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 Kaplan GMAT prep.
Please Excuse My Dear Aunt Sally
Please
Parentheses
(-5 + 2)(-3) = (-3)(-3) = 9
Excuse
Exponents
22 + 4 = 4 + 4 = 8
My
Multiplication
5 + 2 × 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
7 – 5 – 2 = 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
Video Courtesy of Kaplan GMAT prep.
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.
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 | 2 × 4 = 8 (even) | |
even × odd = even | 2 × 3 = 6 (even) | |
odd × odd = odd | 3 × 5 = 15 (odd) |
If k is an odd integer, is the following expression even or odd?
k+ k + 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 + 2k is 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
https://www.youtube.com/watch?v=f4TQU21jF4A
https://www.youtube.com/watch?v=Jd6QMrqoXKA
https://www.youtube.com/watch?v=6Bt9nUTlPwA
https://www.youtube.com/watch?v=XIQm52Csj9k
https://www.youtube.com/watch?v=gpIJWPUe06E
https://www.youtube.com/watch?v=78kcmq6kYPQ
Before attempting these problems, be sure to review this section on data sufficiency questions.
https://www.youtube.com/watch?v=M4sMIHg18Jk
https://www.youtube.com/watch?v=T9GWET9haOs
https://www.youtube.com/watch?v=5FyW40hCQl4
https://www.youtube.com/watch?v=rt26153cBQQ
https://www.youtube.com/watch?v=J5sik_peFMk
https://www.youtube.com/watch?v=W0ryiIh59D8
https://www.youtube.com/watch?v=-u5QkLrzQNs
https://www.youtube.com/watch?v=H9Hj5WSp-lM
https://www.youtube.com/watch?v=xb_LAoLks6g
https://www.youtube.com/watch?v=YdO5GFMBcKQ
https://www.youtube.com/watch?v=vszotK5lnZ8
Video Quiz
Number Rules
Best viewed in landscape mode
11 questions with video explanations
100 seconds per question