Order of Operations

Imagine if you have all of these things you can do with numbers, but didn’t have a set order for how to do them when many different operations are involved.

7 x 2 + 5

What if I had the above problem and it represented millions of dollars that would go into a bank account. I could multiply 7 x 2 first, then add.

7 x 2 + 5 = 14 + 5 = 19

I coud also add 2 to 5, first, then multiply the result by 7.

7 x 2 + 5 = 7 x 7 = 49

If this represented millions of dollars, that would be a big difference. If we had groups of people approaching this problem in different ways, it would be hard to have a baking system, to develop medicines that require measurements, to participate in sporting events that requires statistics, etc.

So, there is a way we approach these problems.

PEMDAS

If we want something to be done first, we put it in parentheses. Then we do exponents next (exponents tell us how many times a number is multiplied by itself). Then we do multiplication and division (they are at the same level and multiplication does not take priority over division, or vice versa). Finally, we do addition and subtraction (they are also at the same level—subtraction does not take priority over addition or vice versa).

If we have many operations at the same level, we work from left to right to get the answer.

The popular acronym that’s used to understand the order in which these operations should be performed is, PEMDAS.

  • Please (parentheses first)

  • Escort (exponents next)

  • My Dear (multiplication and division next)

  • Aunt Sarah (addition and subtraction last)

  • Work from left to right when you have operations at the same level.

Let’s look at some examples.

In the above example: (7 - 3) times 8 ÷ 2, we would do what’s in parentheses first, then multiply and divide from left to right.

(7 - 3) * 8 ÷ 2

4 * 8 ÷ 2 = 32 ÷ 2 = 16

In this example above, we multiply the 3 * 5 first, to get 15, then add 7 to the 15 to get 22.

In the above problem, its all addition and subtraction (which are at the same level), so we start to work from left to right to get the answer above, — 2.

This example shows what can happen when you add exponents into the mix. Here we have parentheses, so we have to do what’s in the parentheses first (2 * 3, this gives us 6). Then we do the exponent, which says multiply 6 * 6 (or to square 6), this gives us 36. We then move on to subtracting 9 from 36 and we get 27 as the final answer.

Summary

PEMDAS

  • Please (parentheses first)

  • Escort (exponents next)

  • My Dear (multiplication and division next)

  • Aunt Sarah (addition and subtraction last)

  • Work from left to right when you have operations at the same level.

Danita Smith