Changing the Subject of the Formula Worked ExamplesIGCSE Maths 0580

Keep the whole of $y - 5$ over the $3$. Writing $y - \frac{5}{3}$ divides only part of the left side and is the standard error here.

$v^2 - u^2$ is not $(v-u)^2$. Leave it as written — factorising to $(v-u)(v+u)$ is valid but gains nothing and risks an error.

Squaring $\frac{T}{2\pi}$ squares the $2$ and the $\pi$ as well, giving $4\pi^2$ — writing $2\pi^2$ is the usual slip. Isolate the root before squaring, never square term by term.

You may only flip both sides once each is a single fraction. Taking reciprocals term by term — turning $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ into $f = u + v$ — is invalid and is the classic error in this question.

Do not cancel the $\pi r^2$ against the $2\pi r$ — the numerator is a difference, not a product, so nothing can be cancelled until it is factorised. This is the same rule as for algebraic fractions.

Whenever the new subject appears more than once, the route is always the same: clear fractions, expand, gather those terms on one side, factorise, divide. Each step earns a method mark, so write one step per line: the June 2025 examiner report found that "many candidates attempted two steps at once, often making errors that invalidated both steps." Check at $x = 5$: $y = \frac{7}{2}$, and $\frac{3(3.5)+2}{3.5-1} = \frac{12.5}{2.5} = 5$. ✓

A common error in (a) is leaving out the two radii and using the arc length on its own as the perimeter. Because (a) is a 'show that', the March 2026 examiner report's advice applies: candidates "should show every step in their method", including the $\frac{\theta}{360} \times 2\pi r$ line. In (b), writing $180P - 2r$ multiplies only part of the left-hand side by $180$. Angles are given to 1 decimal place unless the question says otherwise.

Get the root on its own before squaring. Squaring $\pi r\sqrt{r^2 + h^2}$ as $\pi r(r^2 + h^2)$ forgets to square $\pi r$. Another common error is to square-root term by term: $h = \frac{S}{\pi r} - r$ gives $7.73$, but $\sqrt{a - b}$ is not $\sqrt{a} - \sqrt{b}$. In (c), the June 2025 examiner report warned that candidates who "rounded or truncated intermediate values prematurely" lost marks — keep $12.732\ldots$ in the calculator.

Dividing by $2$ and by $\pi$ separately, for example $r = \frac{P - 2h}{2} - \pi$, is invalid: the only correct route is to factorise $r$ out of both terms. Counting the dashed diameter as part of the frame gives $P = 2h + 4r + \pi r$ and changes every later answer. In (c), the March 2025 examiner report stressed that "intermediate values within longer methods should not be rounded": rounding $r$ to $0.51$ first gives an area of $1.63$. Check (c): $2(1.2) + (2 + \pi)(0.50567\ldots) = 5$. ✓

The June 2025 examiner report noted that some candidates did not realise the best starting point was to work towards isolating the $x^2$ term. Taking a square root early, for example writing $\sqrt{P} = \sqrt{4k} - \sqrt{3}x$, is invalid. Moving $3x^2$ to the other side first avoids dividing by $-3$ and the sign errors that come with it. Check with $k = 7$, $x = 2$: $P = 28 - 12 = 16$, and $\sqrt{\frac{28 - 16}{3}} = \sqrt{4} = 2$. ✓

The March 2026 examiner report noted that a few candidates made a sign error in their first step when rearranging a formula of this type. Here that gives $y^2 + x^2 = 4ab$ and the wrong answer $\frac{x^2 + y^2}{4a}$. Moving the negative term across first, as in step 1, keeps every sign visible. Check with $a = 1$, $b = 2$, $x = 5$: $y^2 = 25 - 8 = 17$, and $\frac{25 - 17}{4} = 2$. ✓

The June 2025 examiner report found that, after clearing the fraction and expanding, the main error was that candidates did not collect the terms containing the subject together on one side. It also noted that "many candidates attempted two steps at once". One step per line earns each method mark. Check with $a = 5$, $b = 2$: $k = \frac{15}{5} = 3$, and $\frac{5 \times 6}{2 \times 3} = 5$. ✓