 # Page 1 Mathematics Knowledge Study Guide for the ASVAB

The Mathematics Knowledge section of the exam measures your knowledge of various math areas, such as algebra and geometry. You may be asked to find the square root of a number or the volume of a brick with given dimensions. Algebraic problems may require finding the value of “y” in a given equation. A review of math symbols—such as ≠, ≤, and √—can help you solve the given problems much faster, and using our ASVAB math study guide to practice answering the algebra and geometry questions on the test can help increase your overall AFQT score. The CAT-ASVAB has 16 questions in 20 minutes; the paper-and-pencil version has 25 questions in 24 minutes.

## Fractions

Like percentages, a fraction is part of a whole but presented differently. It has a numerator and a denominator. In the fraction $\frac{1}{2}$, 1 is the numerator which represents the part, while 2 is the denominator which represents the total number of parts. Fractions, percentages, and decimals are actually different ways of presenting the same concept of parts of a whole. For instance:

The fraction $\frac{1}{100}$ is the same as the percentage 1% and the decimal [0.01].

The fraction $\frac{1}{2}$ is the same as 50% and [0.5].

### Converting Fractions to Percentage or Decimal

ASVAB questions involve converting fractions to percentages or decimals, or the other way around. The methods are quite straightforward, actually, and are as follows:
To convert a fraction to its percentage form: divide the numerator by the denominator, multiply by 100, and then add the % sign.

Thus, converting $\frac{1}{2}$ to a percentage involves the following steps:

Divide the numerator by denominator:

(Note: This is the decimal form of $\frac{1}{2}$.)

Multiply by 100: 0.5 x 100 = 50

To revert 50% to fraction form:

Write 50% as $\frac{50}{100}$.

Reduce $\frac{50}{100}$ to its lowest form, which is $\frac{1}{2}$.

### Simplifying Fractions

Many questions require simplifying fractions. It means reducing a fraction to its simplest form.

Take the following fractions:

They all actually reduce to $\frac{1}{2}$. Did you find it easy simplifying those fractions to $\frac{1}{2}$? If not, here’s how:

Let’s take the fraction: $\frac{7}{14}$

Factor the numerator and denominator:

The 7s in the numerator and denominator cancel out, leaving the reduced fraction: $\frac{1}{2}$.

### Equivalent Fractions

Equivalent fractions look different at first glance, but are actually the same in value. In the previous illustration, for instance, $\frac{1}{2}$ is equivalent to:

$\frac{7}{14}, \frac{4}{8}, \frac{2}{4}, \frac{3}{6}, \frac{5}{10}$.

They are all equivalent fractions.

In a question, you may need to find a number that is equivalent to $\frac{1}{2}$ but has a denominator of 10.
You will get the missing numerator by using this relationship:

You will find the numerator by cross-multiplying 10 and 1 = 10, and dividing 10 by 2 = 5. The numerator is 5.

This gives you the equivalent fraction of $\frac{5}{10}$.

### LCD

The LCD, or least common denominator, is the least common multiple (LCM) of the denominators in two or more given fractions. How’s that again?

To illustrate, let’s look at the fractions:

The denominators are 8, 5, and 10. The multiples of each of those denominators are:

8: 8, 16, 24, 32, 40, 48,…
5: 5, 10, 15, 20, 25, 30, 35, 40, 45,…
10: 10, 20, 30, 40, 50,…

The smallest multiple is 40; hence, it is the LCD of the given fractions. The LCD is a very important concept when learning about fractions, as you will see later.

### Mathematical Operations Involving Fractions

ASVAB questions may ask for addition, subtraction, multiplication, and division of fractions, too. Here are the basic rules:

#### Addition with the same denominators:

Simply add the numerators and copy the common denominator.

Example:

• Determine the LCD.
• Find the equivalent fractions so that all fractions have the same denominator (which is the LCD).

Example:

$\frac{2}{5} + \frac{3}{10} =$ ?

The LCD is 10.

The fraction $\frac{1}{5}$ must be converted so that its denominator is 10.

The equivalent fraction of $\frac{2}{5}$ using the LCD of 10 is $\frac{4}{10}$.

The question is now rewritten as:

$\frac{4}{10} + \frac{3}{10}$.

We get $\frac{7}{10}$.

#### Subtraction with the same denominators:

The procedure is similar to addition involving fractions with the same denominators.

#### Subtraction with different denominators:

The procedure is similar to addition involving fractions with different denominators.

#### Multiplication:

Multiply the numerators (or top numbers).
Multiply the denominators (or bottom numbers).
Simplify or reduce, if the resulting fraction can still be simplified or reduced.

Example:

#### Division:

Turn the second fraction upside down (called the reciprocal form of the fraction), and then proceed to multiplication.

Example: