Writing, simplifying and comparing ratios

Learning outcome

Write ratios in the correct order, simplify whole-number, decimal and fractional ratios, and compare ratios without relying on rounded decimal values.

Understanding ratios

A ratio describes the relative sizes of quantities, not their difference. Throughout this module, all quantities are positive. The notation a:b means “a compared with b” and corresponds to a/b. Reversing the order changes which quantity is being compared with which.

For quantities of the same kind, first convert both measurements to the same unit. Otherwise, the numerical comparison mixes different measuring scales. Once the units match, they cancel in the quotient.

Multiplying or dividing both terms by the same positive number preserves a ratio because that factor cancels in the fraction. For whole-number terms, divide by their greatest common factor. With terminating decimals or fractions, first multiply both terms to clear decimal places or denominators; then reduce the resulting whole numbers. Subtracting the same amount from both terms does not generally preserve a ratio.

To compare a:b with c:d, compare a × d with c × b. This works because multiplying a/b and c/d by the same positive number b × d preserves their order. Equal cross-products mean equal ratios. This method gives an exact comparison, avoiding rounding errors.

A ratio alone does not specify the actual quantities. It tells us their relative sizes; a total or another measurement is needed to recover their individual sizes.

Worked example 1

Two ribbons measure 1.8 m and 75 cm. Find first:second.

Convert 1.8 m to 180 cm. The ratio is 180:75. Dividing both terms by their greatest common factor, 15, gives 12:5. Thus, the first length contains 12 equal-sized parts for every 5 such parts in the second. Comparing 1.8 directly with 75 would incorrectly ignore the units.

Worked example 2

Simplify (3/4):(5/6).

Multiply both terms by 12, a common multiple of the denominators: (3/4) × 12 = 9 and (5/6) × 12 = 10. Hence the ratio is 9:10. The same multiplier clears both denominators without changing the comparison. No further whole-number reduction is possible.

Worked example 3

Compare 7:10 and 9:13.

The cross-products are 7 × 13 = 91 and 9 × 10 = 90. Since 91 > 90, the ratio 7:10 is larger. This compares the first quantity relative to the second, not the unknown totals represented by either ratio.

Common mistakes

Do not reverse the requested order, compare unlike units, or simplify only one term. A larger numerator alone does not establish a larger ratio. In a:b, b is the comparison quantity, not the combined total.

Practice questions

1. Write 2.4 kg:900 g in simplest form.
2. Simplify 0.45:1.2.
3. Simplify (5/8):(15/16).
4. Which ratio is larger: 11:15 or 14:19?

Worked answers

1. Convert 2.4 kg to 2400 g. Then 2400:900, divided by 300 in both terms, becomes 8:3.
2. Multiply both terms by 100 to get 45:120. Divide both by 15 to obtain 3:8.
3. Multiply both terms by 16 to obtain 10:15. Dividing both by 5 gives 2:3.
4. Compare 11 × 19 = 209 with 14 × 15 = 210. Since 209 < 210, the larger ratio is 14:19.