Solving Word Problems with Fractions

1. Reading the problem: add or subtract?

The hardest part of a word problem is often deciding what to do, not the calculation itself. Good problem solvers read the story twice, picture it, and look for clue words. The table below shows the most common clue words — but always think about the story too, because clue words can sometimes mislead.

1. 1Understand: read twice. What do you know? What are you asked to find? Underline the numbers and their units.
2. 2Plan: decide the operation(s) and write a number sentence, e.g. $12\frac{1}{4}$ − $3\frac{1}{3}$ = ?
3. 3Solve: use a common denominator, add or subtract the wholes and fractions, regroup if needed, simplify.
4. 4Check: estimate with whole numbers or benchmarks, then answer in a full sentence with the unit.

2. "What fraction is left?" problems

Many problems describe parts of one whole: a salary, a farm, a tank of water. If you know the fractions that were used, the fraction left is 1 − (sum of the parts used). Remember that 1 can be written as any fraction with the same top and bottom, such as $\frac{12}{12}$.

For example, Mrs Owusu spends $\frac{1}{4}$ of her salary on transport and $\frac{1}{3}$ on food. Fraction spent = $\frac{1}{4}$ + $\frac{1}{3}$ = $\frac{3}{12}$ + $\frac{4}{12}$ = $\frac{7}{12}$. Fraction left = $\frac{12}{12}$ − $\frac{7}{12}$ = $\frac{5}{12}$.

▶ The fraction left is always the whole (12/12 here) minus everything that was used.

3. Measurement problems with mixed numbers

Lengths, distances, masses and times are often given as mixed numbers. The method is the same as before: common denominator, wholes and fractions, regroup when needed. Let us solve Uncle Kojo's problem: the board is $12\frac{1}{4}$ ft and $3\frac{1}{3}$ ft is cut off. The word "cut off" means subtract.

LCM of 4 and 3 is 12, so $12\frac{1}{4}$ − $3\frac{1}{3}$ = $12\frac{3}{12}$ − $3\frac{4}{12}$. We cannot take $\frac{4}{12}$ from $\frac{3}{12}$, so regroup: $12\frac{3}{12}$ = $11\frac{15}{12}$. Then $11\frac{15}{12}$ − $3\frac{4}{12}$ = $8\frac{11}{12}$ ft. That is just short of 9 ft, so the remaining piece is not long enough for a 9-foot bench.

▶ The remaining length lands just before 9 on the line — an estimate that agrees with the exact answer.

4. Checking and reasoning with benchmarks

Always ask: is my answer reasonable? Round each fraction to a nearby benchmark (0, $\frac{1}{2}$ or 1) or each mixed number to the nearest whole. For the board, 12 − 3 = 9, and since we cut a bit more than 3 from a bit more than 12, an answer close to 9 makes sense. If you had got $15\frac{7}{12}$ ft, the estimate would tell you at once that you added instead of subtracting.

Benchmarks also help you reason without calculating. Suppose you add two fractions (each less than 1) and the sum is greater than 1. What can you say about them? If both were $\frac{1}{2}$ or less, the sum could be at most $\frac{1}{2}$ + $\frac{1}{2}$ = 1. So **at least one of the fractions must be greater than $\frac{1}{2}$**. For example $\frac{3}{4}$ + $\frac{2}{5}$ = $1\frac{3}{20}$, and $\frac{3}{4}$ is more than a half.

Worked examples

Try each one yourself first, then open the solution and compare every step.