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Subtracting Mixed Numbers with like fractional parts

This is a free 5th grade lesson about subtracting mixed numbers with like fractional parts (no need to convert the fractions to have a common denominator). The lesson presents two strategies for subtraction: regrouping and subtracting in parts (piece by piece). The next lesson presents yet a third strategy (using a negative fraction).

Strategy 1: Renaming / regrouping

In this method you divide one whole pie into “slices,” and join these slices with the existing slices. After that, you can subtract. It is the same as regrouping in the subtraction of whole numbers.

Rename 3

2

6

 as 2

8

6

, and then subtract 1

5

6

.

   

At first we have three uncut pies and 2/6 more. Then we cut one of the whole pies into sixths. We end up with only two whole (uncut) pies and 8 sixths.

We say that 3 2/6 has been renamed as 2 8/6.
Now we can subtract 1 5/6 easily.

3

2

6

  −  1

5

6

  =   2

8

6

  −  1

5

6

  =  1

3

6

Regroup 1 whole pie as 6 sixths.

Regroup (borrow) 1 whole pie
as 6 sixths.

There were already 2 sixths in
the fractional parts column, so
that is why it becomes 8/6 after
regrouping. Now you can
subtract the 5/6.

Rename 2

1

8

 as 1

9

8

, and then subtract 

5

8

.

 

At first we have two uncut pies and 1/8 more. Then we cut one of the whole pies into eighths. We end up with only one whole (uncut) pie and 9 eighths.

We say that 2 1/8 has been renamed as 1 9/8.
Then we can subtract 5/8 easily.

2

1

8

  −  

5

8

  =    1

9

8

  −  

5

8

  =  1

4

8

Regroup 1 whole pie as 8 eighths.

Regroup (borrow) 1 whole pie
as 8 eighths.

Since there was already one
eighth in the fractional parts
column, it becomes 9/8 after
regrouping. Now you can
subtract 5/8.

1. Do not subtract anything. Just cut up one whole pie into fractional parts. Rename the mixed number.

  
a.  2

1

6

 is renamed as
      
b.  3

1

8

 is renamed as 
    
c.  2

2

9

 is renamed as 
    
d.  2

3

5

 is renamed as
      
e.  3

3

10

 is renamed as 
    
f.  2

1

4

 is renamed as 


2. Rename, then subtract. Be careful. Use the pie pictures to check your calculation.


 

a.    4

2

9

  −   1

8

9

 
 

     
=  3

11

9

  −   1

8

9

 =


 

b.    5

3

12

  −   2

7

12

 
 

     
=  4

 

12

  −   2

7

12

 =

 
c.    5

7

10

  −   3

9

10

 
 

     
=  

 

 

  −   3

9

10

 =

 
d.    4

3

8

  −   1

7

8

 
 

     
=  

 

 

  −   1

7

8

 =

3. Regroup (if needed) and subtract.


a.
  

 

b.

7 

4

9

 

−  2 

7

9

   

 

 

 
c.

12 

9

12

 

−  6 

11

12

   

 

 
 
d.

3

14

 

−  5 

9

14

   

 

 
 
 
 
e.

14 

7

9

 

−  3 

5

9

   

 

 
f.

11 

5

21

 

−  7 

15

21

   

 

 

g.

26 

4

19

 

−  14 

15

19

   

 

 

h.

10 

3

20

  −  5 

7

20

   

 



Strategy 2: Subtract in Parts

First, subtract what you can from the fractional part of the minuend. Then subtract the rest from
one of the whole pies. The examples show two slightly different ways to understand this.

2

1

8

 − 

5

8

= ?

We cannot subtract 5/8 from 1/8. So, first
subtract 1/8, which leaves 2 whole pies.
The rest (4/8) of the 5/8 is subtracted
from one of the whole pies.

   2

1

8

 −  

5

8

 =   2

1

8

 −  

1

8

 −  

4

8

 =  
         =    

2

   −

4

8

 =  1

4

8

3

2

9

 −  2

7

9

= ?

We cannot subtract 7/9 from 2/9. So, first subtract
2 2/9, which leaves 1 whole pie. The rest (5/9)
is subtracted from the last whole pie.
 

   3

2

9

 −  2

7

9

 =   3

2

9

 −  2

2

9

 −  

5

9

 =  
         =    

1

   −

5

9

 = 

4

9

4. Subtract in parts. Remember: you can add to check a subtraction problem.

a.  2

2

6

 − 

5

6

=

b.  3

1

5

 −  2

3

5

c.  3

1

8

 −  1

7

8

=

d.  3

2

7

 −  2

6

7

    e.  5

2

9

 −  

5

9

  −  1

8

9

=

5. Subtract in two parts. Write a subtraction sentence.

a.     Cross out 

3

4

.


 

b.     Cross out 1

5

7

.


 

c.     Cross out 1

5

9

.


 

d.     Cross out 1

11

12

.


 



Example. Look at Mia’s math work:  7

1

6

 − 2

5

6

 = 9

6

6

 = 10.  Can you see why it is wrong?

If you have 7 and a bit and you subtract 2 and some, you cannot get 10 as an answer! In reality, Mia was adding instead of subtracting. (If you have ever done that, you are not alone—it is a common error.)

Always check if your answer is reasonable.

6. Subtract.

a.   8

1

5

  −   3

3

5

  =

 

b.  4

2

8

  −  1

7

8

  =  
c.  12

4

13

  −  9

8

13

  =
d.  11

2

15

  −   6

6

15

  =

 

e.  7

1

20

  −  3

7

20

  =
f.  6

14

100

  −  2

29

100

  =

7. Two sides of a triangle measure 3 5/8 in., and the perimeter of
    the triangle is 10 1/8 in. How long is the third side of the triangle?
 

 

8. Ellie had 4 yards of material. She needed 7/8 yard for making a skirt,
    and she made two. How much material is left?
 

 

9. Harry wants to bake chocolate chip cookies. The recipe calls for 1 3/4 cups of flour
    and he is making a double batch. However, Harry only has 3/4 cup of flour!
    How much more flour would Harry need to borrow from his neighbor?

 

 

  Subtract. The pies may help.

   2

1

2

1

2

3

=        



This lesson is taken from Maria Miller's book Math Mammoth Fractions 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.



Math Mammoth Fractions 1

A self-teaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.

Download ($3.50). Also available as a printed copy.

Learn more and see the free samples!

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