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Introduction to the order of operations

How would you solve this problem?

12 - 2 ⋅ 5 + 1

The answer you get will depend largely on theorderin which you solve the problem. For example, if you work the problem fromlefttoright—12-2, then 10⋅5, then add 1—you'll get 51.

12 - 2⋅ 5 + 1
10 ⋅ 5+ 1
50 + 1
51

On the other hand, if you solve the problem in theoppositedirection—fromrighttoleft—the answer will be 0.

12 - 2 ⋅5 + 1
12 -2 ⋅ 6
12 - 12
0

Finally, what if you did the math in a slightly different order? If youmultiplyfirst, thenadd, the answer is3.

12 -2 ⋅ 5+ 1
12 - 10+ 1
2 + 1
3

It turns out that 3 actuallyisthe correct answer because it's the answer you get when you follow the standardorder of operations. The order of operations is a rule that tells you the rightorderin which to solve different parts of a math problem. (Operationis just another way of sayingcalculation.Subtraction, multiplication, and division are all examples of operations.)

The order of operations is important because it guarantees that people can all read and solve a problem in the same way. Without a standard order of operations, formulas for real-world calculations in finance and science would be pretty useless—and it would be difficult to know if you were getting the right answer on a math test!

Using the order of operations

The standard order of operations is:

  1. Parentheses
  2. Exponents
  3. Multiplication and division
  4. Addition and subtraction

In other words, inanymath problem you must start by calculating theparenthesesfirst, then theexponents, thenmultiplicationanddivision, thenadditionandsubtraction. For operations on the same level, solve fromlefttoright. For instance, if your problem contains more than one exponent, you'd solve the leftmost one first, then work right.

Let's look at the order of operations more closely and try another problem. This one might look complicated, but it's mainly simple arithmetic. You can solve it using the order of operations and some skills you already have.

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 32- 8

Parentheses

Always start with operations contained within parentheses. Parentheses are used togroupparts of an expression.

If there is more than one set of parentheses, first solve for the ones on the left. In this problem, we only have one set:

4 / 2 ⋅ 3 +(4 + 6 ⋅ 2)+ 18 / 32- 8

In any parentheses, you follow the order of operations just like you do with any other part of a math problem.

Here, we have two operations:additionandmultiplication. Because multiplication always comes first, we'll start by multiplying 6 ⋅ 2.
4 / 2 ⋅ 3 + (4 +6 ⋅ 2) + 18 / 32- 8
6⋅2 is 12. Next, we'll add 4.
4 / 2 ⋅ 3 + (4 + 12) + 18 / 32- 8
4+12is 16. So we've simplified our parentheses to16.Since we just have a single number in the parentheses, we can get rid of them all together—they're notgroupingtogether anything now.
4 / 2 ⋅ 3 +16+ 18 / 32- 8

Exponents

Second, solve anyexponents. Exponents are a way ofmultiplyinga number by itself. For instance, 23is2multiplied by itselfthreetimes, so you would solve it by multiplying2⋅2⋅2. (To learn more about exponents, review our lessonhere).

There's only one exponent in this problem:32. 32is3multiplied by itselftwice—in other words,3 ⋅ 3.
4 / 2 ⋅ 3 + 16 + 18 /32- 8
3 ⋅ 3is 9, so 32can be simplified as9.
4 / 2 ⋅ 3 + 16 + 18 /9- 8

Multiplication and division

Next, look for anymultiplicationordivisionoperations. Remember, multiplication doesn't necessarily come before division—instead, these operations are solved fromlefttoright.

Starting from the left means that we need to solve4 / 2first.
4 / 2⋅ 3 + 16 + 18 / 9 - 8
4divided by2is2. That makes our next problem2 ⋅ 3.
2 ⋅ 3+ 16 + 18 / 9 - 8
2 ⋅ 3is 6. Finally, there's only one multiplication or division problem left:18 / 9.
6 + 16 +18 / 9- 8
18 / 9is 2. There's nothing left to multiply or divide, so we can move on to the next and final part of the Order of Operations:additionandsubtraction.
6 + 16 +2- 8

Addition and subtraction

Our problem looks a lot simpler to solve now. All that's left is addition and subtraction.

Just like we did with multiplication and division, we'll add and subtract fromlefttoright. That means that first we'll add6and16.
6 + 16+ 2 - 8
6 + 16is22. Next, we need to add22to2.
22 + 2- 8
22 + 2is24. Only one operation left:24 - 8.
24 - 8
24-8is 16. That's it!
16

We're done! We've solved the entire problem, and the answer is16. In other words,4 / 2 ⋅ 3 + ( 4 + 6 ⋅ 2 ) + 18 / 32- 8 equals16.

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 32- 8 =16

Whew! That was a lot to say, but once we broke it down into the right order it really wasn't that complicated to solve. When you're first learning the order of operations, it might take you a while to solve a problem like this. With enough practice, though, you'll get used to solving problems in the right order.

To see the order of operations in action with another problem, check out this video.