S k i l l
 i n
A R I T H M E T I C

Table of Contents | Home | Introduction

Lesson 25  Section 2

SUBTRACTING MIXED NUMBERS

Back to Section 1

  Example 1. 8 4
5
 − 1 1
5

Solution.  In this example, we may simply subtract the whole numbers and subtract the fractions -- similarly to adding mixed numbers.

  Example 1. 8 4
5
 − 1 1
5
 =  7 3
5
.

But consider the following, in which the fractions are reversed:

  Example 1. 8 1
5
 − 1 4
5
.
How can we deal with that?  We cannot take  4
5
 from  1
5
.

To see how to deal with it, consider the following:

We cannot take 40 minutes from 10 minutes -- we need more minutes. To get them, we will break off 1 of the 7 hours, and decompose it into 60 minutes.  We then regroup them with 10 minutes.

60 minutes + 10 minutes = 70 minutes:

2 hours from 6 hours is 4 hours.  40 minutes from 70 minutes is 30 minutes.

Solution.  We cannot take 11 inches from no inches.  To make inches, then, from 8 feet we will take 1 foot -- which is 12 inches:

5 feet from 7 feet is 2 feet.  11 inches from 12 inches is 1 inch.

We can now return to our problem:

8 1
5
 − 1 4
5
.

We need more fifths. Where will we get them?  From 8.  We will

  break off 1 from 8, and decompose it into  5
5
.  (Lesson 21, Example 4.)  

We will then add those  5
5
 -- we will regroup them -- with the original  1
5
,
  making a total of  6
5
.

Then:

"1 from 7 is 6.   4
5
 from  6
5
 is  2
5
."

Actually, the simplest way to do this problem is mentally

by rounding off 1 4
5
 to 2. That is, add  1
5
 to both. (Lesson 21.)
8 1
5
 − 1 4
5

The problem then becomes

8 2
5
 − 2 = 6 2
5
.
Break off 1 from 4.  Express it as  8
8
, and add it to  1
8
:

Now the mystery, if any, is:  How does that numerator get to be 9?

9 is the sum of the original numerator 1 and denominator 8:

1/8 + 8/8 = 9/8
  Example 5.   Write the missing numerator:  9 2
5
  = 8 ?
5
  Answer.   8 7
5

7 is the sum of denominator plus numerator:  5 + 2.

  Example 6.      9 2
7
 − 3 5
7
  = ?
9 2
7
 becomes 8 9
7
.  The improper numerator 9 is the sum of the
  denominator and numerator of   2
7
:   7 + 2 = 9.
  Example 7. 6 − 1 2
3
  =   5 3
3
 − 1 2
3
 
    =  4 1
3
.
"1 from 5 is 4.    2
3
 from  3
3
 is  1
3
."

Again, the simplest way to do this is mentally

by rounding off 1 2
3
 to 2. That is, add  1
3
 to both.

(Lesson 21.) The problem then becomes

6 − 1 2
3
  =   6 1
3
 − 2 = 4 1
3
.
  Example 8.   Prove:   6 − 1 2
3
 = 4 1
3

Solution.   According to the meaning of subtraction,

1 2
3
 + 4 1
3
 = 5 3
3
 = 6.

Compare Lesson 7, Example 2.

  Example 9.    6 1
2
 − 2 3
4

Solution.  First, we must make the denominators the same:

  Example 10.     6 1
2
 − 2 3
4
 = 6 2
4
 − 2 3
4
We cannot take  3
4
 from  2
4
; therefore, on regrouping   4
4
  from 6, the

fraction becomes 4-fourths + 2 -fourths = 6-fourths:

  Example 10.     6 1
2
 − 2 3
4
 = 5 6
4
 − 2 3
4
   = 3 3
4
.  
"2 from 5 is 3.   3
4
 from  6
4
 is  3
4
."

Mentally:

6 1
2
 − 2 3
4
.
 
"3 from 6 1
2
 is 3 1
2
 
Plus  1
4
 is 3 3
4
."

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

Previous Section


Introduction | Home | Table of Contents


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2014 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com