Why is 1/2 + 1/3 not equal to 2/5?

Because the denominators name different-sized parts. One half plus one third is more than half of one (1/2 is already half), but 2/5 is less than half. The arithmetic of adding tops and bottoms separately doesn't match the size of the parts. Convert to a common denominator first: 1/2 = 3/6, 1/3 = 2/6, so the sum is 5/6.

Why isn't 1/5 equivalent to 2/10 by adding 1 to the top and 5 to the bottom?

Equivalent fractions are made by multiplying (or dividing) the top and the bottom by the same factor, not by adding the same number to both. 1/5 = 2/10 because 1 × 2 = 2 and 5 × 2 = 10. If you check 1/5 = 0.2 and 2/10 = 0.2, the multiplication rule works. The addition rule would tell you 1/5 = 2/6, but 2/6 = 0.33, which is different.

How do I find a common denominator quickly?

If one denominator is a multiple of the other (like 5 and 10), use the larger one. Otherwise, multiply the two denominators together — it always works, even if it's not the smallest option. 1/4 + 1/6: 4 × 6 = 24, so use 24. (Or use 12, the LCM, if you spot it.) Either way, scale both fractions to that denominator before adding the numerators.

GCSE Maths Foundation · AQA · Fractions

Fraction additive: why 1/2 + 1/3 is not 2/5

Adding tops and bottoms separately is the single most common AQA Foundation fraction error. The examiner's report for 1F JUN22 Q12 stated that "more than half of the students simply subtracted the numerator values and subtracted the denominator values instead of finding a common denominator" before computing. The same pattern appears on every series — equivalent fractions by "+1 to the top, +1 to the bottom" (1F NOV24 Q3(b)), mixed-number subtraction with stream-by-stream arithmetic (1F NOV24 Q24). One root cause, three surfaces.

The thirty-second fix: a fraction is one number, not two. The denominator names the size of the parts; the numerator counts them. You can only add or subtract once both fractions are measured in the same parts — that is, once they share a common denominator. Get there first, then add the numerators.

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How to spot it in your own work

You answered $\dfrac{2}{5}$ for $\dfrac{1}{2} + \dfrac{1}{3}$ .
You wrote $\dfrac{6}{5} - \dfrac{3}{10} = \dfrac{3}{10}$ by taking 3 off the top and choosing the bigger denominator.
You ticked $\dfrac{2}{10}$ as equivalent to $\dfrac{1}{5}$ because "+1, +5".
You subtracted a mixed number $1\dfrac{1}{5} - \dfrac{3}{10}$ by treating the whole and the fraction as independent.

An exam question that triggers it

Here is a question identical in shape to AQA 1F JUN22 Q12 — the canonical fraction-additive trigger:

Work out:

\dfrac{2}{3} + \dfrac{1}{4}

The misconception answer is $\dfrac{3}{7}$ , found by adding tops ( $2 + 1 = 3$ ) and adding bottoms ( $3 + 4 = 7$ ). Sanity check it: two-thirds is more than half, so the answer must be bigger than half. But $\dfrac{3}{7}$ is less than half. The procedure has produced an answer smaller than one of the inputs — which can't be right for a sum of two positive fractions.

The correct answer is $\dfrac{11}{12}$ . Common denominator 12: $\dfrac{2}{3} = \dfrac{8}{12}$ and $\dfrac{1}{4} = \dfrac{3}{12}$ . Now add the numerators only — the denominator stays at 12, because the parts are the same size on both fractions: $\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$ .

Why students fall for this

The fraction $\dfrac{2}{3}$ is visually two integers stacked over a line. Adding two of these "stacks" looks like a problem of adding two pairs of integers — and the brain reaches for the only tool it has for adding pairs of integers, which is to add them stream by stream. The denominator looks like a label and the numerator looks like the number, so students think they're "adding the numbers and keeping the labels consistent". They aren't — they're changing the parts.

The same shortcut produces the equivalent-fractions error. If $\dfrac{1}{5} = \dfrac{2}{10}$ by "+1 top, +5 bottom", then by the same rule $\dfrac{1}{5} = \dfrac{2}{6}$ by "+1, +1" — but those two answers aren't the same fraction. Equivalence is made by multiplying by 1 in disguise (like $\dfrac{2}{2}$ ), which preserves the size of the fraction. Addition does not.

The fix — Common denominator

A fraction is one number, not two. The denominator names the parts; the numerator counts them. Once both fractions are measured in the same parts — that is, once they share a denominator — adding them is just counting: $\dfrac{a}{n} + \dfrac{b}{n} = \dfrac{a + b}{n}$ . The denominator stays put because the parts are the same size on both fractions. The numerators add because you're counting how many of those parts you now have.

To convert two fractions to a common denominator, multiply each by $1$ in disguise. To turn $\dfrac{2}{3}$ into something with denominator 12, multiply top and bottom by 4: $\dfrac{2}{3} \times \dfrac{4}{4} = \dfrac{8}{12}$ . The size of the fraction is unchanged (because $\dfrac{4}{4} = 1$ ) but it's now measured in twelfths, which makes it addable with anything else in twelfths.

Worked example

Compute $\dfrac{2}{3} + \dfrac{1}{4}$ .

Find a common denominator. 3 and 4 share no common factors, so the safest common denominator is their product: $3 \times 4 = 12$ .
Scale each fraction to twelfths. $\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$ . $\dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12}$ .
Add the numerators. Keep the denominator.
$\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{8 + 3}{12} = \dfrac{11}{12}$
Sanity check. $\dfrac{11}{12}$ is just under 1. $\dfrac{2}{3} \approx 0.667$ and $\dfrac{1}{4} = 0.25$ , so the sum is about $0.917$ , which is also just under 1. The decimals agree with the fraction answer.

Notice the denominator did not change in step 3. That's the single hardest thing to remember after years of adding integer streams. The 12 stays because the parts stayed the same size.

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Common questions

How do I know which common denominator to pick?: If one denominator divides the other (like 4 and 12), use the larger. If they share factors, use their LCM (lowest common multiple). If you can't spot the LCM, just multiply the two denominators together — it always works. $\dfrac{1}{6} + \dfrac{1}{4}$ can use 24 (=6 × 4) or 12 (=LCM). Both give the same answer once you simplify.
How do I subtract mixed numbers like $1\dfrac{1}{5} - \dfrac{3}{10}$ ?: Convert the mixed number to an improper fraction first: $1\dfrac{1}{5} = \dfrac{6}{5}$ . Then find a common denominator: $\dfrac{6}{5} = \dfrac{12}{10}$ . Subtract: $\dfrac{12}{10} - \dfrac{3}{10} = \dfrac{9}{10}$ . AQA flagged the stream-by-stream wrong answer $\dfrac{3}{10}$ (from $6 - 3 = 3$ and picking the bigger denominator) on 1F NOV24 Q24.
How do I check whether $\dfrac{2}{3}$ is equivalent to $\dfrac{3}{4}$ ?: Multiply top and bottom of each by the same factor to reach a common denominator: $\dfrac{2}{3} = \dfrac{8}{12}$ and $\dfrac{3}{4} = \dfrac{9}{12}$ . $\dfrac{8}{12} \ne \dfrac{9}{12}$ , so they are not equivalent. Equivalence is preserved by multiplication by 1, not by addition.
What about fraction multiplication — does the common denominator rule apply?: No. Multiplication and division of fractions do not need a common denominator. $\dfrac{2}{3} \times \dfrac{1}{4} = \dfrac{2 \times 1}{3 \times 4} = \dfrac{2}{12} = \dfrac{1}{6}$ . You multiply tops and multiply bottoms — that is correct for multiplication and only multiplication. The rule that traps people in addition becomes the correct rule for multiplication, which is part of why the addition rule is so over-learned in primary.

Related misconceptions

Decimal place value— Converting fractions to decimals and ordering them — the next place column reasoning bites.
Ratio: additive vs multiplicative scaling— The same one-number-not-two trap, applied to ratios. Same fix: think multiplicatively.
Percentage change family— Percentages are fractions in another costume — the multiplier method generalises.

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