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The ideas in this fraction lesson are taken from the ebooks that are sold on this website. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.

Dividing fractions by fractions
Free fraction lesson plan from HomeschoolMath.net

As we saw in the last lesson, dividing something (fraction for example) by a whole number
is easily illustrated by dividing that something between that many people.  
When you divide something by a fraction, think how many times 
does the fraction go into the dividend. 

How many times does 6 go into 18?

18 ÷ 6 =

 How many times does 10 go into 40?

40 ÷ 10 =
How many times does 

1

4

  go into  2?
How many times does 

 go into  ?
2 ÷

1

4

  = 
How many times does 

1

2

  go into  3?
How many times does 

 go into  ?
3 ÷

1

2

  = 
How many times does 

2

5

  go into  2?
How many times does 

 go into  ?
2 ÷

2

5

  = 
How many times does 

1

4

  go into 

1

2

 ?
How many times does 

 go into  ?
1 ÷

1

4

  = 

Example problem types

1.  Divide.  Use the thinking pattern "how many times does this (divisor) go into this (dividend)?  Use pictures to help.

a.   2 ÷

1

6

=
 
b.   2 ÷

1

8

=
c.   2 ÷

1

10

=
d.   2 ÷

1

7

=

 

Did you notice anything especial in the last exercise?  There is a shortcut to these problems:
e.   3 ÷

1

4

= 12
same as
3 × 4 = 12
j.   3 ÷

1

5

= 15
same as
3 × 5 = 15
b.   2 ÷

1

8

= 16
same as
2 × 8 = 16
In the last lesson, instead of dividing by a whole number, we multiplied by its reciprocal number.  
The same works if you are dividing by a fraction, too!

Instead of dividing by a fraction, multiply by its reciprocal number.

Instead of dividing by 

1

2

 , multiply by its reciprocal 2.
Instead of dividing by 

1

5

 , multiply by its reciprocal 5.
Instead of dividing by 

1

9

 , multiply by its reciprocal 9, etc.
Examples:

3

4

   ÷ 

1

5

 = 

3

4

   ×

5

 =  

15

4

  = 3 

3

4

goes into
almost 4 times.

7

8

   ÷ 

1

3

 = 

7

8

   ×

3

 =  

21

8

  = 2 

5

8

goes into 
about 2 1/2 times.

1

1

3

   ÷ 

1

4

 = 

4

3

   ×

4

 =  

16

3

  = 5 

1

3

goes into
a little over 5 times.

 

2.  Fill in the dividend from the picture.  Then divide by multiplying by the reciprocal number.  When you get the answer, look at the picture and think if your answer is reasonabe. 

a.    

4

6

  ÷ 

1

3

 =        

 

 
b.      ÷

1

6

=        

 

 
c.      ÷

1

4

=       

 

d.      ÷

1

3

=       

 

Instead of dividing by a fraction, multiply by its reciprocal number.

This rule works ALWAYS - even when the divisor is a more complex looking fraction.
With fractions you will get the reciprocal number by flipping the numerator and denominator.  
Read here to learn more about reciprocal numbers. 

Instead of dividing by  

3

4

 , multiply by its reciprocal  

4

3

 .
Instead of dividing by  

5

12

 , multiply by its reciprocal  

12

5

 .
Instead of dividing by 

7

5

 , multiply by its reciprocal  

5

7

 , etc.
Examples:

4

7

   ÷ 

3

4

 = 

4

7

   × 

4

3

 =  

16

21

 

goes into
almost 3/4 times 
(16/20 would be 3/4).

7

8

   ÷ 

5

12

 = 

7

8

   × 

12

5

 =  

84

40

 = 2 

1

10

goes into 
little over 2 times.

2   ÷ 

7

5

 = 
2   × 

5

7

 =  

10

7

 = 1 

3

7

goes into
almost one and half times.

3.  Practice time!  Divide by multiplying by the reciprocal number of the divisor.  Note: change mixed numbers first to fractions.  As usual, give your answer as a mixed number and in lowest terms, if possible.

a.  

5

8

   ÷ 

2

5

 = 
 

f.   4

   ÷ 

2

5

 = 
b.  

8

5

   ÷ 

3

4

 = 
 
g.  

2

9

   ÷ 

1

7

 = 

Reciprocal numbers

Two numbers are reciprocal numbers of each other if their product is 1.  
In other words, when you multiply them, 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

16

11

   is a reciprocal number of 

11

16

 , because 

16

11

  × 

11

16

  = 

176

176

  = 1

4

11

   is a reciprocal number of  2

3

4

 , because 2

3

4

  = 

11

4

   and 

4

11

 × 

11

4

  = 1

4.  Find reciprocal numbers!

a.  

5

8
b.  

7

9
c.  1

7

8
d.  32
e.  100
f. 

3

1000
g.  1

1

23
h.   3 

12

13
i.  10

3

10

 

Next lesson: Dividing mixed numbers

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



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