Why do you flip fractions when dividing fractions?

Answer 1

Flipped fractions are called reciprocals.

We flip when we reposition a fraction from below the primary dividing line to above the primary, or vice versa, to simplify a calculation.

Short answer: to simplify an expression that includes a fraction in its denominator or in its numerator.

A flipped fraction is a shortcut version of a two step process of moving a fraction from a denominator to a numerator, or vice versa, to help us calculate the answer.

To illustrate the long form of calculation :

you can see that

10 / 1/5 = 10 / 0.2 = 100 / 2 = 50

Restating this example question to help you understand what is going on :

10 x [ 1/ 1/5 ]

which is merely separating the fraction in the denominator from the rest of the question by multiplying the fractional denominator [ alone ] by 1 or,

more accurately, the whole expression given by 1/1. This has not changed the value of the final answer in any way.

Let's make this very clear by considering ONLY the fractional denominator for a moment.

1 / 1/5 times 1 / 1

or

1 x 1 / 1 x 1/5

Now we can simplify this ugly-looking expression by relocating the lower level of denominator, the 5 , up to the primary numerator level, giving :

[ ignoring the dots ]

1 . 5

_ x _ . which is simplified to : 5 / 1 or 5

1 . 1

Now you already know that the final answer to the original question is 50

or 10 x 5

What we did there was MOVE the fraction in the denominator UP to the numerator without changing the overall value of the expression.

We did that by reversing the fraction's numerator and denominator [ flipping both values ] when we repositioned the fraction up to the main numerator position.

By it not changing the final answer when we did this, we have proven that the reciprocal of a fraction can replace [ or substitute for ] the fraction when the fraction is relocated above or below the primary dividing line, without affecting the final answer.

Sounds confusing, doesn't it ?

Just remember that

A over B/C is always equal to A x C/B

C/B is called the reciprocal of B/C

and, of course, B/C can be termed the reciprocal of the fraction C/B.

Dividing by a fraction is the same, mathematically, as multiplying by its reciprocal [flipping the fraction].
There are perhaps two questions here:

• Why does it work?
• Why do we do it?

I'll try to answer both questions:

What you are really doing is multiplying the numerator and denominator by the same number (the flipped denominator). Multiplying this way does not change the value of the original problem by one of the rules of arithmetic (which we can prove). When you choose this flipped fraction to multiple by, you will see that it does something interesting to the denominator, mainly, it turns it into the value of one after you multiply and simplify. Since any number divided by 1 is itself, this simplifies the original fraction and turns the problem into the flipped fraction form.

We do all this because the rule for multiplying fractions together is very simple: multiply the numerators together to get the numerator of the answer and multiply the denominators together to get the denominator of the answer.