lesson 1.1.3

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Lesson 1.1.3Lesson 1.1.3

The Order of OperationsThe Order of Operations

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Lesson

1.1.3The Order of OperationsThe Order of Operations

California Standard:Algebra and Functions 1.2

What it means for you:

Key Words:

Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2.

You’ll learn about the special order to follow when you’re deciding which part of an expression to evaluate first.

• Parentheses• Exponents• PEMDAS

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Lesson


When you have a calculation with more than one operation in it, you need to know what order to do the operations in.

There’s a set of rules to follow to make sure that everyone gets the same answer. It’s called the order of operations — and you’ve seen it before in grade 6.

E.g. if you evaluate the expression 2 • 3 + 7 by doing…

So the order you use really matters.

2 • 3 + 7 = 20“add 7 to 3 and multiply the sum by 2.”

2 • 3 + 7 = 13“multiply 2 by 3 and add 7,”

…you’ll get a different answer from someone who does…

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Finally follow any addition and subtraction instructions from left to right.

First do any operations inside parentheses.

Multiplication or Division

Addition or Subtraction

Parentheses()[]{}

x2 y7

×÷

+–

The Order of Operations is a Set of Rules

Lesson


An expression can contain lots of operations. When you evaluate it you need a set of rules to tell you what order to deal with the different bits in.

Order of operations — the PEMDAS Rule

Exponents Then evaluate any exponents.

Next follow any multiplication and division instructions from left to right.

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Divide first, then multiply.

9 ÷ 4 • 3

Lesson


When an expression contains multiplication and division, or addition and subtraction, do first whichever comes first as you read from left to right.

Following these rules means that there’s only one correct answer. Use the rules each time you do a calculation to make sure you get the right answer.

Multiply first, then divide.

9 + 4 – 3

9 – 4 + 3

9 • 4 ÷ 3

Add first, then subtract.

Subtract first, then add.

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Finally follow any addition and subtraction instructions from left to right.

First evaluate anything grouped by parentheses, fraction bars or brackets

Multiplication or Division

Addition or Subtraction

Grouping()[]{}

x2 y7

×÷

+–

You Can Also Use GEMA For the Order of Operations

Lesson


GEMA is another way to remember the order of operations:

Exponents Then evaluate any exponents.

Next follow any multiplication and division instructions from left to right.

You can use either PEDMAS or GEMA — whichever one you feel happier with.

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Example 1

Lesson


What is 8 ÷ 4 • 4 + 3?

Solution

8 ÷ 4 • 4 + 3

Finally do the addition to get the answer

Then the multiplication

Do the division first

There are no parentheses or exponents

= 11

= 8 + 3

= 2 • 4 + 3 You do the division first as it comes before the multiplication, reading from left to right.

Solution follows…

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Guided Practice

Lesson


Solution follows…

Evaluate the expressions in Exercises 1–6.

1. 3 – 2 + 6 – 1

3. 4 + 3 – 2 + 7

5. 40 – 10 ÷ 5 • 6

2. 6 ÷ 2 + 1

4. 2 + 5 • 10

6. 5 + 10 ÷ 10

6

12

28

4

52

6

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Always Deal with Parentheses First

Lesson


When a calculation contains parentheses, you should deal with any operations inside them first.

You still need to follow the order of operations when you’re dealing with the parts inside the parentheses.

2(4 • 52) + 1

2(4 • 52) + 1

2(4 • 25) + 1

2(100) + 1

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Example 2

Lesson


What is 10 ÷ 2 • (10 + 2)?

SolutionThe order of operations says that you should deal with the operations in the parentheses first — that’s the P in PEMDAS.

= 60

= 5 • 12

Finally do the multiplication

Do the addition in parentheses= 10 ÷ 2 • 12

Then do the division You do the division first here because it comes first reading from left to right.

10 ÷ 2 • (10 + 2)

Solution follows…

First write out the expression

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Guided Practice

Lesson


Solution follows…


7. 10 – (4 + 3)

9. 10 ÷ (7 – 5)

11. 10 • (2 + 4) – 3

13. 6 • (8 ÷ 4) + 11

8. (18 ÷ 3) + (2 + 3 • 4)

10. 41 – (4 + 2 – 3)

12. (5 – 7) • (55 ÷ 11)

14. 32 + 2 • (16 ÷ 2)

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5

57

23

20

38

–10

48

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PEMDAS Applies to Algebra Problems Too

Lesson


The order of operations still applies when you have calculations in algebra that contain a mixture of numbers and variables.

Do the addition in parentheses, then the multiplication

3 • (2 + 4)

a • (2 + 4)

3 • (b + 4)



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Example 3

Lesson


Simplify the calculation k • (5 + 4) + 16 as far as possible.

Solution

= 9k + 16

Do the addition in parentheses= k • 9 + 16

Then the multiplication

k • (5 + 4) + 16 First write out the expression

Solution follows…

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Guided Practice

Lesson


Solution follows…

Simplify the expressions in Exercises 15–20 as far as possible.

15. 5 + 7 • x

17. 3 • (y – 2)

19. 20 + (4 • 2) • t

16. 2 + a • 4 – 1

18. 10 ÷ (3 + 2) – r

20. p + 5 • (–2 + m)

5 + 7x

3y – 6

8t + 20

4a + 1

2 – r

p – 10 + 5m

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Independent Practice

Lesson


Solution follows…

1. Alice and Emilio are evaluating the expression 5 + 6 • 4.

Their work is shown below.

Explain who has the right answer.

Alice5 + 6 • 4= 11 • 4= 44

Emilio5 + 6 • 4= 5 + 24= 29

Emilio has the right answer because he has used the correct order of operations: he has done the multiplication before the addition.

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Lesson


Solution follows…

The local muffler replacement shop charges $75 for parts and $25 per hour for labor.

2. Write an expression with parentheses to describe the cost,

in dollars, of a replacement if the job takes 4 hours.

3. Use your expression to calculate what the cost of the job would be if it did take 4 hours.

75 + (4 • 25)

$175

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Lesson


Solution follows…

8. Paul buys 5 books priced at $10 and 3 priced at $15. He also has a coupon for $7 off his purchase.

Write an expression with parentheses to show the total cost, after using the coupon, and then simplify it to show how much he spent.


4. 2 + 32 ÷ 8 – 2 • 5

6. 7 + 5 • (10 – 6 ÷ 3)

5. 4 + 7 • 3

7. 3 • (5 – 3) + (27 ÷ 3)

–4

47

25

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(5 • 10) + (3 • 15) – 7. He spent $88.

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Lesson


Solution follows…

Simplify the expressions in Exercises 10–12 as far as possible.

10. x – 7 • 2

11. y + x • (4 + 3) – y

12. 6 + (60 – x • 3)

9. Insert parentheses into the expression 15 + 3 – 6 • 4 to make it equal to 48.

(15 + 3 – 6) • 4

x – 14

7x

66 – 3x

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Round UpRound Up

Lesson


This will feature in almost all the math you do from now on, so you need to know it.

Don’t worry though — just use the word PEMDAS or GEMA to help you remember it.

If you evaluate an expression in a different order from everyone else, you won’t get the right answer. That’s why it’s so important to follow the order of operations.

lesson 1.1.3

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