Word problems with fractions

Now let us put fractions to work in everyday situations.

Idea

Some common patterns:

Recipe scaling. "A recipe needs $\tfrac{3}{4}$ cup of flour for $6$ rotis. How much for $10$ rotis?" Per roti: $\tfrac{3}{4} \div 6 = \tfrac{1}{8}$ cup. For $10$: $\tfrac{10}{8} = \tfrac{5}{4} = 1\tfrac{1}{4}$ cups.

Travel. "A bus has covered $\tfrac{2}{5}$ of a $250$ km journey. How much is left?" Covered: $\tfrac{2}{5} \times 250 = 100$ km. Left: $250 - 100 = 150$ km.

Splitting bills. "Three friends share a $\rupee 750$ bill equally. How much each?" $750 \div 3 = \rupee 250$. (Equal share: $\tfrac{1}{3}$ each.)

Combining parts. "A bag contains $\tfrac{1}{4}$ apples, $\tfrac{1}{3}$ oranges, rest bananas. What fraction are bananas?" Apples + oranges $= \tfrac{1}{4} + \tfrac{1}{3} = \tfrac{3+4}{12} = \tfrac{7}{12}$. Bananas: $1 - \tfrac{7}{12} = \tfrac{5}{12}$.

Time and work. "A worker finishes $\tfrac{2}{5}$ of a job in $4$ days. How long for the whole job?" $\tfrac{2}{5}$ in $4$ days means $\tfrac{1}{5}$ in $2$ days, so the whole ($1 = \tfrac{5}{5}$) takes $10$ days.

The key skill: read the problem carefully, identify the fraction, decide whether the operation is add, subtract, multiply, divide, and then compute. Drawing a picture often helps.

Worked examples

Example 1. A jug holds $2$ L. After pouring out $\tfrac{3}{8}$ of its contents, how much is left?

Poured out: $\tfrac{3}{8} \times 2 = \tfrac{3}{4}$ L. Left: $2 - \tfrac{3}{4} = \tfrac{5}{4} = 1\tfrac{1}{4}$ L.

Example 2. Mother distributes $\tfrac{3}{4}$ of a cake equally among $3$ children. What fraction does each child get?

$\tfrac{3}{4} \div 3 = \tfrac{3}{4} \times \tfrac{1}{3} = \tfrac{1}{4}$ of the cake each.

Example 3. A piece of ribbon is $7\tfrac{1}{2}$ m long. It is cut into pieces of $\tfrac{3}{4}$ m. How many pieces?

$\tfrac{15}{2} \div \tfrac{3}{4} = \tfrac{15}{2} \times \tfrac{4}{3} = \tfrac{60}{6} = 10$ pieces.

Example 4. Aman spent $\tfrac{1}{4}$ of his money on books and $\tfrac{2}{5}$ on toys. What fraction is left?

Spent: $\tfrac{1}{4} + \tfrac{2}{5} = \tfrac{5}{20} + \tfrac{8}{20} = \tfrac{13}{20}$. Left: $1 - \tfrac{13}{20} = \tfrac{7}{20}$.

Try it yourself

1. A pizza is divided into $8$ slices. Ravi eats $3$ slices, Mira eats $2$. What fraction is left?
2. A car has travelled $\tfrac{3}{8}$ of a $400$ km trip. How much remains?
3. A bottle holds $\tfrac{3}{4}$ L. Half is poured out. How much is left?
4. A rope $9\tfrac{1}{2}$ m long is cut into pieces of $\tfrac{1}{2}$ m. How many pieces?
5. From a $\rupee 1500$ budget, $\tfrac{1}{3}$ is spent on rent and $\tfrac{1}{6}$ on travel. Money left?
6. A worker does $\tfrac{1}{3}$ of a job in $6$ days. Days to finish?
7. A class has $40$ students; $\tfrac{3}{5}$ are girls. How many boys?
8. Three friends share $5\tfrac{1}{4}$ kg of mangoes equally. Each gets?

Activity

Take any recipe in your kitchen. Scale it up by $\tfrac{3}{2}$ (i.e., make $1.5$ times the recipe). Write the scaled quantity for each ingredient, using fractions or mixed numbers.