Basic MathematicsMixed NumbersProblem SolvingOrder of Operations

David RadcliffeCentury College

3.5 Addition and Subtraction of Mixed Numbersa. Add using mixed numerals.

b. Subtract using mixed numerals.

c. Solve applied problems involving addition and subtraction with mixed numerals.

Visualizing Mixed Numbers

•What mixed number is shown in this picture?

•What improper fraction is shown in this picture?

3 192 and

8 8

Visualizing Addition of Mixed Numbers

•What addition problem does this picture show?

3 4 72 1 3

8 8 8

Visualizing Addition of Mixed Numbers

•What addition problem does this picture show?

1 2 33 1 4

8 8 8

Visualizing Addition of Mixed Numbers

•What addition problem does this picture show?

5 33 1 5

8 8

Visualizing Addition of Mixed Numbers

•What addition problem does this picture show?

6 7 52 1 4

8 8 8

How to add mixed numbers•Write the fractional parts with a

common denominator.•Add the integer parts.•Add the fractional parts.•If the fractional part of the sum is

greater than or equal to one, then regroup.

•Simplify the fractional part.

Practice Problems

Add the mixed numbers.

3 112 8

5 5

2 114 7

9 9

529 17

12

420

5

3 121 21

9 3

546

12

Practice Problems (with regrouping)

Add the mixed numbers.

3 413 9

5 5

4 599 27

9 9

7 1016

11 11

7 2 222 22 1 23

5 5 5

9126 126 1 127

9

17 6 616 16 1 17

11 11 11

Adding mixed numbers with unlike denominators•If the denominators are different, then

you must find a common denominator before adding the fractional parts.

1 224 16

2 3

3 424 16

6 6

740

6

141

6

Practice problems

•Add the mixed numbers.

4 918 38

5 10

1 317 8

6 4

5 1419 1

12 15

8 9 17 718 38 56 57

10 10 10 10

2 9 1117 8 25

12 24 12

25 56 81 27 719 1 20 20 21

60 60 60 20 20

Visualizing Subtraction of Mixed Numbers•What subtraction problem is shown in this

picture?

6 3 3 6 3 33 1 2 or 3 1 2

8 8 8 8 8 8

How to subtract mixed numbers•Write the fractional parts with a common

denominator.•If the first fraction is less than the second

fraction, then regroup.•Subtract the integer parts.•Subtract the fractional parts.•Reduce the fraction, if needed.

Practice problems

•Subtract the mixed numbers.

8 315 7

10 10

7 916 7

10 10

321 7

25

5 18 8

10 2

17 9 8 415 7 8 8

10 10 10 5

25 3 2220 7 13

25 25 25

Application• The tallest person in human

history was Robert Wadlow, who stood 8 feet, 11 inches tall, or 8 11/12 feet. The average American man is 5 feet, 10 inches tall. How much taller than average was Robert Wadlow?

13 ft, or 3 ft 1 in.

12

3.6 Multiplication and Division of Mixed Numbers

a. Multiply using mixed numerals.

b. Divide using mixed numerals.

c. Solve applied problems involving multiplication and division with mixed numerals.

Multiplying Mixed Numbers

•Write the mixed numbers in fraction form.

•Multiply the fractions.

•Convert the answer to mixed number form, if needed.

•Simplify.

Examples

81

32

21

4 33

1

8 7 5628

1 2 2

9 10 90 15 17

4 3 12 2 2

Practice problems• Multiply the mixed numbers.

Don’t forget to simplify!

102

45

2 510

3 8

10 22 22044

1 5 5

4 5 20 26

3 1

3

3 3

2 5

3 8

Can you find the mistake in this calculation?

1 1 14 12

2 3 63

Remember!

•Do not find a common denominator when multiplying mixed numbers – only when adding or subtracting.

•Always convert mixed numbers to fraction form before multiplying.

•Do not cancel before putting the mixed numbers in fraction form.

•Simplify your final answer.

Division of Mixed Numbers

•Write the mixed numbers in fraction form.

•Multiply the first fraction by the reciprocal of the second fraction.

•Simplify completely.•Convert the answer to a mixed

number, if needed.

Examples

42

23

21

3 41

3

4 8 4 3 12 3 11

1 3 1 8 8 2 2

7 7 7 4 4 11

3 4 3 7 3 3

Don’t cancel before taking the reciprocal!

Try it yourself!

252

47

1 24

2 3

253 3

55 5

55

6

36

4

44

7

Applications

•One kilometer is approximately 3/5 of a mile. How many miles is 4 ½ kilometers?

1 3 9 3 27 72 miles

2 5 2 5 104

10

Octomom!

Nadya Suleyman gave birth to octuplets on January 26, 2009. What was their average birth weight? (1 lb = 16 oz)

1. Noah Angel Solomon - male; 2 lbs 8 oz2. Maliyah Angel Solomon - female; 2 lbs 2

oz3. Isaiah Angel Solomon - male; 3 lbs 1 oz4. Nariyah Angel Solomon - female; 2 lbs 3

oz5. Makai Angel Solomon - male; 1 lbs 12 oz6. Josiah Angel Solomon - male; 2 lbs 9 oz7. Jeremiah Angel Solomon - male; 1 lb 13 oz8. Jonah Angel Solomon - male; 2 lbs 7 oz

39 72 lb, or 2 lb 4 oz.

128 8

The Beer Barrel Buster Burger has 11 ½ pounds of meat, 25 slices of cheese, 1 full lettuce, 2 onions, 3 tomatoes and 25,000 calories.

How many servings does this burger make?Answer: 46 servings, if each serving has ¼ lb of meat. (Other answers are possible.)

1 1 23 1 23 446

2 4 2 411

2 1

3.7 Order of Operations; Estimationa. Simplify expressions using the rules for order of

operations.

b. Estimate with fractions and mixed-numeral notation.

Rules for Order of Operations

PEMDAS

1. Do all calculations within parentheses before operations outside.

2. Evaluate all exponential expressions.3. Do all multiplications and divisions in

order from left to right.4. Do all additions and subtractions in

order from left to right.

Helpful hints•Change all mixed numbers to improper

fractions first. (Unless they are being added or subtracted.)

•Change all divisions to multiplications before you start calculating.

•Write each step on a separate line, and copy the parts that do not change at each step.

•Write the auxiliary calculations on the right side, so the don’t interrupt the flow of the argument.

Example

Simplify

1 1 1

2 3 4

Solution

1 1 1 1 1 6 1 7

2 3 4 2 12 12 12 12

Example

•Simplify:

21 1 5 1

3 2 144 102 3 8 2

Solution2

1 1 5 13 2 144 10

2 3 8 2

27 7 5 21

1442 3 8 2

25 2

·1446 21

7·

8

49 144 5 2

· ·36 1 8 21

1049·4

168

84

5 79196 195

84