# Order of Operations - GMAT Math Study Guide

## Table of Contents

## Definitions

- Order of Operations - The sequence (or order) by which computations (or operations) are performed on an expression, term, equation, etc.

## Order of Operations

The order of operations refers to the order in which fundamental mathematical operations must be done. For example, in the question "10 + (5^{2})^{4}," do you raise 2 to the 4th power before raising 5 to the 2nd power or vice versa? The order of operations answers this question. Many times, the acronym PEMDAS ("Please Excuse My Dear Aunt Sally") is used to remember the order.

P - Parentheses

E - Exponents

M - Multiplication

D - Division

A - Addition

S - Subtraction

### P - Parentheses

- Work from the innermost to the outermost parentheses
- Evaluate absolute value and []

### E - Exponents

- Work from left to right

### M & D - Multiplication & Division

- Work from left to right

4 * 6 = 24

Note: Working from right to left would produce the wrong answer: 2 * 6 = 12 and 8 / 12 = .66667

### A & S - Addition & Subtraction

- Work from left to right

4 + 4 = 8

Note: Working from right to left would produce the wrong answer: 8 + 4 = 12 and 12 - 12

## Examples

Consider the following example:

^{2}+ 1

P - There are no parentheses.

E - There is an exponent, so this should be evaluated first.

5

^{2}+ 1 = 25 + 1 [Note: It would be wrong to add the 1 to the 5 and then square it as this would violate the order of operations and result in a wrong answer.]

M - There is no multiplication.

D - There is no division.

A - There is addition.

Add 1 to 25

25 + 1 = 26

S - There is no subtraction.

The final answer: 5

^{2}+ 1 = 26 [and not 6

^{2}= 36, which wrongly adds before taking an exponent].

Consider the following example:

P - There are no parentheses.

E - There is no exponent.

M - There is multiplication, so do this first (before adding 10 to 5).

5 + 10 * 2 = 5 + 20

D - There is no division.

A - There is addition.

5 + 20 = 25

Consider the following example:

P - There are parentheses, so do what is within them first. Do not multiply 5 by -10

(15 - 10) * 5 = 5 * 5

E - There is no exponent.

M - There is multiplication.

5 * 5 = 25

Operations on the same level should be done from left to right--although in some cases they can be done in any order and the result will still be correct. In other words, if the equation in question or term being evaluated only involves division and multiplication, the order of evaluation amongst the same type of terms should be done from left to right even though it will not matter every single time. For example:

P - There are no parentheses.

E - There is no exponent.

M - There is multiplication.

In this case, it does not matter what order you compute 3*5, 2*3, and 4*6. The only potential mistake is to add before multiplying--but this is a violation of the order of operations. To set good habits, we will compute from left to right.

3*5 + 2*3 + 4*6 = 15 + 2*3 + 6*4

15 + 2*3 + 4*6 = 15 + 6 + 6*4

15 + 6 + 6*4 = 15 + 6 + 24

D - There is no division.

A - There is addition.

15 + 6 + 24 = 21 + 24 = 45 [The order within the level did not matter in this case; Right to Left: 24+6=30 and 30 + 15 = 45]

## Types of GMAT Problems

- Multiple Sets of Parentheses
Calculations should start with the innermost parentheses and work outwards.

(2-(11*(3+(5-7))))/9Correct Answer:**B**- Starting with the innermost parentheses and moving out.

(2-(11*(3+(5-7))))/9 = (2-(11*(3+(-2))))/9

(2-(11*(3+(-2))))/9 = (2-(11*(1)))/9

(2-(11*(1)))/9 = (2-(11))/9

(2-(11))/9 = (-9)/9

(-9)/9 = -1

- Starting with the innermost parentheses and moving out.
- Multiple Exponents
What is the value of the following expression?Correct Answer:
**C**- (3
^{2})^{2}= 9^{2}= 81

- (3