Operations - GMAT Math Study Guide
Definitions/Terms
- Sum - The number that results from the addition of two other numbers.
For example, 5 is the sum of 2 and 3 since 2 + 3 = 5
- Difference - The number that results from subtracting two numbers.
For example, 10 is the difference of 25 and 15 since 25 - 15 = 10
- Product - The number that results from multiplying two numbers together.
For example, 12 is the product of 3 and 4 since 3*4 = 12
- Quotient - The number that results from two numbers being divided.
For example, 5 is the quotient of 15 and 3 since 15/3=5
- Dividend - The number that is being divided by another number.
For example, 100 is the dividend in the expression 100/20
- Divisor - The number that is dividing another number.
For example, 20 is the divisor in the expression 100/20
There are four basic operations: addition, subtraction, multiplication and division.
Addition
Addition is done by lining up numbers along their digits and summing up the columns, starting from the right and moving left. If the sum of a column is greater than 10, the tens digit is "carried" to the next column while the units digit of the column sum represents the units digit for the answer. For example:
113
17
23
+16
169
The sum of the ones column is 19 [=3+7+3+6]. Thus, the units digit is 9. Since this number (i.e., 19) is greater than 10, an extra 1 must be placed into the tens column. Hence, the value of the tens digit of the answer is: 6=1+1+2+1+1 [from carrying]. Finally, the sum of the hundredths place is 1. Thus, 113+17+23+16 = 169.
Subtraction
Subtraction is done by lining up numbers so that their digits match and the number being subtracted is on the bottom. Starting from the right, the bottom digit in each column is subtracted from the digit above it. If the digit being subtracted is larger, the next digit of the top number is lowered by one, while the current digit has 10 added to it. This is called "borrowing." For example:
125
-36
89
Since 5<6, "borrow" from the 2 by making it 1 and making the 5 a 15. 15-6 = 9
Since 1<3 [remember: the 2 becomes a 1 from the above borrowing], "borrow" from the 1 by making it 0 and making the 1 an 11. 11-3 = 8.
125-36 = 89
Multiplication
Multiplication is the shorthand way of repeating addition. 4*3 = 12 represents 4+4+4 = 12 or 3+3+3+3 = 12.
Multiplication of numbers with decimals involves a few more steps than multiplication of integers. (1) Multiply the numbers out while ignoring the decimals. (2) Determine the total number of digits to the right of the decimal in all the numbers being multiplied. (3) Place a decimal in the answer so it has an equal number of decimals to the right of it. For example:
1.1
x 1.3
1.43
There is one digit to the right of the decimal in 1.1 and one digit to the right of the decimal in 1.3 for a combined total of two digits to the right of the decimal. Thus, a decimal should be added to 143 so it has two digits to the right of it, yielding: 1.43
However:
11
x 13
143
The answer in this case is 143 since there are no digits in the numbers being multiplied (i.e., there are no digits in the final answer).
Division
Division is the inverse of multiplication. It is the process of determining the number of times a number goes into another number. For example, since four goes into 20 five times, 20 divided by 5 equals 4.
If x (the dividend) is divided by y (the divisor), determine the number of times y will go into x. If x can be divided by y without a remainder, x is said to be divisible by y. However, if y is not a divisor (or factor) of x, then there will be a remainder left over. If the solution is not an integer (has a remainder) or if x and y are large numbers, long division should be used.
Tip: Simplify Before Multiplying
When faced with a division problem, it can often speed up calculations significantly if you: (1) Factor the dividend and divisor (i.e., numerator and denominator). (2) Cancel common (or repeated) terms. Consider the following example:
Factor Shortcut Rules
2 will be a divisor if the dividend is even
3 will be a divisor if the sum of the dividend's digits is divisible by 3
4 will be a divisor if the dividend can be divided by 2 twice (still even after divided by 2 once)
5 will be a divisor if the dividend's last digit is 0 or 5
6 will be a divisor if the dividend is divisible by 2 and 3
8 will be a divisor if the dividend can be divided by 2 thrice (still even after dividing by 2 twice)
9 will be a divisor if the sum of the dividend's digits is divisible by 9
10 will be a divisor if the dividend's last digit is 0
Types of GMAT Problems
- Combination of Multiple Operations
When dealing with multiple operations in the same problem, make sure to follow the order of operations: PEMDAS - parenthesis, exponents, multiplication, division, subtraction
Correct Answer: C