 parts of apple. The whole has $3 + 5 = 8$ parts, so orange is $\frac{3}{8}$ of the blend and apple $\frac{5}{8}$. Keep the two readings apart:

Reading	From $3 : 5$
part to part	orange is $\frac{3}{5}$ of the apple
part to whole	orange is $\frac{3}{8}$ of the blend

The multiplier method

Most ratio problems become one equation if you call the size of one part $k$. With orange to apple $3 : 5$, the amounts are $3k$ and $5k$. Whatever the question tells you (the total, the difference, one of the amounts, or what happens after a change) becomes an equation in $k$. For example, if the total is 24 litres, $8k = 24$, so $k = 3$ and there are 9 litres of orange.

The method shines when a ratio changes. If boys and girls are in the ratio $5 : 3$ and 6 girls join to make it $5 : 4$, write boys $= 5k$, girls $= 3k$, and translate the new ratio: $\frac{5k}{3k + 6} = \frac{5}{4}$. Cross-multiplying gives $20k = 15k + 30$, so $k = 6$: 30 boys, and 18 girls who become 24. Check: $30 : 24 = 5 : 4$.

Linking two ratios

Given $a : b = 3 : 4$ and $b : c = 6 : 5$, you cannot read off $a : c$ directly, because the two $b$ values are on different scales. Rescale both ratios so that $b$ has the same number: the LCM of 4 and 6 is 12, so $a : b = 9 : 12$ and $b : c = 12 : 10$. Now $a : b : c = 9 : 12 : 10$, and $a : c = 9 : 10$. Equivalently, $\frac{a}{c} = \frac{a}{b} \times \frac{b}{c} = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}$.

Direct and inverse proportion

Two quantities are directly proportional when their ratio is constant, $y = kx$: double one and the other doubles. Price and quantity at a fixed unit price behave this way. They are inversely proportional when their product is constant, $xy = k$: double one and the other halves. Speed and time over a fixed distance behave this way, as do workers and days for a fixed job.

Before you set up a proportion, ask which kind it is: "more of this means more of that" or "more of this means less of that". Then use a ratio for the first and a product for the second. If 4 identical printers finish a job in 9 hours, then the job is $4 \times 9 = 36$ printer-hours, so 6 printers take $\frac{36}{6} = 6$ hours, not $\frac{6}{4} \times 9 = 13.5$.

Additive changes break ratios

Adding the same amount to both terms of a ratio changes it: $1 : 2$ becomes $3 : 4$ after adding 2 to each. Only multiplying both terms by the same number leaves a ratio unchanged. This is why "the ratio after a change" questions need the multiplier method rather than guesswork.