Dividing by whole numbers
In a division such as $9 ÷ 2$, the number being divided (9) is called the dividend, the number we divide by (2) is the divisor, and the result is the quotient.
When the divisor is a whole number, we can divide in the usual way. The quotient does not have to be a whole number – it can be a decimal number.
The number 2 goes into 9 four whole times ($4 × 2 = 8$), and 1 is left over. We do not stop at the remainder – we keep dividing. The remaining 1 is half of 2, and half is $0.5$, so the result is $4 + 0.5 = 4.5$.
The same rule works when the dividend is already a decimal number. We divide as usual and keep the decimal point in the quotient directly above the decimal point in the dividend:
Here $4 ÷ 4 = 1$ and $0.8 ÷ 4 = 0.2$, so together we get $1.2$.
Dividing by decimal numbers
When the divisor is a decimal number, dividing directly is awkward. It is much easier to first change the divisor into a whole number. To do this, multiply both the dividend and the divisor by the same number: 10, 100, 1000 and so on – whichever is needed to remove the decimal point from the divisor.
Why does this not change the result? A division is like a fraction: $7 ÷ 3.5$ is the same as $7/3.5$. If we multiply the numerator and the denominator by the same number, the fraction keeps the same value – exactly as $1/2$ and $10/20$ are equal. So the quotient stays the same, only the numbers become easier to work with.
For example:
We multiplied both numbers by 10. The divisor 3.5 became the whole number 35, and $70 ÷ 35$ is easy: the answer is 2. Check: $2 × 3.5 = 7$
Moving the decimal point in both numbers
Instead of thinking about multiplication by 10, 100 or 1000, we can imagine moving the decimal point in both numbers by the same number of places to the right. Multiplying by 10 moves the point one place, by 100 two places, by 1000 three places.
A simple rule: count the decimal places in the divisor. That is how many places you must move the decimal point in both numbers.
For example:
The divisor 0.35 has two decimal places, so we move the decimal point two places to the right in both numbers. $0.35$ becomes $35$, and $0.7$ becomes $70$ (we add a zero, because $0.7 = 0.70$):
It is important to move the decimal point by the same number of places in both the dividend and the divisor. If you move it by different amounts, the result will be wrong.
Dividing as fractions
There is another way to see the same idea. Every decimal number can be written as a fraction with 10, 100 or 1000 in the denominator:
To divide by a fraction, we multiply by its reciprocal – we turn the second fraction upside down and multiply:
We get the same result, $2$, as before. Look at the middle step: the 100 from the denominator of $0.35$ moves up and cancels the 10 from the denominator of $0.7$. That is exactly why "moving the decimal point in both numbers" works – the powers of ten cancel each other.
One more example:
Both denominators are 10, so they cancel completely and only $48 ÷ 12 = 4$ remains.
Finding the easy division first
Sometimes the divisor is a whole number, but the dividend is a small decimal, for example $0.07 ÷ 35$. Nobody can do this directly in their head. Or the numbers become large after we move the decimal point, for example $7 ÷ 0.035 = 7000 ÷ 35$.
In such cases we use a two-step method:
Step 1. Ignore the decimal point and the zeros for a moment and find the easy division hidden inside. In all the examples below it is $70 ÷ 35 = 2$.
Step 2. Fix the decimal point in the result. Two rules tell us how:
| Rule | Example |
| If we made the dividend 10 times smaller, the quotient is also 10 times smaller – move the decimal point one place to the left. Dividing a smaller number gives a smaller result. |
$70 ÷ 35 = 2$ $7 ÷ 35 = 0.2$ $0.7 ÷ 35 = 0.02$ |
| If we made the divisor 10 times smaller, the quotient is 10 times larger – move the decimal point one place to the right. Dividing by a smaller number gives a larger result. |
$70 ÷ 35 = 2$ $70 ÷ 3.5 = 20$ $70 ÷ 0.35 = 200$ |
Now let us use both steps on the two difficult examples.
Example 1: $0.07 ÷ 35$. The easy division is $70 ÷ 35 = 2$. But our dividend is not 70 – it is $0.07$, which is 1000 times smaller ($70 → 7 → 0.7 → 0.07$, three places). So the quotient must also be 1000 times smaller than 2. We move the decimal point three places to the left:
Check: $0.002 × 35 = 0.07$
Example 2: $7 ÷ 0.035$. First we make the divisor a whole number by moving the decimal point three places in both numbers: $7000 ÷ 35$. The easy division is again $70 ÷ 35 = 2$, but our dividend 7000 is 100 times larger than 70 ($70 → 700 → 7000$, two places). So the quotient is 100 times larger than 2 – we move the decimal point two places to the right:
Check: $200 × 0.035 = 7$
Compare this with $7 ÷ 0.35 = 700 ÷ 35 = 20$. Notice that a smaller divisor gives a larger quotient: dividing 7 by 0.035 gives a much larger result than dividing 7 by 0.35. This makes sense: the smaller the pieces we cut 7 into, the more pieces we get.
Long division
When the quotient is not obvious, we use long division. The method is the same as for whole numbers – we only need to take care of the decimal point.
First, we make the divisor a whole number. The divisor 3.2 has one decimal place, so we multiply both numbers by 10:
Now we can use long division. Since 32 is larger than 6, the number 32 goes into 6 zero times, so the quotient starts with 0 followed by a decimal point.
To continue, we write 6 as
Therefore:
Check: $0.1875 × 3.2 = 0.6$
