Dividing Fractions
Has A Weird Rule
Dividing fractions can be a little tricky. It's the only operation that requires using the reciprocal. Using the reciprocal simply means you
flip it over, or invert it.
For example, the reciprocal of 2/3 is 3/2.
After we give you the rule, we will attempt to explain WHY you have to
use the reciprocal in the first place. But for now...
Here's the Rule for division...
To divide fractions, convert the division process to a multiplication
process by using the following steps.
Change the "÷" sign to "x" and invert the fraction
to the right of the sign.
Multiply the numerators.
Multiply the denominators.
Re-write your answer in its simplified or reduced form, if needed
Once you complete Step #1 for dividing fractions, the problem actually changes from division to multiplication.
1/2 ÷ 1/3 = 1/2 x 3/1
1/2 x 3/1 = 3/2
Simplified Answer is 1 1/2
Now that's all there is to it. The main thing you have to remember when
you divide is to invert the fraction to the right of the division sign, and change the sign to multiplication.
The "divisor" (like 1/3 in our example) has some other consideration that you should keep in mind...
Special Notes!
- Remember to only invert the divisor.
- The divisor's numerator or denominator can not be "zero".
- We must convert the operation to multiplication BEFORE performing an cancellations.
I promised to try to explain why the rule requires inverting the divisor.
Here goes..
Why Dividing Fractions
Requires Inverting The Divisor
Let's use our simple example to actually validate this strange Rule for
division.
If you really think about it, we are dividing a fraction by a fraction, which forms what is called a "complex fraction". It actually looks like this...
When working with complex fractions, what we want to do first is get rid of the denominator (1/3), so we can work this problem easier.
You may recall that any number multiplied by its reciprocal is equal to 1.
And since, 1/3 x 3/1 = 1, we can use the reciprocal property of 1/3 (3/1) to make the value of the denominator equal to 1.
But, you might also remember that whatever we do to the denominator, we must also do to
the numerator, so as not to change the overall "value".
So let's multiply both the numerator and denominator by 3/1...
Which gives us...
Here's what happened...
By multiplying the numerator and denominator by 3/1, we were then able to use the reciprocal property to eliminate the
denominator. Actually, without our helpful Rule,
we would have to use all of the steps above.
So, the Rule for dividing fractions really saves us a lot of steps!
Now that's the simplest explanation I could come up with for WHY and HOW we end up with a
Rule that says we must invert the divisor!
Best regards,
