Algebraic Fractions Worked ExamplesIGCSE Maths 0580

Cancelling is division, so it applies to factors, not terms. Here the whole numerator and denominator are single products, which is what makes it valid.

The June 2025 examiner report noted that a small minority did not attempt to factorise and instead cancelled the squared terms. Cancelling requires factors, and neither expression is a product until it has been factorised. Factorise the top and the bottom first, every time.

Leave the denominator factorised — expanding it to $x^2 + x$ gains nothing. Then stop: the June 2025 examiner report found that "some candidates then incorrectly cancelled x from terms in the numerator and denominator." $\frac{5x + 2}{x(x+1)}$ has no common factor to cancel. Check at $x = 1$: $2 + 1.5 = 3.5$, and $\frac{7}{2} = 3.5$. ✓

Bracket the second numerator before expanding — writing $5x + 5 - 3x - 6$ gives $2x - 1$. The June 2025 examiner report found that "incorrect expansion of the numerator was common", and that sign errors there "meant that the correct final answer was not obtained".

Flip the second fraction before factorising or cancelling — cancelling across a division sign is invalid. Check at $x = 3$: $\frac{5}{4} \div \frac{1}{12} = 15$, and $3(5) = 15$. ✓

Multiplying through by the common denominator clears the fractions in one step. The June 2025 examiner report noted that "the most common error when multiplying through to remove the denominators of the fractions was to forget to multiply the 1" — here the right-hand side must become $x(x - 1)$, not stay as $1$. Neither root makes a denominator zero ($x \neq 0, 1$), so both are valid; always check this when you have cleared fractions.

Multiply each whole numerator by its factor: $4(x - 2) = 4x - 8$, not $4x - 2$. A common denominator of $24$ also works but leaves a fraction to cancel at the end. In (b), stopping at $x = 14$ loses the final mark, because the question asks for a length.

The June 2025 examiner report found that "many candidates struggled to form the initial equation due to errors in the relationship between speed, distance and time in an algebraic context". Write time $=$ distance $\div$ speed for each stage before combining. Writing $2$ hours $30$ minutes as $2.3$ is another common error. In (c), the rejection needs a reason tied to the context, not just 'it is too small'. Use the unrounded root for the check.

In (a), taking reciprocals term by term to get $R_P = x + (x + 3)$ is invalid. You can only flip both sides once each is a single fraction, the same rule as for the lens formula. In (b), $2$ must be written as $\frac{2(2x + 3)}{2x + 3}$ before adding; writing $\frac{2 + x(x + 3)}{2x + 3}$ is the usual error.

The March 2025 examiner report said fully correct answers were less common, "primarily because of inefficient methods or sign errors". Using $4 \times 6 \times 8 = 192$ as the denominator works but leaves big numbers and a fraction to cancel — use the LCM. Writing $-8x - 4$ for the middle bracket gives $\frac{x - 1}{24}$. Check at $x = 1$: $1 - \frac{1}{6} - \frac{1}{2} = \frac{1}{3}$ and $\frac{8}{24} = \frac{1}{3}$. ✓

The March 2026 examiner report found that many candidates were unable to correctly factorise the numerator. Expand your brackets to check before cancelling: $(2x - 1)(x + 3) = 2x^2 + 5x - 3$. ✓ Factorising both parts earns most of the marks, and you should expect a common bracket to appear — if none does, recheck your factorising.

The June 2025 examiner report noted that "some candidates only took out a factor of 2 rather than 2x", and that "a common incorrect approach was to cancel terms from the numerator and denominator in a pairwise fashion". It also found that candidates with factors such as $x(7 - x)$ and $(x - 7)(x + 7)$ tended to make further errors — spot that $6 - x = -(x - 6)$ before cancelling. Check at $x = 1$: the original is $\frac{10}{-35} = -\frac{2}{7}$, and $-\frac{2}{1 + 6} = -\frac{2}{7}$. ✓

The June 2025 examiner report found that forming the product was fine but "fully simplifying it was a problem for many." It also noted that "a minority of candidates did not leave their answer as a fraction". Writing $\frac{2}{15}a^{-3}b^2$ is correct but not the expected form. Cancelling before multiplying ($3$ with $9$, $4$ with $10$) keeps the numbers small.