Order of Operations (PEMDAS)

We are going to start with PEMDAS, which tells you the order to add, subtract, and multiply.

5 + 2 × 3 = (5 + 2) × 3 = (7) × 3 = 21  Add first, then multiply?
or
5 + 2 × 3
= 5 + (6) = 11   Multiply first, then add?

Which answer is correct?

The correct Order of Operations is: Parentheses – Exponents – Multiplication – Division – Addition – Subtraction.

 

Video Courtesy of (site-affiliate) Kaplan GMAT prep.

Please Excuse MDear Aunt Sally

Please

Parentheses

(-5 + 2)(-3) = (-3)(-3) = 9

Excuse

Exponents

22 + 4 = + 4 = 8

My

Multiplication

5 + × 3 = 5 + 6 = 11

Dear

Division

\dfrac{\bold{8}}{\bold{2}} + 2 = 4 + 2 = 6

Aunt

Addition

4 + 6 – 10 = 10 – 10 = 0

Sally

Subtraction

 5 – 2 =  2 = 0

Step 1: Parenthesis

Parentheses can be used for notating fractions and negative numbers.

\dfrac{8 + 4}{5 \,–\, 2} = \dfrac{(8 + 4)}{(5 \,-\, 2)} = \dfrac{12}{3} = 4

25 × (\dfrac{1}{5}) = 5

5 + (-2) = 3

Step 2: Combine all like terms in an expression.

2x + x – y + 4y

3x + 3y

The expression is simplified by combining the like terms (x‘s and y’s).

Step 3: Distribute numbers to eliminate parentheses.

2(x – y) = 2x – 2y
-3(x – y) = -3x + 3y       Be sure to distribute the negative to each term.

Step 4: Evaluate expressions’ grouping symbols from the inside out.

x(x + 2(3x + 4)  3)

x(x + 6x + 8  3)

Distribute 2 into the inner parentheses: 2(3x + 4) = 6x + 8

x(7x + 5)

Combine like terms (the x‘s) inside the parentheses.

7x2 + 5x

Distribute to remove the last parentheses.

Addition and Subtraction

  • Addition and subtraction have equal precedence.
  • Go from left to right. (8 – 6 is not the same as 6 – 8).
  • Subtraction is the same as addition of a negative number:
    5 – 3 = 5 + (-3) = 2.

Multiplication and Division

  • Multiplication and division have equal precedence
  • Division is the same as multiplying by the reciprocal:

18 × 2 ÷ 3 ÷ 2 is easier when you do:

18 × 2 × \dfrac{1}{2} × \dfrac{1}{3}

= 18 × 1 × \dfrac{1}{3} = \dfrac{18}{3} = 6.

  • Make sure you first change division to multiplication by reciprocals if you change the order:(\dfrac{18}{3} is not the same as \dfrac{3}{18}).
  • Make sure you first change division to multiplication by reciprocals if you change the order: (\dfrac{18}{3} is not the same as:
    \dfrac{3}{18}).

\dfrac{16}{(2(8 \,-\, 3(4 \,-\, 2)))} + 3 =

Solution

\dfrac{16}{(2(8 \,-\, 3(4 \,-\, 2)))} + 3

\dfrac{16}{(2(8 \,-\, 3\bold{(2)}))} + 3       …Combine the inner parentheses: 4 – 2 = 2

\dfrac{16}{(2(\bold{2}))} + 3       …Combine the next parentheses: 8 – 3(2) = 2

\dfrac{16}{\bold{4}} + 3       …Multiply out 2(2) = 4

4 + 3 = 7       …Division, \dfrac{16}{4} = 4, before addition.

\dfrac{10 \,-\, 3(8 \,-\, 3)^{\displaystyle{2}}}{15} =

Solution

\dfrac{10 \,-\, 3(8 \,-\, 3)^{\displaystyle{2}}}{15}\\[1ex]
\dfrac{10 \,-\, 3(\bold{5})^{\displaystyle{\bold{2}}}}{15}       …Combine inside the parentheses.

\dfrac{10 \,-\, 3(\bold{25})}{15}       …Calculate the exponent.

\dfrac{10 \,-\, \bold{75}}{15}       …Multiply.

10 \,-\, \bold{5} = \bold{5}       …Division, \dfrac{75}{15} = \bold{5}, before subtraction.

Types of Integers

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Positive number

a number greater than zero, such as +5 (usually written simply as 5).

Negative number

a number less than zero, such as -5.

Positive and Negative Numbers: Addition and Subtraction

All numbers, except zero, are positive or negative.

Adding two numbers can give a positive or a negative number.
positive + positive = positive
negative + negative = negative

Positive + negative can give a positive, a negative, or zero.
examples
-7 + 9 = 2             
4 + (-5) = 4 – 5 = -1              
2 + (-2) = 0

Subtracting two numbers can give a positive, a negative, or zero.
positive – positive        
9 – 7 = 2             
7 – 9 = -2       
7  7 = 0
negative – negative     
-9 – (-7) = -2       
-7 – 9 = -16     
-9  (-9) = 0
positive – negative       
9 – (-7) = 9 + 7 = 16
negative  positive       
-9 – 7 = -16

Be sure to distribute the negative sign to all terms inside parentheses.

9 – (14 – 8)

Solution

9 – (14 – 8) = 9 – 14 + 8 = 3
You can use the order of operations to check:
9 – (14 – 8) = 9 – 6 = 3

10 – (5 + 2)

Solution

10 – (5 + 2) = 10 – 5 – 2 = 3
You can use the order of operations to check:
10 – (7)  = 3

Multiplication and Division

When multiplying or dividing: if the two numbers have the same sign, then the product is positive. If the two numbers have different signs, the product is negative.  A way to remember this is Same Signs product PoSitive.

positive × positive = positive
6 × 2 = 12

positive × negative = negative
6 × (-2) = -12          
(-6) × 2 = -12

negative × negative = positive
(-6) × (-2) = 12

\dfrac{positive}{positive} = positive       

\dfrac{6}{2} = 3

\dfrac{positive}{negative} = negative       

\dfrac{6}{(-2)} = -3   

\dfrac{(-6)}{2} = -3

\dfrac{negative}{negative} = positive       

\dfrac{(-6)}{(-2)} = 3

800score Strategies

Use Plug-In or Backsolving on odd/even and positive/negative questions.
Make up numbers to substitute in for variables on questions that ask if variables result in a positive or negative number. For example, if x < 0, is x2 positive? Substitute -2 for x and you can see that (-2)2 is positive. This is also very useful for double checking.

Make a list
If a positive, negative or prime number question asks “How many two digit prime numbers” or “How many two digit numbers less than 50” then you have a finite (small) number of possibilities. Here it is useful to make a list of all the number possibilities rather than try to solve the question using algebra.

Only integers can be odd, even, or prime numbers.

Odd and Even: Adding, Subtracting and Multiplying

Even number

an integer that is divisible by 2
(…, -6, -4, -2, 0, 2, 4, 6, 8, 10, …).
No, don’t email us saying that’s a typo; zero is an even integer.

Prime number

a positive integer with exactly two different positive divisors: 1 and itself. For example, 2, 3, 5, 7 are prime numbers.
Note: 1 is not considered a prime number since it has only one positive divisor.

a number that can be expressed as a fraction of two integers.

Odd number

an integer not divisible by 2
(…, -5, -3, -1, 1, 3, 5, 7, 9, …).

Looking at examples is the easiest way to know the rules for odd and even.

Rule

Example

even + even = even 4 + 4 = 8 (even)
odd + even = odd 3 + 4 = 7 (odd)
odd + odd = even 3 + 3 = 6 (even)
even – even = even 16 – 8 = 8 (even)
even – odd = odd 16 – 5 = 11 (odd)
odd – odd = even 9 – 5 = 4 (even)
even × even = even × 4 = 8 (even)
even × odd = even × 3 = 6 (even)
odd × odd = odd × 5 = 15 (odd)

If k is an odd integer, is the following expression even or odd?

kk + k

Solution

(k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd.

If k is an odd integer, is the following expression even or odd?

k × k × k

Solution

k × k is odd. Thus (k × k× k is an odd times an odd, which is odd.

If k is an odd integer, is the following expression even or odd?

k+ 2k

Solution

k + 2is an odd plus an even, which is odd.

If k is an odd integer, is the following expression even or odd?

2k × k

Solution

2k is even. An even times an odd is even.

Dividing Odd and Even:

examples:

Dividing even by even can be even, odd, or not an integer.

\dfrac{40}{4} = 10       

\dfrac{40}{8} = 5       

\dfrac{8}{40} = \dfrac{1}{5}

Dividing even by odd can be even, odd, or not an integer.

\dfrac{30}{5} = 6       

\dfrac{30}{6} = 5       

\dfrac{15}{30} = \dfrac{1}{2}

Dividing odd by odd cannot be even.

\dfrac{15}{3} = 5

Dividing odd by even cannot be an integer.

\dfrac{15}{2} = 7.5

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Number Rules

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