### Factors

Factors are whole numbers that are multiplied together to create another number.

Examples

2 × 4 = 8 (2 and 4 are factors of 8.)

1 × 8 = 8 (1 and 8 are factors of 8.)

So 1, 2, 4, and 8 are all factors of 8.

1 × 32 = 32 (1 and 32 are factors of 32.)

2 × 16 = 32 (2 and 16 are factors of 32.)

8 × 4 = 32 (8 and 4 are factors of 32.)

So 1, 2, 4, 8, 16, and 32 are all factors of 32.

Types of Factors   Knowing about factors will be very useful when solving equations.

Common Factors   Common factors are factors that are shared by two or more numbers. As shown above, 8 and 32 share the common factors 1, 2, 4, and 8.

### Multiples

Multiples of a number are found by multiplying the number by 1, 2, 3, 4, 5, 6, 7, 8, and so on. So multiples of 5 are 10, 15, 20, 25, 30, 35, 40, and so on.

Common Multiples   Common multiples are multiples that are shared by two or more numbers.

Least Common Multiple   The least common multiple is the smallest multiple that two numbers share. For example, 6, 12 and 18 are all common multiples of 2 and 3 because they are all evenly divisible by both 2 and 3. 6 is the least common multiple because it is the smallest number that 2 and 3 both divide evenly.

### Fractions

Fractions have two numbers, a numerator and a denominator.

The denominator tells how many parts something is divided into. The numerator tells how many of those parts you have. The fraction as a whole tells the proportion of the parts you have to the parts there are in all.

So if a pie is divided into 8 pieces and you take 2 of them, the fraction telling the proportion of the pie you have is . The denominator (8) tells how many pieces of pie there are in all. The numerator (2) tells how many of those pieces you have.

or a proportion of 2 to 8.

Fractions can be positive or negative. A negative fraction is written like this:

Types of Fractions   For the ASVAB, you'll need to know about proper fractions, improper fractions, and mixed numbers.

Proper Fractions   Fractions representing amounts smaller than 1 are called proper fractions. Fractions smaller than 1 have numerators that are smaller than the denominators.

Examples

Improper Fractions   Improper fractions are greater than 1. The numerator is larger than the denominator.

Examples

### Mixed Numbers

Expressions that include both whole numbers and fractions are called mixed numbers.

Examples

Renaming an Improper Fraction as a Mixed Number   To rename an improper fraction as a mixed number, simply divide the numerator by the denominator.

Examples

Renaming a Mixed Number as a Fraction   To rename a mixed number as an improper fraction, simply multiply the whole number by the denominator and add the numerator. Put that number over the denominator.

Examples

### Working with Fractions and Mixed Numbers

ASVAB problems are likely to require you to find equivalent fractions, reduce fractions to lowest terms, and add, subtract, multiply, and divide fractions and mixed numbers.

Finding Equivalent Fractions   Two fractions are said to be equivalent (the same in value) if they use different numbers but represent the same proportion. For example, the following fractions are equivalent:

To change a fraction into an equivalent fraction, multiply or divide the numerator and denominator by the same number.

Examples

Reducing Fractions to Lowest Terms   Fractions are commonly shown in their lowest terms, that is, the smallest numbers that still represent the original proportion. When a fraction is not in its lowest terms, you can reduce it to its lowest terms by dividing the numerator and denominator by the largest number that will divide into both evenly.

Examples

Adding and Subtracting Fractions   To add and subtract fractions with like denominators, simply add or subtract the numerators. The result is often shown in lowest terms.

Examples

To add and subtract fractions with unlike denominators, you first need to find equivalent fractions that all have the same denominator. To do this, you need to find the least common denominator (LCD), the least common multiple of the denominators of all the fractions. Use the LCD to create new fractions equivalent to the original ones. Then add the new numerators.

Examples

The least common denominator is 8, so does not need to be changed. However, is changed to the equivalent fraction .

Find the least common denominator. The least common multiple of 8, 3, and 2 is 24, so 24 is the least common denominator. Use the LCD to create new fractions equivalent to the original ones:

Use the same approach to add and subtract positive and negative fractions. Refer to the section earlier in this math review on adding and subtracting positive and negative numbers.

Adding and Subtracting Mixed Numbers   Add and subtract mixed numbers by following the same rules previously outlined. Change the fractions to equivalent fractions using the least common denominator. Add or subtract the fractions. Add or subtract the whole numbers. If the fractions add up to more than a whole number, add that to the whole numbers.

Example (Addition)

Examples (Subtraction)

Caution: Remember that subtraction means moving left on the number line. Start at 3 and move left 5 on the number line. The correct answer is –1.

Multiplying Fractions   To multiply fractions, just multiply the numerators and multiply the denominators. Reduce the resulting fraction to lowest terms.

Examples

Multiply the numerators: 5 × 2 = 10

Multiply the denominators: 12 × 3 = 36

Result:

Multiply the numerators: 4 × 7 = 28

Multiply the denominators: 5 × 8 = 40

Result:

Dividing Fractions   To divide fractions, invert the second fraction and multiply. Reduce the result to lowest terms.

Examples

Multiplying Mixed Numbers   To multiply mixed numbers, change each mixed number to a fraction and then multiply as usual. Reduce to lowest terms.

Examples

Dividing Mixed Numbers   To divide mixed numbers, rename the mixed numbers as fractions and then follow the rule for dividing fractions: invert the second fraction and multiply.

Example