# Math Study Guide for the NLN PAX

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The Math section of the National League for Nursing (NLN) Pre-Admission Examination (PAX) covers a variety of topics in mathematics, but generally, only the basics are covered. There are 40 questions, and you will have 40 minutes in which to complete them. This means you will have about one minute per question.

All questions on the NLN PAX are typical multiple-choice questions with four answer choices.

## Basic Number Concepts

These basic number concepts are the foundation upon which mathematical proficiency is built. They are fundamental principles that provide the framework for more advanced mathematical operations and problem-solving skills. The concepts discussed in this section encompass the understanding of numbers, their relationships, and their application in various real-world scenarios.

### Place Value

Place value describes the value of the digits in a multi-digit number with respect to the position of the digit in the number. Reading large numbers becomes manageable when we understand the place value system. For instance, in the number $$8\text{,}529$$, the $$8$$ represents thousands, the $$5$$ represents hundreds, the $$2$$ represents tens, and the $$9$$ represents units or ones. The number $$8\text{,}529$$ is read “eight thousand five hundred twenty-nine.” To visualize and comprehend the place value of each digit, a place value chart for whole numbers can be immensely useful. Below is a place value chart:

This type of chart organizes digits into columns, such as billions, millions, thousands, and ones, providing a clear structure for understanding the value of a given number. In this chart the first digit is in the ones place, the next digit (to the left) is in the tens place, and so on.

### Operations with Numbers

Understanding and performing operations with numbers are foundational skills with broad applications in daily life and academic pursuits. These operations, including addition, subtraction, multiplication, and division, are not only about solving mathematical problems but are also essential for making informed decisions in various real-world scenarios.

Addition is the process of combining two or more numbers. Below is a simple addition problem:

Here, the numbers being added are called addends. The result of the operation is known as the sum. The operator symbol for addition is a plus sign ($$+$$).

#### Subtraction

Subtraction is the process of finding the difference between two or more numbers. Below is a simple subtraction problem:

The number being subtracted is called the subtrahend. The number from which the subtrahend is being subtracted (usually the bigger one) is called the minuend. The result is known as the difference. The operator symbol for subtraction is a minus sign ($$-$$).

#### Multiplication

Multiplication is used to find the product of two or more numbers. It is a process of repeated addition. Below is a simple multiplication problem:

The first number is called the multiplicand. The second number being multiplied by the multiplicand is called the multiplier. The result is known as the product. We can also call both the numbers being multiplied factors. The operator symbol for multiplication is a times sign ($$\times$$).

#### Division

Division is a mathematical operation that represents the process of distributing (or partitioning) a quantity into equal parts. Below is a simple division problem:

The number being divided is called the dividend. The number doing the division is called the divisor. The result of the operation is known as the quotient. The operator symbol for division is the division sign ($$\div$$).

### Rounding

Rounding is a mathematical technique that is employed to simplify numerical values while maintaining a reasonable level of accuracy. This is particularly useful when precise figures are not required and a general estimate is sufficient for practical purposes.

These are the steps for rounding:

• Identify and determine which digit (place value) you want to round a number to. We can refer to this as the “rounding digit.”

• Look at the digit immediately to the right of the rounding digit.

• If the digit to the right is $$5$$ or greater, round the rounding digit up by adding $$1$$ to it. If the digit to the right is less than $$5$$, keep the rounding digit unchanged.

• Make all the numbers to the right of the rounding digit $$0$$.

Here is an example:

Round the number $$51\text{,}783$$ to the thousands place.

The rounding digit would be $$1$$.

The digit to the right of the rounding digit is $$7$$, so the $$1$$ will be rounded up.

The rounded number will be $$52\text{,}000$$.

### Divisibility

Divisibility plays a key role in simplifying numbers and identifying patterns within them. This concept is important for simplifying fractions and determining the factors of a given number. Divisibility allows us to break down complex numerical relationships into more manageable components.

#### Prime and Composite Numbers

Numbers can be classified into two main categories: prime and composite.

Prime numbers are integers greater than $$1$$ that have only two distinct positive divisors, $$1$$ and the number itself. Examples include $$2$$, $$3$$, and $$5$$. Note that $$2$$ is the only even prime number.

On the other hand, composite numbers have more than two distinct positive divisors, making them divisible by numbers other than $$1$$ and themselves. Examples include $$4$$, $$6$$, and $$9$$.

#### Prime Factorization

When you need to find all of a number’s prime factors, there is a simple method to determine all of them. Simply start with the number and break it down, one prime factor at a time, continuing until there are only prime numbers left. Here is how that might look:

The prime factors of $$36$$ are $$3$$ and $$2$$. There are two of each, so we could express this as $$36=3^2 \times 2^2$$.

#### Divisibility Rules

If a number, $$x$$, goes evenly into another number, $$y$$, when divided, with no remainder, we say that $$y$$ is divisible by $$x$$. To check whether a number is divisible by a certain number, there are certain shortcuts/rules for common numbers. They are discussed below:

• divisibility by $$2$$—If the last digit of a number is even, then that number is divisible by $$2$$. For example, $$48$$ is divisible by $$2$$ since $$8$$ is an even number.

• divisibility by $$3$$—A number is divisible by $$3$$ if the sum of the digits in the number is divisible by $$3$$. For example, $$93$$ is divisible by $$3$$ since the sum of the digits of the number is $$9 + 3 =12$$, which is a number that is divisible by $$3$$. (Notice that $$12$$ is also divisible by $$3$$ because $$1+2=3$$.)

• divisibility by $$5$$—If the last digit of a number is $$0$$ or $$5$$, then the number is divisible by $$5$$. For example, the numbers $$50$$ and $$125$$ are both divisible by $$5$$.

• divisibility by $$6$$—A number is divisible by $$6$$ if it is divisible by both $$2$$ and $$3$$. For instance, the number $$18$$ is divisible by both $$2$$ and $$3$$. Consequently, we can say that it is also divisible by $$6$$ ($$18 \div 6 = 3$$).

• divisibility by $$9$$—A number is divisible by $$9$$ if the sum of the digits in the number is $$9$$ (or divisible by $$9$$). For example, $$27$$ is divisible by $$9$$ since the sum of the digits of the number is $$2 + 7 =9$$. Likewise, $$369$$ is divisible by $$9$$ because $$3+6+9=18$$ and $$18$$ is $$1+8=9$$.

• divisibility by $$10$$—If a number ends with a $$0$$, it is divisible by $$10$$. The numbers $$20$$, $$90$$, and $$18\text{,}760$$ are all divisible by $$10$$.

• divisibility by $$12$$—A number is divisible by $$12$$ if it is divisible by both $$3$$ and $$4$$. For instance, the number $$48$$ is divisible by both $$3$$ and $$4$$. Consequently, we know that it is also divisible by $$12$$ ($$48 \div 12 = 4$$).

### Average

The average of a set of numbers is the sum of the numbers divided by the number of values in the set. This statistical measure provides a single value around which all the other values of the data set are centered. If your last four test scores are $$81$$, $$92$$, $$93$$, and $$88$$, then your average score is:

$\text{average} = \frac{\text{sum}}{\text{number of values}} = \frac{81+92+93+88}{4} = \frac{354}{4} = 88.5$

The average is also known as the mean.

### Exponents

Exponents represent the number of times a base is multiplied by itself. For example, in the expression $$2^4 = 16$$, the number $$2$$ is the base and the number $$4$$ is the exponent (or power).

The square of a number is when that number is multiplied by itself. For example, three squared is $$3^2 = 3 \times 3 = 9$$. The cube of a number is when that number is multiplied by itself two times. For example, two cubed is $$2^3 = 2 \times 2 \times 2 = 8$$. For expediting mental calculations, you should be familiar with the first $$10$$ squares and the first six cubes, shown below:

### Roots

While exponents involve repeated multiplication, roots are the inverse operation, by which we find the original value that was repeatedly multiplied. The most common roots are the square root (the symbol is $$\sqrt{}$$) and the cube root (the symbol is $$\sqrt[3]{}$$).

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, $$\sqrt{36} = 6$$ since $$6\times 6 = 36$$.

The cube root of a number is a value that, when multiplied by itself twice, results in the original number. For example, $$\sqrt[3]{8} = 2$$ since $$2 \times 2 \times 2 = 8$$. You should know the following common square roots and cube roots for faster mathematical calculations:

Note: A square root can be symbolized by $$\sqrt[2]{}$$, but $$\sqrt{}$$ means the same thing and is more commonly used.

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