# Page 1 Arithmetic Study Guide for the ACCUPLACER® test

## How to prepare for the ACCUPLACER Arithmetic Test

### General Information

The ACCUPLACER Arithmetic test has 17 questions, each of them designed to gauge your knowledge in the area of basic mathematical principles. These are multiple-choice questions and require you to utilize mathematical thinking in three separate areas: mathematical operations with whole numbers and fractions, mathematical operations utilizing decimals and percentages, and operations requiring problem-solving skills. Most of these questions are simple in nature and require only a rudimentary knowledge of mathematical concepts.

### Operations with Whole Numbers and Fractions

Mathematical operations using whole numbers and fractions vary somewhat in their delivery but focus on using whole numbers and fractions both separately and together. This may be in the form of addition, subtraction, multiplication, or division. When studying for this portion of the test, pay attention to different rules applied to these mathematical operations—particularly the order of operations and how it affects different mathematical principles. Simple equations can be made far more difficult when whole numbers and fractions are combined, so be sure to practice operations with whole numbers and fractions regularly.

You will be required to work accurately with numbers containing both whole numbers and fractions, including skills in these areas:

#### Addition and Subtraction of Fractions

In order to combine fractions with addition or subtraction, the fractions must be of the same type, or denominator. Fractions that do share the same denominator can be combined by adding or subtracting the numerators and placing the result above the original denominator. For example:

$\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}$ and $\frac{5}{9} - \frac{2}{9} = \frac{5-2}{9} = \frac{3}{9} = \frac{1}{3}$

Fractions that do not share a denominator must first be manipulated before addition or subtraction can be performed. When adding or subtracting only 2 fractions with different denominators, a common denominator can be found by multiplying the first fraction (top and bottom) by the denominator of the second fraction, and multiplying the second fraction (top and bottom) by the denominator of the first fraction. For example:

#### Multiplication of Fractions

To multiply fractions, begin by reducing any of the fractions in the problem. Next, calculate the product of every numerator and place this value over the product of every denominator. Reduce the resulting fraction, if possible.

#### Division of Fractions

To divide one fraction by another fraction, replace the divisor (the second fraction) with its reciprocal, and multiply the resulting fractions. The reciprocal of a fraction is a fraction in which the numerator and denominator have been swapped.

The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. The reciprocal of $5$ is $\frac{1}{5}$.

Consider the problem, $\frac{7}{8} ÷ \frac{3}{4}$

Replace the second fraction with its reciprocal: $\frac{7}{8} ÷ \frac{4}{3}$

Replace the division symbol with the multiplication symbol: $\frac{7}{8} \cdot \frac{4}{3}$

Place the product of the numerators above the product of the denominators: $\frac{7 \cdot 4}{8 \cdot 3}$

Reduce: $\frac{7 \cdot 1}{2 \cdot 3}$

Multiply: $\frac{7}{6}$

#### Recognizing Equivalent Fractions and Mixed Numbers

Equivalent fractions are those that have the same value but contain different numerators and denominators.

$\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions, because $\frac{2}{4}$ can be divided by $\frac{2}{2}$ to arrive at $\frac{1}{2}$. Likewise, $\frac{1}{2}$ can be multiplied by $\frac{2}{2}$ to arrive at $\frac{2}{4}$.

To confirm the equivalency of two fractions, test to see whether the numerator and denominator have both been multiplied or divided by the same number.

$\frac{3}{5} = \frac{9}{15}$, because $3 \cdot 3 = 9$ and $5 \cdot 3 = 15$

$\frac{4}{5} \neq \frac{16}{25}$, because $4 \cdot 4 = 16$, but $5 \cdot 4 \neq 25$

Mixed numbers are those that include a whole number and a proper fraction. Improper fractions (fractions that represent a quantity larger than 1) can be written as mixed numbers.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The whole number of times the denominator goes into the numerator will be the whole number beside the proper fraction, and the remainder will be the numerator of the fraction. The denominator remains the same. For example:

$\frac{19}{3} = 6\frac{1}{3}$, because 3 goes into 19, 6 times with a remainder of 1.

To convert a mixed number to an improper fraction, multiply the whole number with the denominator, add the numerator to this value, and place the resulting number above the original denominator. For example: