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This pdf covers following topics of fractions:

Adding with like denominators

Adding with unlike denominators

Subtracting with like denominators

Subtracting with unlike denominators

Adding with like denominators

Adding with unlike denominators

Subtracting with like denominators

Subtracting with unlike denominators

1.
Adding and Subtracting Fractions

There are four main operations that we can do with numbers: addition (+), subtraction (–),

multiplication (x), and division (÷). You will be assessed on your ability to add and

subtract fractions.

In order to add or subtract, fractions must have common denominators.

ADDING FRACTIONS

1. Adding with Common Denominators

To add fractions, if the denominators are the same, we simply add the

numerators and keep the same denominators.

Add

1 and 5

e.g.

12 12

Since the denominators are common, simply add the numerators. 1 + 5 = 6

12 12 12

Notice that we must reduce the answer, if possible. = 1

2

2. Adding When One Denominator is a Multiple of the Other

2 5

e.g. Add and

9 27

Notice that the denominators are not common. Also notice that 27 is a multiple

of 9 (since 9 x 3 = 27). So, we make the lowest common denominator ( LCD)

27.

2 (x3) = 6

9 27

Therefore: 2 + 5 is the same as 6 + 5

9 27 27 27

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 1

There are four main operations that we can do with numbers: addition (+), subtraction (–),

multiplication (x), and division (÷). You will be assessed on your ability to add and

subtract fractions.

In order to add or subtract, fractions must have common denominators.

ADDING FRACTIONS

1. Adding with Common Denominators

To add fractions, if the denominators are the same, we simply add the

numerators and keep the same denominators.

Add

1 and 5

e.g.

12 12

Since the denominators are common, simply add the numerators. 1 + 5 = 6

12 12 12

Notice that we must reduce the answer, if possible. = 1

2

2. Adding When One Denominator is a Multiple of the Other

2 5

e.g. Add and

9 27

Notice that the denominators are not common. Also notice that 27 is a multiple

of 9 (since 9 x 3 = 27). So, we make the lowest common denominator ( LCD)

27.

2 (x3) = 6

9 27

Therefore: 2 + 5 is the same as 6 + 5

9 27 27 27

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 1

2.
We can now add the top 2 numbers, so the answer is = 11

27

3. Adding Any Fraction

Add 7 and 13

12 15

We must find a common denominator by examining multiples of the

largest denominator. In the example above we find that the LCD = 60.

7 (x5) + 13 (x4) = 35 + 52

12 15 60 60

= 87 Then simplify = 1 9

60 20

When adding mixed numbers, add the whole numbers and the fractions

separately. Find common denominators and add.

Add 1

5 and 2 3

6 8

1 5 ( = 1 20 ) + 2 3 (= 2 9 ) total equals 3 29

6 24 8 24 24

If an improper fraction occurs in the answer, change it to a common

fraction by doing the following.

3 29 = 3 + 1 5 = 4 5

24 24 24

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 2

27

3. Adding Any Fraction

Add 7 and 13

12 15

We must find a common denominator by examining multiples of the

largest denominator. In the example above we find that the LCD = 60.

7 (x5) + 13 (x4) = 35 + 52

12 15 60 60

= 87 Then simplify = 1 9

60 20

When adding mixed numbers, add the whole numbers and the fractions

separately. Find common denominators and add.

Add 1

5 and 2 3

6 8

1 5 ( = 1 20 ) + 2 3 (= 2 9 ) total equals 3 29

6 24 8 24 24

If an improper fraction occurs in the answer, change it to a common

fraction by doing the following.

3 29 = 3 + 1 5 = 4 5

24 24 24

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 2

3.
SUBTRACTING FRACTIONS

1. Common Fractions

As in addition, we must have common denominators in order to subtract.

Find the LCD; change the fractions to equivalent fraction with the LCD as

the denominator. Then subtract the numerators, but keep the same

5 3 2 1

- = or

8 8 8 4

2 3 16 9 7

- = - =

3 8 24 24 24

2. Mixed Numbers

When subtracting whole numbers, subtract the whole numbers, and then subtract

the fractions separately.

5 3 2

3 -1 = 2

9 9 9

However, if the common fraction we are subtracting is smaller than the other

common fraction, we must borrow the number “1” from the large whole number.

2 7 2 9

i.e. 4 =3+ + , or 3

7 7 7 7

2 5 9 5 4

4 - 2 = 3 -2 = 1

7 7 7 7 7

3 2

To subtract 1 from 6 , first change the common fractions to equivalent

4 3

8 9 -

fractions with the LCD. Since is smaller than , borrow from 6.

12 12

2 3 8 9

6 - 1 =6 -1

3 4 12 12

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 3

1. Common Fractions

As in addition, we must have common denominators in order to subtract.

Find the LCD; change the fractions to equivalent fraction with the LCD as

the denominator. Then subtract the numerators, but keep the same

5 3 2 1

- = or

8 8 8 4

2 3 16 9 7

- = - =

3 8 24 24 24

2. Mixed Numbers

When subtracting whole numbers, subtract the whole numbers, and then subtract

the fractions separately.

5 3 2

3 -1 = 2

9 9 9

However, if the common fraction we are subtracting is smaller than the other

common fraction, we must borrow the number “1” from the large whole number.

2 7 2 9

i.e. 4 =3+ + , or 3

7 7 7 7

2 5 9 5 4

4 - 2 = 3 -2 = 1

7 7 7 7 7

3 2

To subtract 1 from 6 , first change the common fractions to equivalent

4 3

8 9 -

fractions with the LCD. Since is smaller than , borrow from 6.

12 12

2 3 8 9

6 - 1 =6 -1

3 4 12 12

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 3

4.
8 12 8 20

6 = 5 + = 5

12 12 12 12

20 9 11

5 -1 = 4

12 12 12

ADDING AND SUBTRACTING FRACTIONS WITH FEET AND INCHES

The most important thing to remember when you are working with feet and inches is that they do

NOT follow the decimal system. Your measurements will not be correct if you try to make an

inch into a decimal or vice versa.

There are 12 inches (“) in 1 foot (‘)

You will be asked to calculate measurements in feet and inches; for example, in the practice test,

you were asked to calculate the length of a board that has been cut into 3 pieces. The pieces

3 1 ,6 3 and 7 4 . You are also told that is used up for each saw cut (kerf)

So, you need to calculate:

3 1 +6 3 +7 4 + +

Add the feet first: 3’ + 6’ + 7’ = 16’

Now add the inches: 1” + 3” + 4” = 8”

Now add the fraction of inches: + + + +

We know that we have to make all of these fractions have the same denominator before we can

add them. The only one that is different is . To covert this into sixths, we can multiply the top

and bottom by 2. (x2) =

So, + + + + =

The final answer is 16’ + 8” + ” = 16’ 9 ”

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 4

6 = 5 + = 5

12 12 12 12

20 9 11

5 -1 = 4

12 12 12

ADDING AND SUBTRACTING FRACTIONS WITH FEET AND INCHES

The most important thing to remember when you are working with feet and inches is that they do

NOT follow the decimal system. Your measurements will not be correct if you try to make an

inch into a decimal or vice versa.

There are 12 inches (“) in 1 foot (‘)

You will be asked to calculate measurements in feet and inches; for example, in the practice test,

you were asked to calculate the length of a board that has been cut into 3 pieces. The pieces

3 1 ,6 3 and 7 4 . You are also told that is used up for each saw cut (kerf)

So, you need to calculate:

3 1 +6 3 +7 4 + +

Add the feet first: 3’ + 6’ + 7’ = 16’

Now add the inches: 1” + 3” + 4” = 8”

Now add the fraction of inches: + + + +

We know that we have to make all of these fractions have the same denominator before we can

add them. The only one that is different is . To covert this into sixths, we can multiply the top

and bottom by 2. (x2) =

So, + + + + =

The final answer is 16’ + 8” + ” = 16’ 9 ”

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 4

5.
Practice: 1: Adding Fractions

a) Add the following:

1 2 4 3 4 2

1) + 2) + 3) +

5 5 5 5 9 9

3 3 3 2 1 1 3

4) + + 5) + 6) +

4 4 4 3 9 2 8

1 5 2 4 1 1

7) + 8) + 9) 3 +4

4 16 3 15 2 4

2 1 1 4 3 1

10) 9 + 3 11) 8 + 4 12) 2 +6

3 6 2 5 4 2

1 5 1 3 2 4

13) 4 + 6 14) 6 + 8 15) 7 +

3 6 3 4 3 5

2 1 3

16) 8 + 6 + 1

3 4 8

Practice 2: Subtracting Fractions

a) Subtract the following:

9 1 14 1 5 3

1) - 2) - 3) -

12 8 15 6 6 8

7 2 2 1 1 1

4) - 5) 9 - 6 6) 4 -1

9 3 3 6 2 4

3 5 11 2 1 2

7) - 8) - 9) 6 - 2

4 8 12 3 3 3

1 3 5 6 3 11

10) 13 - 5 11) 5 - 4 12) 4 - 1

4 4 7 7 4 12

Answers on the next page

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 5

a) Add the following:

1 2 4 3 4 2

1) + 2) + 3) +

5 5 5 5 9 9

3 3 3 2 1 1 3

4) + + 5) + 6) +

4 4 4 3 9 2 8

1 5 2 4 1 1

7) + 8) + 9) 3 +4

4 16 3 15 2 4

2 1 1 4 3 1

10) 9 + 3 11) 8 + 4 12) 2 +6

3 6 2 5 4 2

1 5 1 3 2 4

13) 4 + 6 14) 6 + 8 15) 7 +

3 6 3 4 3 5

2 1 3

16) 8 + 6 + 1

3 4 8

Practice 2: Subtracting Fractions

a) Subtract the following:

9 1 14 1 5 3

1) - 2) - 3) -

12 8 15 6 6 8

7 2 2 1 1 1

4) - 5) 9 - 6 6) 4 -1

9 3 3 6 2 4

3 5 11 2 1 2

7) - 8) - 9) 6 - 2

4 8 12 3 3 3

1 3 5 6 3 11

10) 13 - 5 11) 5 - 4 12) 4 - 1

4 4 7 7 4 12

Answers on the next page

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 5

6.
Remember that your answers may be slightly different from those given below, because of rounded

decimals and the route you took to reach your answer.

If you find any errors on the study material, please email [email protected]

Practice 1: Adding Fractions

3 7 2 2 9 1 7 7 9 14 3 5

1) 2) or 1 3) 4) or 2 5) 6) 7) 8) 9) 7 10) 12

5 5 5 3 4 4 9 8 16 15 4 6

3 1 1 1 7 7

11) 13 12) 9 13) 11 14) 15 15) 8 16) 16

10 4 6 12 15 24

Practice 2: Subtracting Fractions

5 23 11 1 1 1 1 1 2 1

1) 2) 3) 4) 5) 3 6) 3 7) 8) 9) 3 10) 7

8 30 24 9 2 4 8 4 3 2

6 5

11) 12) 2

7 6

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 6

decimals and the route you took to reach your answer.

If you find any errors on the study material, please email [email protected]

Practice 1: Adding Fractions

3 7 2 2 9 1 7 7 9 14 3 5

1) 2) or 1 3) 4) or 2 5) 6) 7) 8) 9) 7 10) 12

5 5 5 3 4 4 9 8 16 15 4 6

3 1 1 1 7 7

11) 13 12) 9 13) 11 14) 15 15) 8 16) 16

10 4 6 12 15 24

Practice 2: Subtracting Fractions

5 23 11 1 1 1 1 1 2 1

1) 2) 3) 4) 5) 3 6) 3 7) 8) 9) 3 10) 7

8 30 24 9 2 4 8 4 3 2

6 5

11) 12) 2

7 6

Carpentry: Adding and Subtracting Fractions – Study Guide 1FB/2014 Page 6