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A R I T H M E T I C

Lesson 25 Section 2

SUBTRACTING MIXED NUMBERS

Example 1.

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.

4
5

− 1

1
5

3
5

But consider the following in which the fractions are reversed:

Example 1.

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:

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

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.)

1
5

− 1

4
5

The problem then becomes

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:

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.

2
7

5
7

2
7

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

1
3

Again, the simplest way to do this is mentally

2
3

(Lesson 21.) The problem then becomes

6 − 1

2
3

1
3

− 2 = 4

1
3

Example 8. Prove: 6 − 1

2
3

= 4

1
3

Solution. According to the meaning of subtraction,

2
3

+ 4

1
3

= 5

3
3

= 6.

Compare Lesson 7, Example 2.

Example 9.

1
2

− 2

3
4

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

Example 10.

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

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.

Continue on to the next Lesson.

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