# Page 1 Mathematics Study Guide for the PRAXIS® test

## How to Prepare for the PRAXIS® Core Mathematics Test

### General Information

The Mathematics section of the PRAXIS test is different from the other two sections in the format of its questions. In addition to regular “selected-response” (multiple-choice) questions, you will encounter two additional question formats. One is like the selected-response questions, but you will be asked to select one or more answers. The other alternative question type requires you to fill your answer in boxes provided, with no given choices. This is called a “numeric entry” type of question.

As for the content of the Mathematics test, it is spread over four basic math areas, in these approximate percentages:

Number and Quantity 30%
Algebra and Functions 30%
Geometry 20%
Statistics and Probability 20%

There is a great emphasis on solving real-world problems on this test, in other words, “word problems.” Many times, the problem-solving process will involve designing a method of combining several steps to reach the final answer.

You will have 85 minutes to answer 56 questions during this PRAXIS test.

### Number and Quantity

Number covers arithmetic operations with real numbers, especially relationships using fractions, multiplication and division. Quantities have dimensions such as seconds, feet, pounds, etc. So, 3 is a number while 3 miles is a quantity.

#### Ratios and Proportions

Ratios and proportions describe relationships between quantities in a specific order. They can be written in different ways. For example, if a classroom has 8 boys and 10 girls, the ratio of boys to girls is 8:10 or $\frac{8}{10}$.

Ratio

Ratios describe a comparison of quantities in a specified order, such as “apples : oranges is 3:2.” The fact that ratios can be expressed as fractions makes it possible to compare the ratios by comparing the fractions.

For example, the ratios 8:12 and 12:18 are the same, because the fractions $\frac{8}{12}$ and $\frac{12}{18}$ are equal. Both can be reduced to $\frac{2}{3}$.

Proportion

In a proportion, two ratios are declared to be equal. This information is useful because the equal ratios can be written as an equation, which can then be solved.

For example, in a recipe, 1 cup of milk is needed to make batter for 15 cupcakes. Suppose you need to make 60 cupcakes and want to know how much milk is needed. The ratio of milk to cupcakes is the same in both cases so the equation can be written:

Ratio and Proportion in Problem-Solving

A proportion can be solved by writing the ratios as fractions and setting them equal. Usually one of the quantities is unknown and shown as a variable. The resulting equation can be solved by various methods.

In the preceding example, $\frac{1}{15} = \frac{x}{60}$, it is easiest to notice that the denominator increased by a factor of 4 and therefore the numerator must increase by the same factor to keep the fractions equal. Thus 4 cups of milk are needed.

In a more complex example, such as $\frac{7}{8} = \frac{x}{20}$, the technique of cross multiplication may be used. The result of cross multiplication is:

Then,

or

#### The Real Number System

The real number system consists of the rational and irrational numbers. Rational numbers are integers, such as 7, 0, and -3, as well as ratios of integers, such as $\frac{2}{3}$.

In decimal form, real numbers either terminate, such as 0.125, or they repeat in an infinite pattern, such as 0.181818. . . , the decimal form of $\frac{2}{11}$.

Least Common Multiple

The least common multiple of two numbers is the smallest non-zero number that is a multiple of both numbers. The least common multiple is often needed when adding or subtracting fractions with different denominators. For example, the least common multiple of 4 and 6 is 12.

Greatest Common Factor

The greatest common factor of two numbers is the largest non-zero factor that is a factor of both numbers. The greatest common factor is often needed when simplifying fractions. For example, the greatest common factor of 12 and 18 is 6.

Operations with Fractions

Arithmetic operations with fractions are different than with whole numbers. When operating on fractions, it is important to consider whether the fractions have identical denominators, and all divisions are transformed to multiplications before proceeding.

Fractions must have the same denominator before adding or subtracting them. The operation is done on the numerators only. The denominator remains the same. For example:

If the denominators are not the same, then one or both fractions must be converted to an equivalent fraction with matching denominators. For example:

To multiply fractions:

Multiplication is the most straightforward of the fraction operations. To multiply fractions, simply multiply the numerators to get the numerator of the answer, and multiply the denominators to get the denominator of the answer. The resulting fraction may need to be simplified.

To divide fractions:

To divide fractions, remember that multiplication and division are inverse operations. Dividing by a number is the same as multiplying by its inverse.

For example, the inverse of 4 is $\frac{1}{4}$.

$24\div4 = 6$ is the same as $24 \cdot \frac{1}{4} = 6$

Thus, fractions are never truly divided. Instead, invert the second number and multiply, instead. The result may need to be simplified. For example:

Another example:

Irrational Numbers

Irrational numbers cannot be expressed as a ratio of integers. They are decimal numbers that extend infinitely and have no repeating pattern. The number pi, or π, is the most famous example. If a number is not a perfect square, its square root is an irrational number, such as $\sqrt{2}$. Irrational numbers can be approximated with rational numbers. For example, $\pi \approx \frac{22}{7}$.

Exponents

Exponents provide a way to indicate that a number is to be repeatedly multiplied times itself. The base number is the number to be multiplied and the exponent shows how many times to multiply it. For example, $2^{3} = 2 \cdot 2 \cdot 2 = 8$. For this test, you will only have to work with exponents that are integers.

Radicals are the inverse operation to exponents. The most common type is the square root.

$\sqrt{7}$ (“square root of 7”) means to find a number, x, so that $x \cdot x = 7$.

A cube root, $\sqrt[3]{12}$ (“cube root of 12”), means to find a number x so that $x \cdot x \cdot x = 12$.

#### Quantities

A quantity is an amount of a measurable unit. One quantity is 17 feet. When working with quantities, the units can be used to see how the problem is worked out and what the resulting units will be.

For example, if you travel at $55 \frac{miles}{hour}$ for 5 hours, the distance traveled is:

It is as if the units can be treated as variables in a fraction that can be simplified.