Dividing Quantities by Fractions

1. How many pieces fit? Quantity ÷ fraction

In the last lesson you learned that dividing by a fraction asks how many of that fraction fit into the number. Real problems use the same idea with quantities: the stack weighs 10 kg and each plate is $\frac{1}{4}$ kg, so the number of plates is $10÷\frac{1}{4}$.

Multiply the dividend and the divisor by the reciprocal of the divisor (which is 4): $10÷\frac{1}{4}=(10\times 4)÷(\frac{1}{4}\times 4)=40÷1=40$. In short, $10\times 4=40$ plates. That makes sense: every kilogram holds 4 quarter-kilogram plates, and there are 10 kilograms. Dzifa has enough plates.

1. 1Decide what is being divided (the whole quantity) and what it is divided by (the size of one piece or one share).
2. 2Write the division with both numbers as common fractions.
3. 3Multiply the first fraction by the reciprocal of the second, cancel and simplify.
4. 4Check by multiplying back, and write the answer with its unit or in words (plates, packets, learners).

2. Sharing a quantity equally

Sometimes we know how many people share and want the size of each share. *A class was given $9\frac{1}{2}$ litres of fruit juice to share equally. If there are 38 learners, how many millilitres does each learner get?* Each share is $9\frac{1}{2}÷38$.

Write $9\frac{1}{2}=\frac{19}{2}$. Then $\frac{19}{2}÷38=\frac{19}{2}\times \frac{1}{38}=\frac{19}{76}=\frac{1}{4}$ litre. The question asks for millilitres, and 1 litre = 1,000 ml, so each learner gets $\frac{1}{4}\times 1000=250$ ml. Another good route is to change to millilitres first: 9,500 ml ÷ 38 = 250 ml.

▶ Change $9\frac{1}{2}$ litres to 9,500 ml first, then long division by 38 gives 250 ml for each learner.

3. Finding the whole from a part

A powerful use of division by a fraction is finding a whole when you only know a part. Suppose a graph of the ages of learners in a Primary 5 class shows that the twelve 10-year-olds make up $\frac{3}{8}$ of the class. How many learners are in the class?

We know that $\frac{3}{8}$ of the class = 12, so the class size is $12÷\frac{3}{8}=12\times \frac{8}{3}=32$. The bars below show why: 3 of the 8 equal parts hold 12 learners, so one part holds 12 ÷ 3 = 4 learners, and all 8 parts hold 8 × 4 = 32.

▶ Notice the two moves: divide by the numerator (3) to find one part, then multiply by the denominator (8) to rebuild the whole. That is the same as multiplying by $\frac{8}{3}$.

The same idea works for money. If GH₵45 is $\frac{3}{5}$ of Kojo's savings, his total savings are $45÷\frac{3}{5}=45\times \frac{5}{3}=$ GH₵75. Check: $\frac{3}{5}$ of 75 = 45 ✓.

4. A fraction ÷ a fraction, and answers that are not whole

The quantity being divided can itself be a fraction. *Ama has $\frac{3}{4}$ of a cake. Each guest gets $\frac{1}{8}$ of a cake. How many guests can she serve?* $\frac{3}{4}÷\frac{1}{8}=\frac{3}{4}\times \frac{8}{1}=6$ guests.

Real answers are not always whole numbers, and then you must think about the situation. A tailor in Kejetia has 10 m of cloth and each dress needs $2\frac{1}{3}$ m. $10÷2\frac{1}{3}=10\times \frac{3}{7}=\frac{30}{7}=4\frac{2}{7}$. She cannot sell $\frac{2}{7}$ of a dress, so she can make 4 complete dresses. They use $4\times 2\frac{1}{3}=9\frac{1}{3}$ m, leaving $10-9\frac{1}{3}=\frac{2}{3}$ m of cloth.

Worked examples

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