# Dividing Fractions: Reciprocal Numbers

In this lesson we study reciprocal numbers and how to use them in fraction divisions. We also try to make sense of the reciprocal numbers and division of fractions visually.

The video below explains the reciprocal numbers and the "rule" for dividing fractions conceptually.

First, let’s review a little.

How many times does one number go into another?

From this situation, you can always write a division.

Yes — EVEN if the numbers are fractions! Ask:
“How many times does the divisor go into the dividend?”

How many times does  go into ?

 Three times. We write the division:  2 ÷ 2 3 = 3.
 Then check the division:  3 × 2 3 = 6 3 = 2.

1. Solve. Write a division. Then write a multiplication that checks your division.

 a. How many times does go into ?
 3 ÷ 1 3 = _____
 Check: ____ × 1 3 =
 b. How many times does go into ?
 ÷ = _____
 Check: ____ × =
 c. How many times does go into ?
 ÷ = _____

Check:

 d. How many times does go into ?
 ÷ = _____

Check:

2. Solve. Think how many times the divisor goes into the dividend. Can you find a pattern or a shortcut?

 a.   3 ÷ 1 6 =
 b.   4 ÷ 1 5 =
 c.   3 ÷ 1 10 =
 d.   5 ÷ 1 10 =
 e.   7 ÷ 1 4 =
 f.    4 ÷ 1 8 =
 g.   4 ÷ 1 10 =
 h.   9 ÷ 1 8 =

The shortcut is this:
 5 ÷ 1 4 ↓ ↓ 5 × 4 = 20
 3 ÷ 1 8 ↓ ↓ 3 × 8 = 24
 9 ÷ 1 7 ↓ ↓ 9 × 7 = 63

That is, multiply the number by the reciprocal of the divisor. Notice that 1/4 upside down,
or inverted, is 4/1 or just 4. We call 1/4 and 4 reciprocal numbers, or just reciprocals.

Does the shortcut make sense to you? For example, consider the problem 5 ÷ (1/4).
Since 1/4 goes into 1 exactly four times, it must go into 5 exactly 5 × 4 = 20 times.

Two numbers are reciprocal numbers (or reciprocals) of each other if, when multiplied, they make 1.
 3 4 is a reciprocal of 4 3 , because 3 4 × 4 3 = 12 12 =  1.
 1 7 is a reciprocal of 7, because 1 7 × 7 = 7 7 =  1.
 You can find the reciprocal of a fraction m n by flipping the numerator and denominator: n m .
 This works, because m n × n m = n × m m × n =  1.

To find the reciprocal of a mixed number, first write it as a fraction, then “flip” it.

 Since 2 3 4 = 11 4 , its reciprocal number is 4 11 .

3. Find the reciprocal numbers. Then, write a multiplication with the given number and its reciprocal.

 a. 5 8
 5 8 × =  1
 b. 1 9
 × =  1
 c.  1 7 8
 × =  1

d.   32

 32 × =  1
 e.  2 1 8
 × =  1

4. Write a division sentence to match with each multiplication above.

 a. 1 ÷ =
 b. 1 ÷ =
 c. 1 ÷ =
 d. ___ ÷ =
 e. ___ ÷ =

Read, and try to understand. This is important!

Let’s now try to make sense of the reciprocal numbers and division of fractions visually.

Thinking of the division problem 1 ÷ (2/5), we ask, first of all, how many times does 2/5 fit into 1?

 Using pictures: How many times does go into ?
 From the picture we can see that goes into two times, and then we have 1/5 left over.
 But, how many times does 2 5 fit into the leftover piece, 1 5 ? How many times does go into ?

That is like trying to fit a TWO-part piece into a hole that holds just ONE part.
Only 1/2
of the two-part piece fits! So, 2/5 fits into 1/5 exactly half a time.

 In total, we find that 2/5 fits into one exactly 2 1/2 times. We can write the division  1 ÷ 2 5 =  2 1 2 or 5 2 .
 Notice, we got  1 ÷ 2 5 = 5 2 . Checking that with multiplication, we get 5 2 × 2 5 = 1. Reciprocals again!

One more example. Thinking of the division problem 1 ÷ (5/7), we ask how many times does 5/7 fit into 1?

 Using pictures: How many times does go into ?
 From the picture we can see that goes into just once, and then we have 2/7 left over.
 But, how many times does 5 7 fit into the leftover piece, 2 7 ?  How many times does go into ?
The FIVE-part piece fits into a hole that is only big enough for two parts just 2/5 of the way.
 So in total, 5/7 fits into one exactly 1 2/5 times. The division is  1 ÷ 5 7 =  1 2 5 or  1 ÷ 5 7 = 7 5 .

5. Solve. Think how many times the given fraction fits into one whole. Write a division.

 a.  How many times does go into ?

 1 ÷ =
 b.  How many times does go into ?

 1 ÷ =
 c.  How many times does go into ?

 1 ÷ =
 d.  How many times does go into ?

 1 ÷ =
 e.  How many times does go into ?

 1 ÷ =
 f.  How many times does go into ?

 1 ÷ =

6. Solve. Think how many times the given fraction fits into the other number. Write a division.

 a.  How many times does go into ?

 2 ÷ =
 b.  How many times does go into ?

 ÷ =
 c.  How many times does go into ?

 3 ÷ =
 d.  How many times does go into ?

 ÷ =

### SHORTCUT: instead of dividing, multiply by the reciprocal of the divisor.

Study the examples to see how this works.

How many times
does go into ?

 3 4 ÷ 1 3 ↓ ↓ 3 4 × 3 = 9 4 =  2 1 4

Answer: 2 1/4 times.

Does it make sense?

Yes, fits into a little
more than two times.

How many times
does go into ?

 7 4 ÷ 2 5 ↓ ↓ 7 4 × 5 2 = 35 8 =  4 3 8

Answer: 4 3/8 times.

Does it make sense?

Yes. goes into 1 3/4
over four times.

How many times
does go into ?

 2 9 ÷ 2 7 = ↓ ↓ 9 × 7 = 7 9

Answer: 7/9 of a time.

Does it make sense?

Yes, because does not
go into even one full time!

Remember: There are TWO changes in each calculation:

1. You need to change the division into multiplication.
2. You need to use the reciprocal of the divisor.

7. Solve these division problems using the shortcut. Remember to check if your answer makes sense.

 a. 3 4 ÷ 5 ↓ ↓ 3 4 × 1 5 =
 b. 2 3 ÷ 6 7
 c. 4 7 ÷ 3 7
 d. 2 3 ÷ 3 5
 e.  4 ÷ 2 5
 f. 13 3 ÷ 1 5

8. a. Write a division to match the
situation on the right.

b. Check your division by multiplication.

 How many times does fit into ?

We have 8/5, which is eight pieces, trying to fit into five
pieces... so they fit 5/8 of the way.

9. Fill in.

 2 ÷ 3 4 = ?
 Or, how many times does go into ?
 First, let’s solve how many times goes into .
 Since 1 ÷ 3 4 = , it goes into one times.

If 3/4 fits into _____ times, then it fits

into double that many times, or _____ times.

We get the same answer by using the shortcut:

 2 ÷ 3 4 ↓ ↓ 2 × =

10. Fill in.

 ÷ = ?
 Or, how many times  does go into ?
 First, let’s solve how many times goes into .
 Since 1 ÷ = , it goes into one times.

If 2/7 fits into _____ times, then it fits

into exactly 5/6 as many times as it fits into 1,

 which is 5 6 × = =
We get the same answer by using the shortcut:

 ÷ ↓ ↓ × =

This lesson is taken from my book Math Mammoth Fractions 2.

#### Math Mammoth Fractions 2

A self-teaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.

Download (\$5.75). Also available as a printed copy.

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