Arithmetic
Expect to deal with arithmetic in some form or another during the math portion of the GMAT. As a result, you will need to understand basic arithmetic definitions.
The following are the different kinds of numbers that appear on the GMAT.
• Real numbers are numbers on the number line. All of the numbers used on the GMAT are real numbers.
• Rational numbers are numbers expressed as the ratio of two integers, and include all integers and fractions.
• Irrational numbers are real numbers with non-terminating decimals places. They can be either positive or negative.
• Integers are numbers with no fractional or decimal areas. They are all multiples of one. The distinction between an integer and a number (which is not necessarily an integer) is important on the GMAT.
Operations
Operations essentially dictate how the entire mathematical sequence reads on paper. Operations include everything from parenthesis and exponents to simple arithmetic expressions.
The order for operations is as follows:
• Parentheses;
• Exponents;
• Multiplication / Division (from left to right);
• Addition / Subtraction (from left to right).
(An easy way to remember this is the acronym PEMA!)
Let’s look at a sample equation that requires you to follow the above order:
27 + 5 x 22 + (100/5) – 19 = ?
If we don’t follow the order, and just work from left to right, the answer is 26.6.
However, doing it in correct operation order yields the following:
1. Solve the numbers in the parenthesis first:100/5 = 20
2. Solve the exponent: 22 = 4
3. Do the multiplication and division from left to right: 5 x 4 = 20
4. Now the formula is as follows: 27 + 20 + 20 – 19
5. Now do the addition and subtraction from left to right
6. This yields the result of 48.
This answer is a bit different from the answer we got by not following the order of the operations.
Laws of Operations
You should remember the following mathematics laws and their operations so that you can correctly do the calculations.
Commutative Law
Addition and multiplication are commutative. This means that no matter what order they are done in, the result is the same.
10 + 15 = 25
15 + 10 = 25
9 ×2 = 18
2 × 9 = 18
Division and subtraction, however, are not commutative.
9 – 2 = 7
2 – 9 = –7
10/2 = 5
2/10 = .2
Associative Law
Addition and multiplication, as above, are also associative. This means that these operations can be regrouped in any way without changing the end result. Division and subtraction, on the other hand, are not associative.
(10 + 10) + 9 = 10 + (10 + 9)
(5 × 5) × 2 = 5 × (2 × 5)
20 + 9 = 10 + 19
25 × 2 = 5 × 10
29 = 29
50 = 50
Distributive Law
Distributive law allows you to distribute a factor among the terms that are added or subtracted, often seen as a(b + c) = ab + ac.
4(10 + 2) = 4 × 10 + 4 × 2
4 x 12 = 40 + 8
48 = 48
Division can be done in a similar format.
Fractions
Fractions form a significant portion of the math portion of the GMAT. Let’s review the basics of fractions, such as:
3 / 5
3 is the numerator, with the fraction bar below, meaning divided by, and 5 is the denominator.
There is more to remember than simply numerator and denominator, so let’s get into the different types of fractions.
Equivalent Fractions
In this type of fraction, its fractional value remains unchanged when you multiply it by one. When dealing with fractions, multiplying the numerator and denominator by the same non-zero number is equivalent to multiplying them by one; the fractional value remains unchanged.
You also get the same result when dividing by the same non-zero number.
Canceling and Reducing
When dealing with fractions, especially on the GMAT, you have to work to put fractions in the lowest terms. This means that the numerator and denominator cannot be divisible by any other common integer except for one.
The best way to think about this is to look at 6/24. This fraction can be reduced by dividing both the numerator and the denominator by 3, to get 2/8. We can divide the numerator and denominator once more by 2, and get 1/4 .
An example of a question that you will face on the GMAT concerning this is this as follows:
Reduce 50/110 to its lowest terms.
To do this, find a common factor of the numerator and denominator, such as 5, and divide by it to get 10/22 .
At this point, divide by another common factor, 2, to get 5/11
Addition and Subtraction
You can add or subtract two fractions only if they have the same denominator. We first need to find a common denominator, which is generally the lowest common denominator. The common denominator is an important concept in understanding fractions.
Here, the denominators are 5, 3, and 4. To find the lowest common denominator, you can usually just multiply the denominators together.
Lowest common denominator = 5 × 3 × 4 = 60
Then, determine what value each denominator should be multiplied by in order to reach the common denominator, and multiply both the numerator and denominator by that value.
This yields:
Now, all we have to do is add the numerators together over the common denominator.
Multiplication
With fractions, multiplication is different from addition and subtraction. First, no common denominator needs to be in place for multiplication to work. For multiplication, you simply need to reduce both diagonally and vertically, and then multiply numerators together and denominators together.
This is reduced to:
Then, multiply it all together:
Division
Division works just like multiplication and, in fact, all you do is multiply the reciprocal of the divisor, the number going into the other number. To get the reciprocal, invert the fraction by changing the position of the numerator and denominator. Let's look at an example:
We first have to find the reciprocal of 4/9, which is 9/4.
Now we multiply the reciprocal of the divisor:
We then reduce this to:
which equals 3.
Decimal Fractions
Decimal fractions are simply fractions in decimal form. To find the decimal form of a fraction, multiply to the power of ten in the denominator.
In addition, each digit in the decimal has a name. The GMAT will sometimes test you on this, so here is a quick guide. For example,
527.236
5 = the hundreds digit
2 = the tens digit
7 = the units digit
2 = the tenths digit
3 = the hundredths digit
6 = the thousandths digit
Now, let’s look at examples of how to change decimal fractions into actual fractions using an example question from the GMAT.
Arrange in order from smallest to largest: 0.5, 0.55, 0.05, 0.505 and 0.055
.5 = .500 = 500/1000
.55 = .550 = 550/1000
.05 = .050 = 50/100
.505 = .505 = 505/1000
.055 = .055 = 55/1000
Therefore, the order should be: .05 < .055 < .5 < .505 < .55
When adding or subtracting these, make sure that all of the decimal points line up properly, one on top of the other. This will ensure that the tenths add with the tenths, the hundredths add with the hundredths, and so on.
Let’s look at this example:
.9 + .09 + .009
In this case, you will convert it into vertical form:
0.9
-
0.09
-
0.009
0.999
The same holds true when subtracting, so whenever using addition or subtraction, always remember to line up the decimal points to ensure that you get the proper answers.
In terms of multiplication, multiply as you would with any other integer. Then decide where the decimal point should be. The number of decimal places for the answer equals the sum of the number of decimal places from each of the two decimals you are multiplying together.
For example:
0.5 × 0.3 = 5 × 3 = 15 and two decimal spaces = 0.15
1.2 × 1.7 = 12 × 17 = 204 and 2 decimal spaces = 2.04
When dividing a decimal with another decimal, multiply each by the power of 10 so that the divisor becomes an integer. Then, simply carry out the division as you would with integers, placing the decimal point in the quotient directly above the decimal point in the dividend.
For example:
8 ÷ 0.4 =
(8 × 10) ÷ (0.4 × 10) =
80 ÷ 4 = 20
Ø More Practice from the GMAT**®** Review 13****th Edition: Questions 27, 46, 76, 97, 226 (Fractions)
Ø More Practice from the GMAT**®** Review 13****th Edition: Questions 20, 41, 42, 85, 156 (Decimals)