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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 does it provide 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.

4/2 = 2

(i.e.) If we split the number 4 into groups of 2, how many such groups can be created?

▢ ▢ ▢ ▢ | divide by 2 | ▢ ▢ | ▢ ▢ |
---|---|---|---|

4 | → | 1 | 2 |

Four divided by two equals two.

4/4 = 1

(i.e.) If we split the number 4 into groups of 4, how many such groups can be created?

▢ ▢ ▢ ▢ | divide by 4 | ▢ ▢ ▢ ▢ |
---|---|---|

4 | → | 1 |

Four divided by four equals one.

4/1 = 4

(i.e.) If we split the number 4 into groups of 1, how many such groups can be created?

▢ ▢ ▢ ▢ | divide by 1 | ▢ | ▢ | ▢ | ▢ |
---|---|---|---|---|---|

4 | → | 1 | 2 | 3 | 4 |

Four divided by one equals four.

So far, everything is clear. What happens when we divide the number by a fraction?

4/(1/2) = 8

(i.e.) If we split the number 4 into groups of halves, how many such groups can be created?

What is a group of half numbers? Simple. Split a single number into two equal parts. Put one half in one group.

▢ ▢ ▢ ▢ | divide by half (1/2) | ◸ | ◿ | ◸ | ◿ | ◸ | ◿ | ◸ | ◿ |
---|---|---|---|---|---|---|---|---|---|

4 | → | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Four divided by 1/2 equals eight.

Please ignore the shape and size of the objects. Just count the numbers.

When we divide a number, we are not just grouping the numbers. We split the numbers first and then grouped them. So, the number increases (multiplied).

Let us see what happens when we invert the fraction and multiply.

4 / (1/2) = 4 X 2 = 8

▢ ▢ ▢ ▢ | multiply by 2 | ▢ ▢ ▢ ▢ + ▢ ▢ ▢ ▢ |
---|---|---|

4 | → | 8 |

Multiplication adds the number to itself. Division by a fraction splits the numbers and groups it. So, when we divide a number by a fraction or inverse the fraction and multiply the number, the result is the same.

I hope this explanation helps.

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