Division by Fraction

02 Feb 2020

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I stumbled upon a question recently, When dividing a number by a fraction, why do we invert and multiply?. We all know the answer intuitively - It produces the same result.

Do we know why it provides the same result?

Let us get the basics straight.

Multiplication is the repeated addition of the same number

Division is the splitting of numbers into equal parts

Let us use even numbers to explain the concepts.

Problem 1

4 ÷ 2 = 2

If we split the number 4 into groups of 2, how many such groups can be created?

4 divided by 2 equals 2 groups.

Problem 2

4 ÷ 4 = 1

If we split the number 4 into groups of 4, how many such groups can be created?

4 divided by 4 equals 1 group.

Problem 3

4 ÷ 1 = 4

If we split the number 4 into groups of 1, how many such groups can be created?

4 divided by 1 equals 4 groups.

So far, everything is clear.

What happens when we divide the number by a fraction?

Problem 4

4 ÷ (1/2) = 8

If we split the number 4 into groups of halves, how many such groups can be created?

Group of halves? We only have whole numbers.

Ok then, let’s split the whole number in to halves and then group them by halves.

Check the shapes in the below table.

4 divided by 1/2 equals 8 groups.

When we divide a number with a whole number, we just group by the whole number.

When we divide a number with a fraction, we split the number to fractions first and then group them.

PS: Replace the word divide with group. Instead of saying, “divide by 2”, we could say, “group by 2”, which is more intuitive in English.

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