ABE Math Tutorials/Whole numbers/Order of operations

Order of operation

You may remember the yard sale problem in the lesson on "Introduction to Word Problems":

One weekend, Susan's family had a garage sale with Marika's family. Together they made $64 on Saturday, and $102 on Sunday. They decided to share the money they earned equally. How much money did each family receive from the garage sale?

First we had to add:

64 + 102 = $166

and then we divided:

166 ÷ 2 = $83

to get the answer: Each family will receive $83 from the yard sale.}}

It is time-consuming to write the two operations (adding, then dividing) separately each time. Usually, when we have to do 2 (or 3, or 4, etc.) operations to complete a problem we write them together:

30 + 250 - 208 = 72

However, this can be confusing. Although we often do arithmetic problems like this working from left to right, a mathematician (or a calculator) would do the division first (102 ÷ 2) and then add 64. This would give a completely different answer! To make sure that these kinds of mistakes don't happen, mathematicians have a rule:

Key points

ALWAYS PUT BRACKETS ( ) AROUND THE OPERATION THAT YOU'RE SUPPOSED TO DO FIRST !!!

so in this last example, it would be best to write it this way:

(64 + 102) ÷ 2

This way we are absolutely sure to do the adding inside the brackets (some people call them parentheses) first, before we do the division.

Activity

Ready for more? Try these: The Mountain View Community College had 4500 students enrolled last year. The fall term had 1500 enrollments; the winter term had 1350 enrollments; and the spring term had 1150 enrollments. How many students enrolled for the summer term? Martin pays $585 for car insurance every year. His wife pays $40 per month for her car insurance. How much does this couple pay in total, per year, for their car insurance? To earn some extra money, Karen types letters and answers the phone for a marketing company. She gets paid $3 for each page she types, plus $7 per hour to answer the phone. In a regular 4-hour shift, she can type 10 pages . How much money will she earn, in all, during a 4-hour shift?

And here are the answers:

1. We know that there were 4500 student enrollments for all 4 terms. If we add up the number that enrolled for the 3 terms, then subtracted this amount from the total for the year, we will know how many enrollments there were for the summer term.

• First: we put the numbers in order: 4500 - (1500 + 1350 + 1150)
• Math Step 1: we do the part inside the brackets: 4500 - (4000)
• Math Step 2: we do the subtraction: 4500 - 4000 = 500

So we know that there were 500 enrollments for the summer term.

1. Martin pays his insurance by the year, but his wife pays hers by the month. To find out how much she pays for a year, we will have to multiply her monthly insurance payment times 12 (for 12 months in a year). Once we know how much she pays per year, we can add that amount to what Martin pays for the year, to come up with the total.

• First: we put the numbers in order: (40 x 12) + 585
• Math Step 1: we do the part inside the brackets: (480) + 585
• Math Step 2: we do the addition: 480 + 585 = 1065

It will cost them a total of $1065 for their car insurance.

1. This is a more complicated problem! Karen is sort of doing two things at once: typing letters and answering the phone when it rings. She gets paid for both "jobs" during her shift: $3 for each page , plus $7 for each hour. We'll have to figure out how much she will make for each job during her shift, then add the two together:

• First, we put the numbers in order: (10 x 3) for the typing, + (4 x 7) for telephone answering
• Math Step 1: we do the part inside the brackets: (30) + (28)
• Math Step 2: do the addition: 30 + 28 = 58

She can earn $58 in a regular 4-hour shift.

Order of Operations:

When we did the question above, we first put the numbers in order:

(10 x 3) + (4 x 7)

Preknowledge

You may remember from the lesson on "Different Ways of Expressing an Operation" that there are other ways to express a multiplication.

We could also have written it like this: 

(3)(10) + (4)(7)

or even like this:

3(10) + 4(7)

Whenever you see a number right next to brackets, with no "+" or "-" sign in between, it automatically means that you must multiply whatever is inside the brackets by whichever number is beside it. You still must do the operation (if there is one) inside the brackets first. For example:

5(6 + 2) means that we first must add the 6 + 2; and then multiply that result by 5. The final answer will be 8 x 5 = 40. Sometimes, even with these rules, it could be confusing to figure out which operation to do first. Look at this example:

(2)4 + 3(5)

Which operation should we do first? Should we add the 4 and 3, and then multiply; or should we do the multiplications (2 x4 and 3 x 5) first? In a case like this, we do any multiplications (or divisions) first; and then do any additions (or subtractions) last. These rules are called "The Order of Operations", and so far they look like this: First do any operations that are inside brackets Then, working from left to right, do any multiplications or divisions Finally, working from left to right, do any additions or subtractions that are left.

Activity

Let's try a few: 4(10 + 2) (10 + 2)4 (3 - 1)4 + 3(2 + 3) 5(10) - 3(10)

Ready for the answers?

1. 4(10 + 2) is the same as 4 x 12. The answer is 48.
2. (10 + 2)4 is the same as 12 x 4. The answer is 48.
3. (3 - 1)4 + 3(2 + 3) is the same as 2 x 4 added to 3 x 5. The answer is 23.
4. 5(10) - 3(10) is the same as 5 x 10 subtract 3 x 10. The answer is 20.

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