Fraction Word Problems: Choosing the Right Operation

$\tfrac{5}{8}$ of the pizza

“Together” means add. $\tfrac{1}{4} = \tfrac{2}{8}$, so $\tfrac{2}{8} + \tfrac{3}{8} = \tfrac{5}{8}$.

$\tfrac{1}{3}$ of a litre

“Remains” means subtract. Over an LCD of $6$: $\tfrac{5}{6} - \tfrac{3}{6} = \tfrac{2}{6} = \tfrac{1}{3}$.

Look for the word “of”.

$24$ students

$\tfrac{3}{4} \times 32 = \tfrac{96}{4} = 24$.

$3$ pieces

“How many fit” means divide. $\tfrac{9}{10} \div \tfrac{3}{10} = \tfrac{9}{10} \times \tfrac{10}{3} = \tfrac{90}{30} = 3$.

$1\tfrac{1}{4}$ litres

Sharing equally means divide: $5 \div 4 = \tfrac{5}{4} = 1\tfrac{1}{4}$ litres per room.

Half of what she had, not half a cake.

$\tfrac{1}{3}$ of a cake

She gave away $\tfrac{1}{2} \times \tfrac{2}{3} = \tfrac{2}{6} = \tfrac{1}{3}$.

So she has $\tfrac{2}{3} - \tfrac{1}{3} = \tfrac{1}{3}$ left. Giving away half of something always leaves the other half, which is a quicker route to the same answer.

$\tfrac{1}{3}$ of a cup

$\tfrac{1}{2} \times \tfrac{2}{3} = \tfrac{2}{6} = \tfrac{1}{3}$ of a cup.

The part is known and the whole is not.

$800$ grams

$600 \div \tfrac{3}{4} = 600 \times \tfrac{4}{3} = \tfrac{2400}{3} = 800$ grams.

Check: $\tfrac{3}{4} \times 800 = 600$, which matches.

$\tfrac{5}{8}$ of a mile

“How much further” means subtract. Over an LCD of $8$: $2\tfrac{2}{8} - 1\tfrac{5}{8}$.

Borrow one whole: $2\tfrac{2}{8} = 1\tfrac{10}{8}$, so $1\tfrac{10}{8} - 1\tfrac{5}{8} = \tfrac{5}{8}$ of a mile.

Two steps: how much water, then how many bottles.

$16$ bottles

Step 1, the amount of water: $\tfrac{2}{3} \times 12 = 8$ litres.

Step 2, the number of bottles: $8 \div \tfrac{1}{2} = 8 \times 2 = 16$.

The first answer shrank because you took a part of the tank. The second grew because you counted how many small bottles fit. Both movements are correct, and noticing them is the fastest check on the work.