Ratios and Proportions Worked ExamplesIGCSE Maths 0580

Always check your shares sum to the original amount. That catches a common slip: dividing by one of the ratio numbers instead of by their total.

Identify carefully what the given figure represents: a difference, one share, or the total. In a similar question, the June 2025 examiner report noted that successful candidates realised the first step was to divide the difference by the difference in parts, while a common error was to divide it by the total number of parts or by one of the ratio numbers. Here that means $24 \div 4$, not $24 \div 10$, $24 \div 7$ or $24 \div 3$.

Scale the whole ratio, not just the shared term — multiplying only the $3$ to make $12$ destroys the relationship. Always check the final three for a common factor. The March 2026 examiner report noted that some candidates, asked for a ratio in simplest form, tried to make one of the parts equal to $1$. Simplest form means whole numbers with no common factor, so $8 : 12 : 15$ stands, not $1 : 1.5 : 1.875$.

Write $y = kx$ and find $k$ explicitly. Scaling directly ($x$ went up by $2.5$, so $y$ does too) gets the right answer but can forfeit method marks when the question says 'find a formula'.

Sense-check the direction: in inverse proportion, $x$ going down must make $y$ go up. In (a) $x$ halved from $5$ to $2.5$-ish and $y$ rose — consistent. If your answer moves the same way as $x$, you have used direct proportion by mistake.

Read the relationship precisely: 'proportional to the square of $x$' gives $kx^2$; 'proportional to $x$' gives $kx$; 'inversely proportional to the square' gives $\frac{k}{x^2}$. Note $x$ rose by a factor of $2.5$ but $y$ rose by $2.5^2 = 6.25$ — that ratio is a quick sanity check.

A common error in (c) is to scale the map area by the length factor: $4.8 \times 0.5 = 2.4\text{ km}^2$ is wrong because area scales by the square of the length factor. On a plan-area question, the June 2025 examiner report found candidates unsure whether to multiply or divide by the scale, and even fewer squared it. Convert the lengths first, then multiply. In (b), do not stop at the map measurement: the March 2025 report noted that "A few measured the line and gave it as their answer." In (a), $2 : 1$ or $1 : 0.5$ mixes units and scores nothing.

Adding the ratios as $1 : 5 + 3 : 5 = 4 : 10$ is a common wrong method — it ignores that the jugs hold different amounts. In (c), the key idea is that the water does not change, so it fixes the size of one part. Setting up $\frac{0.5 + x}{2 + x} = \frac{2}{7}$ also works but is slower.

Two ratios work together here: the length ratio fixes $AP$, and the area ratio fixes $DQ$. A common error is to use $\frac{1}{2}$ of $96$ (reading $1 : 2$ as 'one half'), but a ratio $1 : 2$ means $\frac{1}{3}$ of the total. In (c), a general statement is not enough — a calculation that gives a different ratio is needed.

In (a), the March 2026 examiner report noted that some candidates tried to make one of the parts equal to $1$ — simplest form means whole numbers with no common factor, not $1 : 1.33 : 1.67$. In (b), the ratio is applied to the remaining $300$, not to all $540$. On a similar question, the same report found that some candidates treated the full total as the number to be shared in the ratio.

On a similar question, the June 2025 examiner report found that candidates were unsure whether to multiply or divide by the scale number, and even fewer squared it. Areas scale by the square of the length scale ($40^2 = 1600$), and $\text{m}^2$ to $\text{cm}^2$ is $\times 10\,000$, not $\times 100$. Sense-check: a real floor of $24\text{ m}^2$ drawn on a page as $150\text{ cm}^2$ (about $12$ cm by $12$ cm) is sensible.