Division With Number Line

As we know multiplication can be viewed as repeated addition of the same number. For example,

Today we will see how to view division as a repeated subtraction. Let us try to understand how.

We know that $23 \div 5$ gives a quotient of $4$ and a remainder of $3$.

Let try to subtract 5 from 23 for as many times as we can.

Now we are left with $3$, which is less than $5$, subtracting $5$ anymore from it will give a number less than $0$, so we will not subtract it any more.

How many times could we subtract $5$ from $23$? We subtracted if $4$ times. Therefore the quotient is $4$ and we were left with $3$ when we stopped subtracted, that is how we get $3$ as the remainder.

Using the number line, we can draw these operations as shown below:

$14 - 9 = 5$ Now we are left we $5$ and we cannot subtract $9$ anymore from it.

As we can count from above, we could do the subtraction $4$ times.

Therefore we get a quotient of $4$ and a remainder of $5$, and we know that

If you use the number line to draw the same division operation, you get a figure like below:

We can find the closest divisible number to a given divisor, either greater or lesser, using the following ways:

$\underline{Closest\ divisible\ number,\ lesser\ than\ the\ divisor}$

- By subtracting the extra value (the remainder) from the dividend. That way on the number line there will be an exact number of leaps starting at $0$ up to the dividend. In this case we are subtracting the remainder which is always less than the divisor.

$\underline{Closest\ divisible\ number,\ greater\ than\ the\ divisor}$

- By adding the difference of the divisor and the remainder, that is, $divisor - remainder$ to the dividend. That way also, there will be an exact number of leaps from $0$ to the dividend. In this case we are adding $divisor - remainder$ which should always be less than the divisor.

You should observe that in either of the ways, the number to be subtracted or added will be less than the divisor.

In our example of $41 \div 9$ the remainder is $5$. The closest number, less than $41$, that is divisible by $9$ is $41 - 5 = 36$.

The closest number, greater than $41$, that is divisible by $9$ is $41 + 4 = 45$. In this case we added $divisor - remainder = 9 - 5 = 4$ to the dividend $41$. In both cases the number to be added or to be subtracted should always be less than the $divisor$.