OPERATIONS ON FRACTIONS

MSJC ~ San Jacinto CampusMath Center Workshop Series

Janice Levasseur

Review Mixed Numbers & Improper Fractions

To rewrite a Mixed Number as an Improper Fraction, multiply the denominator to the whole number and add it to the numerator.

135

8011

3711

To rewrite an Improper Fraction as a Mixed Number, divide the denominator into the numerator, which is the whole number and write remainder as the numerator.

165

5 3 1 x

711 80

Addition of Fractions & Mixed Numbers

Fractions with the same denominator are added by adding the numerators and placing that sum over the (common) denominator.

We are literally counting how many parts in total there are where all the parts are the same size.

Ex: Add 2/5 + 1/5 Draw each fraction.

2/5 1/5+ =5

2 + 1= 3/5

When the fractions have the same denominator, the parts of the whole are of the same size so adding fractions is literally counting up the parts.

When the fractions have different denominators, the parts of the whole are not the same size so we cannot add fractions by counting the parts “adding apples and oranges”

To add fractions with different denominators, first rewrite the fractions as equivalent fractions with the same denominator “adding apples and apples”

The common denominator we will use for the equivalent fractions is the LCM of the denominators, called the LCD, Least Common Denominator.

Ex: Add 1/4 + 5/8Draw each fraction.

Note: different denominators

1/4

5/8

Can we further divide each of the pieces so that the pieces of each whole are of the same size?

YES! LCM(4, 8) = LCD = 8 divide each whole into 8 pieces

What times 4 is 8? 2

Divide each part into two pieces Consider the first fraction 1/4 :

1/4 = 2/8

Consider the second fraction 5/8 : It is already divided into 8 pieces

5/8 1/4 + 5/8 = 2/8 + 5/8 = 7/8

Mathematically:

LCD = LCM(4, 8) = 8

41

141

4122

82

Therefore,

85

41

85

82

87

Ex: Add 1/5 + 1/2 Note: different denominatorsDraw each fraction.

1/5

1/2

Can we further divide each of the pieces so that the pieces of each whole are of the same size?

YES! LCM(5, 2) = LCD = 10 divide each whole into 10 pieces

Consider the first fraction 1/5 : What times 5 is 10? 2

Divide each part into two pieces 1/5 = 2/10

Consider the second fraction 1/2 : What times 2 is 10? 5

Divide each part into five pieces 1/2 = 5/10

1/5 + 1/2 = 2/10 + 5/10 = 7/10

Your turn to try a problem.

Ex: Subtract 175/9 - 115/12 Use a vertical format

12511

9517

-

Start right (with the fractions) and work left

To subtract fractions, we need aCommon Denominator LCM(9,12) = 36

361511

362017

-

Subtract 20/36 – 15/36 = 5/36

365

6

Subtract whole numbers 17 - 11= 6

Find equivalent fractions 3 * 3 2 * 2 * 3

5 2 2173 3 2 2

5 3112 2 3 3

Ex: Subtract 7 – 4 2/5 Use a vertical format

524

7

-

Start right (with the fractions) and work left

The minuend does not have a fraction part so we have to borrow take one whole

524

556

-Subtract 5/5 – 2/5 = 3/5

53

2 Subtract whole numbers 6 - 4 = 2

6

Cut the borrow whole into parts . . .

How many parts? 5

How many parts do we have? 5/5

Practice: Rewrite the mixed number 5 2/7 as a mixed number with an improper fraction part with denominator 21.

725 Find equivalent fraction with

denominator 21 2/7 = 6/21

2165 Now borrow a whole4

Cut the borrowed whole into parts . . .How many parts? 21

How many parts do we have?21 from the chops plus the original 6 21 + 6 = 27 27/2121

274

Your turn to try a problem.

Ex: Multiply ½ x ¾ read “ ½ of ¾” and draw it

Take ½ of three-fourths by chopping the whole in 2 parts (the other direction) and shading 1 part

How many parts are there now? 8How many parts represent ½ of ¾ ? (i.e. how many parts are doubly-shaded?) 3

Therefore, ½ of ¾ is 3/8 ½ x ¾ = 3/8

Multiplication of Fractions & Mixed Numbers

Ex: Multiply 9/11 x 2/3

32x

119

3 3 2=11 3

611

Can we factor the numerator and the denominator?

Reduce any factor in the numerator with the same factor in the denominator?

Notice that the multiplication of the numerators and/or denominators can get more complicated.

We multiplied first and then simplified.

We can simplify first and then multiply.

Charge!

Tidy Up First!

Ex: Multiply 9/11 x 2/3

32x

119

3x112x9

3318

116x

33

116

Can we simplify the fraction?

32x

119

32x

113x3

112x3x

33

116x1

116

• Since we know how to multiply fractions, we can now multiply fractions, whole numbers, and mixed numbers together

• To multiply whole numbers, mixed numbers, and fractions first turn every factor into a fraction.

fraction multiplication: multiply and simplify OR simplify and multiply

Ex: Multiply 2 x 6/7

First rewrite the question as a multiplication of fractions write the whole number 2 as a fraction

76x

122 = 2/1

712

751

improper fraction mixed number

Consider this . . . 2 x 6/7 can be read two times 6/7 “twice” 6/7 draw 6/7 then double it!

How many parts are shaded?

How many parts make a whole?

12

7

Ex: Multiply 3 1/5 x 2 3/11

First rewrite the question as a multiplication of fractions write mixed number as a fraction

516

513

1125x

516

1180

1125

1132and

115x5x

516

115x16x

55

1180x1

1137

Your turn to try a problem.

Division of Fractions & Mixed Numbers

Ex: Divide What are we doing with the division?4

121

The answer will be how many chunks of size ¼ we can make out of a part of size ½ ?

Divide the whole into fourths?

How many fourths are in ½?

½ divided by ¼ = 2

Ex: Divide What are we doing with the division?4

132

The answer will be how many chunks of size ¼ we can make out of a part of size 2/3 ?

Divide the whole into fourths?

How many fourths are in 2/3?

2/3 divided by 1/4 = 2 and 2/3 = 2 + 2/3 = 2 2/3

1 2 But now what?Put the two pink pieces together . . .

How much of a fourth do we have? 2/3

What is the process for dividing fractions?

First, a definition: the reciprocal of a fraction is the fraction with the numerator and denominator interchanged (“flip it!”)

Ex: Find the reciprocals of the following:

115

51

515

511

15

= 5

51

• “divided by” mathematically is the same operation as “times the reciprocal of”

• To divide fractions, multiply the first fraction (the dividend) by the reciprocal of the second fraction (the division)

Ex: Divide41

21

41

21

14

21

Reciprocal of ¼ is __? 4/1

122

21

121

22 21 = 2

Ex: Divide53

98

53

98

35

98

Reciprocal of 3/5 is __? 5/3

2740

27131

The division is asking, “how many chunks of size 3/5 can be made from a part of size 8/9?”

Answer: 1 whole chunk (of size 3/5) and 13/27 of another chunk (of size 3/5)

• Since we know how to divide fractions, we can now divide fractions, whole numbers, and mixed numbers together

• To divide whole numbers, mixed numbers, and fractions first turn every number into a fraction.

fraction division: multiply the first fraction by the reciprocal of the second fraction.

Ex: Divide324

The division is asking, “how many chunks of size 2/3 can be made from a 4 wholes?”

1 2 3 4 Now divide the wholes into 3

How many chunks of size 2/3 are there?

1 2 3 4 5 6

= 63

= 4•2

Ex: Divide651

943

651

943

116

931

Reciprocal of 11/6 is __? 6/11

3362

611

931

1132

3331

113231

33

1132311

33291