USING THE ORDER OF OPERATIONS WITH INTEGERS

The order of operations is a rule that tells the correct  sequence of steps for evaluating a math expression.

We can remember the order using PEMDAS :

Important Notes :

1. In a particular simplification, if you have both multiplication and division, do the operations one by one in the order from left to right.

2. Multiplication does not always come before division. We have to do one by one in the order from left to right. 

3. In a particular simplification, if you have both addition and subtraction, do the operations one by one in the order from left to right.

Examples : 

16 ÷ 4 x 3 = 4 x 3 = 12

18 - 3 + 6 = 15 + 6 = 21

In the above simplification, we have both division and multiplication. From left to right, we have division first and multiplication next.

So we do division first and multiplication next.

Practice Problems

Problem 1 :

Evaluate :

3 + 6 x (5 + 4) ÷ 3 -7

Solution :

Problem 2 :

Evaluate :

56 - 2(20 + 12 ÷ 4 x 3 - 2 x 2) + 10

Solution :

Problem 3 :

Hannah made four withdrawals of $20 from her checking account. She also wrote a check for $215. By how much did the amount in her checking account change ?

Solution :

Analyze Information :

We have to find the total change in Hannah’s account. Since withdrawals and writing a check represent a decrease in her account, use negative numbers to represent these amounts.

Formulate a Plan :

Write a product to represent the four withdrawals.

-20 + (-20) + (-20) + (-20) = 4(-20)

Add -215 to represent the check that Hannah wrote.

4(-20) + (-215)

Solve :

Evaluate the expression to find by how much the amount in the account changed.

4(-20) - 215 = -80 - 215 (Multiply first)

= -295 (Then, subtract)

The amount in the account decreased by $295.

Justify and Evaluate :

The value -295 represents a decrease of 295 dollars. This makes sense, since withdrawals and writing checks remove money from the checking account.

Problem 4 :

Reggie lost 3 spaceships in level 3 of a video game. He lost 30 points for each spaceship. When he completed level 3, he earned a bonus of 200 points. By how much did his score change?

Solution :

Analyze Information :

We have to find the total change in Reggie's score. Since the lost represents a decrease in his score, use negative numbers to represent the points he lost.

Formulate a Plan :

Write a product to represent the lost of 3 spaceships.

-30 + (-30) + (-30) = 3(-30)

Add 200 to represent the bonus

3(-30) + 200

Solve :

Evaluate the expression to find by how much his score changed.

3(-30) + 200 = -90 + 200 (Multiply first)

= 110 (Then, add)

The points in his score increased by 110.

Justify and Evaluate :

The value 110 represents a increase 110 points in the score. Even though he lost 3 spaceships, finally he got a bonus of 200 points. So, this makes sense.

Problem 5 :

Travis goes to the book fair where paperback books are $1.50 and hardback books are $3. Travis buys five paperback and two hardback books. How much change will Travis receive from a $20 bill?

Solution :

Cost of one paperback book is = $1.50

Cost of one hard book is = $3

Number of paperbook books = 5

Number of hardback books = 2

Amount given = $20

Change = 20 - [3(1.50) + 2(3)]

= 20 - [4.50 + 6]

= 20 - 10.50

= $9.5

Amount of change is $9.5

Problem 6 :

Kim sales necklaces to earn extra money. She charges $10 per necklace for material and $2.75 per hour to make unique gifts. How much would two necklaces cost together if one takes her an hour to make and the other three hours to make?

Solution :

Material cost = $10 per necklace

Making charges = $2.75 per hour

Number of necklace to be made = 2

Number of hours required to make it = 3

= 2(10) + 3(2.75)

= 20 + 8.25

= $28.25

So, the required cost is $28.25

Problem 7 :

A certain Math Club makes 35 bars of laundry soap a week and sells these at $20 each. Before the soap can all be sold, the pupils found out that 6 bars were destroyed by mice. How much will be the total sale at the end of a four-week month? Write an expression and show your work and answer.

Solution :

Number of bars sold = 35

Number of soaps sold = $20

Number of bars destroyed = 6

Number of bars made = 35(4)

= 140

Total number of bars now = 140 - 6

= 134

Total sale = 134(20)

= $2680

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