A ratio compares part to part (e.g. boys to girls); a fraction compares part to whole (e.g. boys out of all students). For 13 boys and 17 girls, the ratio of boys to girls is $13:17$, but the fraction of students who are boys is $\frac{13}{30}$ (using the total, $13+17=30$). Mixing these two up is one of the most common sources of lost marks on this topic.

The order quantities are mentioned in the question must match the order in your ratio — not the order the numbers happen to appear.

Example: A class has 13 boys and 17 girls. The ratio of girls to boys is $17:13$ — not $13:17$, even though $13$ is mentioned first in the setup.

Divide every term by their highest common factor (HCF) — the largest number dividing all terms exactly.

Example: Simplify $16:36$. Factors of $16$: $1,2,4,8,16$. Factors of $36$: $1,2,3,4,6,9,12,18,36$. HCF $=4$, so $16:36 = 4:9$.

This extends to three-part ratios: $15:30:24$ has HCF $3$, simplifying to $5:10:8$.

First convert every term into the same whole-number units, then find the HCF as normal.

• Decimals/fractions: multiply every term by the same power of 10 (or common denominator). $2.5:3.75 \to 250:375 \to 2:3$ (HCF 125).
• Mixed units: convert to a single unit first. $1.2 \text{ kg} : 200\text{ g} \to 1200\text{ g}:200\text{ g} = 6:1$.

Ratio → fraction: add the parts to get the total (this becomes the denominator); the part you want is the numerator. If Ann and Bob share in the ratio $3:4$, the total is $7$ parts, so Bob's fraction of the whole is $\frac{4}{7}$.

Fraction → ratio: subtract the fraction from $1$ to find the "other part", then state both parts in order. If $\frac{3}{4}$ of fish are male, the female fraction is $1-\frac{3}{4}=\frac{1}{4}$, so males:females $=3:1$.

Given $a:b$ and $b:c$, scale each ratio so the shared term ($b$) matches in both, using the LCM of its two values.

Example: $a:b=3:2$ and $b:c=3:5$. The $b$-values are $2$ and $3$; LCM $=6$. Scale the first ratio by $3$: $9:6$. Scale the second by $2$: $6:10$. Since $b=6$ in both, combine directly: $a:b:c = 9:6:10$.

A fitness class has 30 people in total; 23 are female. Find the simplified ratio of males to females.

1. Number of males: $30-23=7$.
2. Ratio of males to females: $7:23$.
3. Check for a common factor: $7$ and $23$ share no common factor other than $1$, so $7:23$ is already in its simplest form.