Dividing mixed fractions is the same as multiplying by the reciprocal or inverse.

## Dividing Fractions:

To divide a fraction by a fraction to divide a fraction by a whole number (except for zero!) or divide a fraction by a mixed number, the procedure is straightforward:

*“Invert (turn upside down) the divisor and then multiply.”*

When you look at the division symbol, **÷, **you are looking at the structure of a fraction, because fractions are division problems. The bottom number in a fraction, the denominator, tells you what group is being used. The top number, the numerator, tells you the quantity of that group that is present. So, the fraction 1/2 (one-half), can be said in words such as, “How many groups of 2 are there in 1?” The answer is “one half a group of 2 is in 1.” This is why all fractions that have the same number in their numerator and denominator always equal one. How many groups of 2 are there in 2? There is one group of 2 in 2. How many groups of a million are there in a million? There is one group of a million in a million. Conversely, the fraction 10/5 = 2 because there are two groups of five in the number ten. Using division nomenclature, we can also express this as “1 divided by 2 equals 1/2” or “10 divided by 5 equals 2”.

So, now let’s demonstrate why dividing a fraction by a fraction yields a larger number. We can divide 1/2 by 1/8 for example. Using the words above, this can be stated as “How many 1/8ths are there in 1/2?” There are four groups of 1/8 in one-half. Having a pizza handy would instantly make this clear. A pizza cut into 8 slices means each slice is 1/8th of the pie. How many slices (1/8ths) are there in 1/2 the pizza? There are 4, of course. The bottom line is, given the same numerator, the smaller the denominator is, the higher the result will be; since you are making the group smaller, there will be more complete groups in the numerator. Let’s go back to the pizza again. This time, the pizzeria cut the pie you ordered into 16 slices. Now, one slice is 1/16th of the pie. So, how many slices (1/16ths) are there in 1/2 the pie? 8! Because the group is now smaller (1/16th instead of 1/8th), the same amount of pizza (1/2 the pie) will have more complete groups (8 instead of 4). If the pie was cut into 100 slices, how many 1/100ths would there be in 1/2 the pie? You’re right, there would be 50.

*Examples*:

1/2 ÷ 2/3 = 1/2 × 3/2 = (1 × 3)/(2 × 2) = **3/4**

1/2 ÷ 6 = 1/2 × 1/6 = (1 × 1)/(2 × 6) = **1/12**

3/4 ÷ 1¾ = 3/4 ÷ 7/4 = 3/4 × 4/7

= 3/4 × 4/7

= (3 × 1)/(1 × 7) (the 4’s cancelled each other)

= **3/7**

## Dividing Mixed Fractions:

Consider the example of dividing mixed fractions, Justin’s favorite hiking trail is 6^{1}_{2} miles long, and takes 234 hours^{3}_{4}hours to complete. How many miles is Justin covering, on average, per hour?

Since we are looking for an answer with the units of miles per hour, we must divide the given miles by the given hours:

6^{1}/_{2} ÷ 2^{3}/_{4}

let’s convert each mixed number to an improper fraction and rewrite the problem:

^{13}/_{2} ÷^{11}/_{4}

we change the division of a fraction to multiplication of the inverse (or reciprocal):

^{13}/_{2}• ^{4}/_{11}

Noting that we can cross-cancel the 2 with the 4, we can simplify the problem to

^{13}/_{1}•^{2}/_{11} = 26/11

= 2^{4}/_{11} miles/hour

**Note:** when performing a division of fractions, you should not perform any cross cancelling until you have converted the problem to a multiplication of fractions.

## Dividing Fractions with Variables:

26 ÷ ^{𝑥}/_{3}= 26•^{3}/_{x} = ^{78}/_{x}

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