Dividing by a Half Makes Things Bigger

Ask how many half-rotis are in 3 rotis. The answer is 6 — more than you started with, because each roti gives two halves. Written as maths that is $3÷\frac{1}{2}=6$.

Division asks how many of these fit into that, and when "these" are small pieces, many of them fit. This page covers everything in the CBSE Class 7 Mathematics chapter's second half: reciprocals, dividing by whole numbers and by fractions, dividing mixed numbers, and checking the size of your answer.

The reciprocal of a fraction is that fraction turned upside down — numerator and denominator swapped.

So the reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$, and the reciprocal of $\frac{1}{4}$ is $\frac{4}{1}=4$.

The defining property is that a number times its reciprocal always gives 1:

$$\frac{3}{5}\times \frac{5}{3}=\frac{15}{15}=1$$

For a whole number, write it over 1 first. The reciprocal of 7 is $\frac{1}{7}$, since $7=\frac{7}{1}$.

For a mixed number, convert to an improper fraction before flipping. The reciprocal of $2\frac{1}{3}$ is not $2\frac{3}{1}$ — convert to $\frac{7}{3}$, then flip to get $\frac{3}{7}$.

One number has no reciprocal at all: zero. Flipping $\frac{0}{1}$ would give $\frac{1}{0}$, and division by zero is undefined — which is also why you can never divide by zero in any later chapter.

The rule is: to divide by a fraction, multiply by its reciprocal.

$$\frac{2}{3}÷\frac{4}{5}=\frac{2}{3}\times \frac{5}{4}=\frac{10}{12}=\frac{5}{6}$$

The reason is easiest to see with a simple case. $1÷\frac{1}{4}$ asks how many quarters fit into one whole, and the answer is 4 — which is exactly $1\times 4$, one times the reciprocal.

The same logic scales up. $3÷\frac{1}{4}$ asks how many quarters are in 3 wholes: each whole holds 4, so three wholes hold 12, and $3\times \frac{4}{1}=12$.

So flipping is not an arbitrary trick. Dividing by a quarter and multiplying by 4 are the same question asked two ways.

One caution about order: only the second fraction flips. For $\frac{2}{3}÷\frac{4}{5}$, the $\frac{2}{3}$ stays exactly as it is. Flipping both, or flipping the first, changes the question.

Every case uses the same two steps: write both numbers as fractions, then multiply by the reciprocal of the second.

Whole number ÷ fraction. Write the whole number over 1:

$$6÷\frac{2}{3}=\frac{6}{1}\times \frac{3}{2}=\frac{18}{2}=9$$

Fraction ÷ whole number. Write the whole number over 1 and flip it, which divides the fraction into that many parts:

$$\frac{3}{4}÷2=\frac{3}{4}\times \frac{1}{2}=\frac{3}{8}$$

Mixed numbers. Convert both to improper fractions before anything else:

$$2\frac{1}{2}÷1\frac{1}{4}=\frac{5}{2}÷\frac{5}{4}=\frac{5}{2}\times \frac{4}{5}=\frac{20}{10}=2$$

A word problem shows why this matters. If a ribbon $4\frac{1}{2}$ metres long is cut into pieces of $\frac{3}{4}$ metre each, the number of pieces is

$$\frac{9}{2}÷\frac{3}{4}=\frac{9}{2}\times \frac{4}{3}=\frac{36}{6}=6$$

Six pieces — a whole number, as it must be when counting pieces.

Compare the divisor with 1, and the direction is fixed:

- Dividing by a number less than 1 gives a larger result: $5÷\frac{1}{2}=10$.
- Dividing by a number greater than 1 gives a smaller result: $5÷2=2\frac{1}{2}$.
- Dividing by exactly 1 leaves it unchanged.

This is the mirror image of the multiplication rule, and the two together catch most errors in the chapter.

The reason is the "how many fit" reading. Small divisors are small pieces, and many small pieces fit into a given amount — so the count is large.

Use it as a final line. If a question asks how many $\frac{1}{3}$ litre glasses fill a 4 litre jug, your answer must be more than 4, and $4\times 3=12$ passes the check. An answer of $\frac{4}{3}$ would mean you multiplied instead of dividing.

It also explains this page's title: dividing by a half doubles, because two halves fit inside every whole.