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starMay 2012

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Dividing Fractions 2b: Reciprocal Numbers

The video below explains about reciprocal numbers conceptually, and about the "rule" (how-to) for dividing fractions:

We know that 

1

5

  goes into 1 exactly five times. But how many times does  

2

5

  go into 1?
Let's think with pictures.  

 goes into    two times, and then we have 1/5 left over.
How many times does 

2

5

   fit into 

1

5

  ? Or, how many times does  

 go into   ? 

That is like trying to fit a two-piece part into a hole that just fits one piece. Only 1/2 of the part fits. So, 2/5 goes into 1/5 half a time. All in all, 2/5 fits into one exactly 2 1/2 times.

We can write a division from this:  1 ÷

2

5

  =  2

1

2

  

Let's look at another similar example.

We know that  

1

4

  goes into 1 exactly four times. But how many times does  

3

4

  go into 1?
Let's look at pictures.  

 goes into   once, and we have 1/4 left over.
How many times does  

3

4

   fit into  

1

4

  ? Or, how many times does  

 go into    ? 

We have a three-piece (3/4) part trying to fit into one piece (1/4), so the part fits 1/3 of the way. 

Putting it all together, 3/4 fits into one 1 1/3 times. As a division:  1 ÷

3

4

  =  1

1

3

  


1. Solve. First think how many times the fraction fits into 1 completely, and then think how many
    times it fits into the leftover part. Write a division.

a.  How many times

     does 

 go into  ?

 

1  ÷   = 

b.  [available in the book]

i. 

[available in the book]

j.  How many times

     does 

 go into  ?

 

1  ÷   = 


2. Write the divisions from the previous exercise

  

    so that the answer is written
    as a fraction, and not as a
    mixed number. For example,
    2 1/3 (the answer to #a)
    is written as 7/3. I want you
    to notice something special!

    a.  1  ÷  

3

7

   =  

7

3

 

    b.  1  ÷   = 
c.  1  ÷   = 

 

d.  1  ÷   = 

 

e.  1  ÷   = 
 
f.  1  ÷   = 
g.  1  ÷   = 

 

h.  1  ÷   = 
 
i.  1  ÷   = 
 
j.  1  ÷   = 
     

3. Did you notice the special
    thing? Now, let's turn it
    around.

    Write each division sentence
    from above “backwards” as
    a multiplication. The first
    one is done for you.

    a.   

4

9

  × 

9

4

    =  1

 

    b.   ×   =  1
c.   ×   =  1

 

d.   ×   =  1

 

e.   ×   =  1
 
f.   ×   =  1

 

g.   ×   =  1
 
h.   ×   =  1
 
i.   ×   =  1
 
j.   ×   =  1

These kinds of numbers are called reciprocal numbers!

Reciprocal numbers

Two numbers are reciprocal numbers of each other if, when multiplied, you get 1.

3

4

   is a reciprocal number of  

4

3

 , because  

3

4

  × 

4

3

  = 

12

12

  = 1.

1

7

   is a reciprocal number of 

7

 , because 

1

7

  × 

7

  = 

7

7

  = 1.
You can find a reciprocal of a fraction  

a

b

   by flipping the numerator and denominator: it is  

b

a

 .
This works because  

a

b

 × 

b

a

   = 1 (a's and b's cancel each other in simplification.)

For mixed numbers, first write them as fractions, then find the reciprocal number.

Since 2

3

4

  = 

11

4

 , its reciprocal number is 

4

11

.


4. Find reciprocal numbers!

a.  

5

8
b.  

1

9
   
f.  1

1

23
  [available in the book]
j.  

3

1000

5. Solve the division problems.

a.   1 ÷ 

3

4
[available in the book]    

 

 How many times does  

 

  go into  ?

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

As a division, we write  1  ÷ 

8

5

  = 

5

8

 .  The idea of reciprocal numbers works here as well.

If you check the division by multiplication, it checks:  

5

8

  ÷ 

8

5

  =  1.

6. Solve the division problems.

a.   1 ÷ 

7

4
     

7. [available in the book]

8. [available in the book]

The ideas in this fraction lesson are taken from the Math Mammoth Fractions 2 book. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.

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.

Available as an affordable download ($5.00), and also as a printed copy.

=> Learn more and see the free samples!

See more topical Math Mammoth books

Next lesson: The Shortcut for fraction division

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