1. Order of Operations
Lesson 2.1.4

2. Order of Operations 2.1.4 California Standards: What it means for you:
Lesson
2.1.4
Order of Operations
California Standards:
Algebra and Functions 1.3
Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Algebra and Functions 1.4
Solve problems manually by using the correct order of operations or by using a scientific calculator.
What it means for you:
You’ll see that it matters in which order you evaluate an expression, and you’ll learn the correct order.
Key Words:
parentheses
exponent
multiply and divide
add and subtract
expression
evaluate

3. Lesson
2.1.4
Order of Operations
Sometimes you’ll meet expressions containing different combinations of operations (such as ×, ÷, +, and –).
2 × 4 – 6
4 –
2 – x ÷ 3
a × 4 + b × 3
When you do, you need to be sure to do all of them in the right order.

4. Lesson
2.1.4
Order of Operations
The Order In Which You Evaluate Makes a Difference
The expression × 7 looks like it could give two different answers, depending on how you work it out.
Working out “2 + 3” first gives: × 7 = 5 × 7 = 35
Working out “3 × 7” first gives: × 7 = = 23
To avoid this situation, this rule is used for all math expressions:
Always do multiplication before addition, unless parentheses tell you to do otherwise.

5. Order of Operations 2.1.4 Evaluate 2 + 3 × 7, and (2 + 3) × 7.
Lesson
2.1.4
Order of Operations
Example 1
Evaluate × 7, and (2 + 3) × 7.
Solution
2 + 3 × 7
Work out the multiplication first =
= 23
The parentheses tell you to do the addition first
(2 + 3) × 7
= 5 × 7
= 35
You do all multiplications and divisions before additions and subtractions (unless parentheses tell you otherwise).
Solution follows…

6. Order of Operations 2.1.4 Evaluate: (i) 32 – 3 × 8 (ii) 56 – 8 ÷ 2
Lesson
2.1.4
Order of Operations
Example 2
Evaluate: (i) 32 – 3 × 8 (ii) 56 – 8 ÷ 2
Solution
(i) 32 – 3 × 8
Work out the multiplication first
= 32 – 24
= 8
(ii) 56 – 8 ÷ 2
Work out the division first
= 56 – 4
= 52
Solution follows…

7. Lesson
2.1.4
Order of Operations
You do multiplications and divisions working from left to right.
You also do additions and subtractions from left to right — after you’ve finished all the multiplications and divisions.
For example, 3 × 7 – ÷ 3
= (3 × 7) – 2 + (9 ÷ 3)
= (21) – 2 + (3) =
= 22

8. Order of Operations 2.1.4 Evaluate: 45 + 34 ÷ 17 – 2 × 3. Solution
Lesson
2.1.4
Order of Operations
Example 3
Evaluate: ÷ 17 – 2 × 3.
Solution
÷ 17 – 2 × 3
× and ÷ first, from left to right
= – 2 × 3
= – 6
Then + and –, from left to right
= 47 – 6
= 41
Solution follows…

9. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 1–4.
× 5
× 4
3. 12 ÷ 2 + 4
4. 8 × 7 + 1
4 + 2 × 5 = = 14
3 + 7 × 4 = = 31
12 ÷ = = 10
8 × = = 57
Solution follows…

10. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 5–8.
5. 11 – 1 × 4
÷ 2
7. (88 – 3) ÷ 5
÷ 3 × 5 – 3
11 – 1 × 4 = 11 – 4 = 7
÷ 2 = = 98
(88 – 3) ÷ 5 = (85) ÷ 5 = 17
8 + 9 ÷ 3 × 5 – 3
= × 5 – 3
= – 3
= 23 – 3
= 20
Solution follows…

11. Lesson
2.1.4
Order of Operations
Remember... multiplications and divisions before additions and subtractions — unless parentheses tell you otherwise.
More generally, the order you should perform operations is as follows:
Order of Operations:
( ) [ ] 1. Evaluate anything inside parentheses (or brackets)
y2 2. Evaluate exponents
× ÷ 3. Multiply and divide from left to right
+ – 4. Add and subtract from left to right

12. Lesson
2.1.4
Order of Operations
An easier way to remember the order of operations is to remember the word PEMDAS.
This word stands for:
( ) [ ] 1. Parentheses
y2 2. Exponents
× ÷ 3. Multiplication / Division
+ – 4. Addition / Subtraction

13. Order of Operations 2.1.4 Evaluate: 32 × 5 – (20 – 2) ÷ 32. Solution
Lesson
2.1.4
Order of Operations
Example 4
Evaluate: 32 × 5 – (20 – 2) ÷ 32.
Solution
32 × 5 – (20 – 2) ÷ 32
= 32 × 5 – 18 ÷ 32
Parentheses first
= 9 × 5 – 18 ÷ 9
Then exponents
= 45 – 18 ÷ 9
× and ÷, from left to right
= 45 – 2
= 43
+ and –, from left to right
Solution follows…

14. Lesson
2.1.4
Order of Operations
Example 5
Alan and Jessica are each trying to calculate the expression 24 ÷ 3 – 22.
Their work is shown below. Who has the correct answer?
Alan: 24 ÷ 3 – 22 = 8 – 22 = 62 = 36
Jessica: 24 ÷ 3 – 22 = 24 ÷ 3 – 4 = 8 – 4 = 4
Solution follows…

15. Order of Operations 2.1.4 Solution
Lesson
2.1.4
Order of Operations
Example 5
Solution
Alan: 24 ÷ 3 – 22 = 8 – 22 = 62 = 36
Jessica: 24 ÷ 3 – 22 = 24 ÷ 3 – 4 = 8 – 4 = 4
Alan performed the division, then the subtraction, then evaluated the exponent.
That is incorrect — he should have found the exponent first.
Jessica evaluated the exponent first, then performed the division, then the subtraction.
This is the right order, so Jessica has the right answer.

16. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 9–18.
(10 – 2) ÷ 4
10. 4 – 2 × 16
11. 5 × × 5
12. 7 × 3 – 8 ÷ 2 + 6
× 9
8 + (10 – 2) ÷ 4 = ÷ 4 = = 10
4 – 2 × 16 = 4 – 32 = –28
5 × × 5 = × 5 = = 70
7 × 3 – 8 ÷ = 21 – 8 ÷ 2 + 6
= 21 – = = 23
× 9 = × 9 = = 45
Solution follows…

17. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 9–18.
– 9 ÷ 3
15. (4 + 1)2 – 12 ÷ 4 – 1
16. 3 × (42 – 5) + 11
17. (62 – 32) ÷ (6 – 3)2
18. (15 ÷ 3 – 2) × (33 – 5 × 4)
23 – 9 ÷ 3 = 8 – 9 ÷ 3 = 8 – 3 = 5
(4 + 1)2 – 12 ÷ 4 – 1 = 52 – 12 ÷ 4 – 1
= 25 – 12 ÷ 4 – 1 = 25 – 3 – 1 = 21
3 × (42 – 5) + 11 = 3 × (16 – 5) + 11
= 3 × = = 44
(62 – 32) ÷ (6 – 3)2 = (36 – 9) ÷ (6 – 3)2
=27 ÷ 32 = 27 ÷ 9 = 3
(15 ÷ 3 – 2) × (33 – 5 × 4) = (5 – 2) × (27 – 20)
= 3 × 7 = 21
Solution follows…

18. Lesson
2.1.4
Order of Operations
Order Rules Also Apply to Expressions with Variables
The order rules apply to all types of expressions, including those with variables.

19. Lesson
2.1.4
Order of Operations
Example 6
Evaluate x + yz when x = –3, y = 9, and z = 10.
Solution
Substitute in your values for x, y, and z
–3 + 9 × 10
There are no parentheses or exponents
–3 + 9 × 10
Carry out the multiplication
–3 + 90
Carry out the addition, giving the answer 87
Solution follows…

20. Lesson
2.1.4
Order of Operations
Example 7
Evaluate 19 – f 2 × (w + q) when f = 3, w = 5, and q = 2.
Solution
Substitute in your values for f, w, and q
19 – 32 × (5 + 2)
Evaluate the parentheses
19 – 32 × 7
Evaluate the exponents
19 – 9 × 7
Carry out the multiplication
19 – 63
Carry out the subtraction, giving the answer
–44
Solution follows…

21. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 19–21,
given that f = 2, j = 3, and g = 19.
19. g – j × 8
20. (f + g) × j + 4
21. ( j 2 – f 2 ) ÷ 5
19 – 3 × 8 = 19 –24 = –5
(2 + 19) × = 21 × = = 67
(32 – 22) ÷ 5 = (9 – 4) ÷ 5 = 5 ÷ 5 = 1
Solution follows…

22. Order of Operations 2.1.4 Guided Practice
Lesson
2.1.4
Order of Operations
Guided Practice
Evaluate the expressions shown in Exercises 22–24,
given that s = 10, k = 3, and q = 1.
22. s × k + qs + 23
23. (k 2 – 1) ÷ 4q
24. q – sk 3
10 × × = 10 ×
= = 63
(32 – 1) ÷ 4 × 1 = (9 – 1) ÷ 4 × 1 = 8 ÷ 4 = 2
1 – 10 × 33 = 1 – 10 × 27 = 1 – 270 = –269
Solution follows…

23. Order of Operations 2.1.4 Independent Practice
Lesson
2.1.4
Order of Operations
Independent Practice
Evaluate Exercises 1–4, given that w = 2, b = 8, c = –0.5.
1. b × (w – c)
2. w × b + w × c
3. (6 – w)2
4. (b + w + 2 × c) ÷ (b – 2 × c) 20 15 16 1
Solution follows…

24. Order of Operations 2.1.4 Independent Practice Felipe: 2 × 18 – 6
Lesson
2.1.4
Order of Operations
Independent Practice
5. Felipe and Sylvia are trying to evaluate the expression 2 × 18 – 6.
Felipe: 2 × 18 – 6
= 36 – 6
= 30
Sylvia: 2 × 18 – = 2 × = 24
Who is correct?
What has the other person done wrong?
Felipe is correct.
Sylvia’s mistake was to do subtraction before multiplication.
Solution follows…

25. Order of Operations 2.1.4 Independent Practice
Lesson
2.1.4
Order of Operations
Independent Practice
Which of the expressions in Exercises 6–11 are true, and which are false? For those that are false, explain what is wrong.
6. (2 + 3) × 6 = × 6
7. (7 + 5) – 3 = – 3
8. (100 ÷ 2) ÷ 25 = 100 ÷ 2 ÷ 25
9. 10 ÷ 2 × 5 = 10 ÷ (2 × 5)
(6 × 9) = × 9
11. 6 × 52 = (6 × 5)2
False. On left, + is first — equals 30.
But on right, × is first — equals 20.
True — both sides equal 9.
True — both sides equal 2.
False. On left, ÷ is first — equals 25.
But on right, × is first — equals 1.
True — both sides equal 57.
False. On left, 52 is first — equals But on right, × is first — equals 900.
Solution follows…

26. Order of Operations 2.1.4 Independent Practice
Lesson
2.1.4
Order of Operations
Independent Practice
12. Carly wants to exercise for a total of 50 hours this month.
She exercised 1.5 hours for each of the first eight days.
Then she exercised 2 hours for each of the next four days.  
Write and evaluate an expression for the remaining number of hours she has to exercise.
50 – (1.5 × 8) – (2 × 4) = 30
Solution follows…

27. Order of Operations 2.1.4 Independent Practice
Lesson
2.1.4
Order of Operations
Independent Practice
13. Mr. Chang earns $12 per hour baking bread at the bakery.
He worked 7 hours on Thursday and c hours on Friday.
Write an expression of the form a × (b + c) that can be used to calculate how much Mr. Chang earned on Thursday and Friday.
14. Given that Mr. Chang worked for 8.5 hours on Friday, find the value of the expression you wrote for Exercise 13.
12 × (7 + c)
12 × ( ) = $186
Solution follows…

28. Order of Operations 2.1.4 Independent Practice
Lesson
2.1.4
Order of Operations
Independent Practice
Which of the expressions in Exercises 15–19 are true, and which are false? For those that are false, explain what is wrong.
15. h – r + q = h – (r + q)
16. a × (5 + g) = a × 5 + g
× f ÷ = 16 × f ÷ (3 + 2)
j – w = (5 + j) – w
19. 18y – 2j = (18 – 2) × (y – j)
False. On left, subtraction is first. But on right, + is first.
False. On left, + is first. But on right, × is first.
False. On left, order is ×, ÷, +. But on right, order is +, ×, ÷.
True
False — completely different equation.
Solution follows…

29. Order of Operations 2.1.4 Round Up
Lesson
2.1.4
Order of Operations
Round Up
Hopefully you can now see how important the order of operations is when evaluating expressions.
You’ll need to be able to get the order right from now on — so make sure you memorize it.