Dividing decimals by decimals

When dividing decimals by decimals, such as 45.89 ÷ 0.006, we are told to move the decimal point in both the dividend and the divisor so many steps that the divisor becomes a whole number. Then, you use long division. But why?

Schoolbooks often don't tell us the "why", just the "how".

This video explores this concept.


Divide decimals - why do we move the decimal point?


It has to do with the fact that when we move the decimal point, we are multiplying both numbers by 10, 100, 1000, or some other power of ten. When the dividend and the divisor are multiplied by the same number, the quotient does not change. This principle makes sense:

0.344 ÷ 0.004 can be thought of, "How many times does four thousandths fit into 344 thousandths?"

The same number of times as what four fits into 344!

So, 0.344 ÷ 0.004 can be changed into the division problem 344 ÷ 4 without changing the answer. Both 0.344 and 0.004 got multiplied by 1000.

When we simplify fractions or write equivalent fractions, we use the same principle. Remember, fractions are like division problems.

3/7 is 3 ÷ 7.

And, 3/7 = 6/14. We can multiply the numerator and the denominator (or the dividend and the divisor) by some same number, without changing the value of the fraction (the quotient).

Or, 90/100 = 9/10. Or 0.9/0.2 = 9/2.