Why Multiplying Can Make a Number Smaller

Because multiplying by a number less than 1 takes only a part of what you started with. Half of 10 is 5, and $10\times \frac{1}{2}$ is the same calculation — so the product is smaller than 10, even though the word "multiply" suggests growth.

This page covers everything in the CBSE Class 7 Mathematics chapter's first half: multiplying fractions by whole numbers and by each other, the area model, cancelling before multiplying, and predicting whether a product will be larger or smaller.

Multiply the whole number by the numerator and keep the denominator, then simplify.

$$\frac{2}{5}\times 3=\frac{6}{5}=1\frac{1}{5}$$

It works in either order, since $3\times \frac{2}{5}$ means three lots of two-fifths.

For mixed numbers, convert to an improper fraction first:

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

In context, the meaning matters as much as the arithmetic. If one glass holds $\frac{3}{4}$ litre of water, then 6 glasses hold

$$\frac{3}{4}\times 6=\frac{18}{4}=4\frac{1}{2}\text{ litres}$$

Here the answer is larger than $\frac{3}{4}$, because we multiplied by 6, a number greater than 1.

So "of" and "×" mean the same thing in these problems. Two-thirds of 15 is $\frac{2}{3}\times 15=10$.

Picture a square as one whole. Shade $\frac{1}{2}$ of it by dividing it vertically. Now take $\frac{1}{3}$ of that shaded half by dividing the square horizontally into three.

The overlap — the part shaded both ways — is $\frac{1}{6}$ of the whole square. So

$$\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$$

The same thing happens with paper folding: fold a sheet in half, then fold it into three the other way, and unfolding shows six equal parts.

The area model shows exactly why the rule is what it is: the vertical cuts multiply with the horizontal cuts, giving $2\times 3=6$ pieces in total.

So for any two fractions, multiply the numerators and multiply the denominators:

$$\frac{2}{3}\times \frac{4}{5}=\frac{2\times 4}{3\times 5}=\frac{8}{15}$$

No common denominator is needed — that requirement belongs to addition and subtraction, and mixing the two procedures is a frequent source of error.

Cancelling common factors before multiplying keeps the numbers small and the simplifying easy.

Take $\frac{4}{9}\times \frac{3}{8}$. The 4 and the 8 share a factor of 4, so they become 1 and 2. The 3 and the 9 share a factor of 3, so they become 1 and 3:

$$\frac{4}{9}\times \frac{3}{8}=\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$$

Multiplying first would have given $\frac{12}{72}$, which reduces to the same $\frac{1}{6}$ — but through much larger numbers.

Cancelling works diagonally or vertically across the multiplication, because everything on top is multiplied together and everything below is multiplied together.

Write the final answer in lowest terms, and convert an improper result to a mixed number unless told otherwise. So $\frac{20}{8}$ becomes $\frac{5}{2}$, then $2\frac{1}{2}$.

The restriction to remember: cancelling like this is only valid for multiplication. You cannot cancel across a $+$ sign.

Compare the multiplier with 1, and you can predict the answer before calculating.

- Multiplying by a number greater than 1 gives a larger result: $\frac{2}{3}\times 4=\frac{8}{3}$, which is bigger than $\frac{2}{3}$.
- Multiplying by a number less than 1 gives a smaller result: $\frac{2}{3}\times \frac{1}{2}=\frac{1}{3}$, which is smaller than $\frac{2}{3}$.
- Multiplying by exactly 1 leaves it unchanged: $\frac{2}{3}\times 1=\frac{2}{3}$.

For example, taking $\frac{3}{4}$ of a 200 g packet leaves 150 g — less than you started with, because you took a part of it rather than several copies.

This is the most useful checking tool in the chapter. If you compute $\frac{5}{6}\times \frac{2}{3}$ and get an answer bigger than $\frac{5}{6}$, something has gone wrong — both multipliers are under 1, so the product must be smaller than either.

It also explains the chapter's title question: multiplication only "makes things bigger" when the multiplier exceeds 1, which is simply not true of proper fractions.