Math Study Guide for the NLN NEX

General Information

The Math section of the National League for Nursing (NLN) Nursing Entrance Exam (NEX) covers a variety of topics in mathematics, but it generally focuses on the basics. There are 45 questions, and you will have 60 minutes in which to complete them. This means you will have a little more than one minute per question. Only 40 of the questions will be scored, but you will not know which ones these are. All questions on the NLN NEX are typical multiple-choice questions with four answer choices.

The NLN NEX assesses math skills in four areas in approximately these percentages:

• numbers and operations: 30%
• measurement (including conversions): 35%
• algebra: 17.5%
• data and information: 17.5%

Numbers and Operations

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 their position 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 as “eight thousand five hundred twenty-nine.”

To visualize 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. As this chart shows, the first digit is in the ones place, the next digit (to the left) is in the tens place, and so on.

Types of Numbers

Understanding the different types of numbers is imperative to being able to do higher-level mathematics. It provides a framework for categorizing and working with numerical values. Here, we will look at whole numbers, integers, and rational numbers.

Whole Numbers

Whole numbers encompass all the natural numbers along with zero. They are the set of counting numbers, starting from \(0\) and extending infinitely (\(0, 1, 2, 3…\)). Whole numbers allow us to represent quantities in a discrete, non-fractional manner.

Integers

Integers consist of all positive and negative whole numbers along with zero (\(...-3, -2, -1, 0, 1, 2, 3…\)). They are a versatile set of numbers, allowing the representation of quantities above and below zero. We can perform basic arithmetic operations (addition, subtraction, multiplication, and division) with integers.

Adding Integers

Adding integers, like adding whole numbers, is a fairly simple procedure. However, the inclusion of positive and negative numbers means you need to know some extra steps. Here are rules to guide you:

• same signs—When adding integers with the same sign (positive or negative), add their absolute values and keep the common sign. For example:

\[(+3) +(5) = +(3+5) = +8\] \[(-3) + (-5) = -(3+5) = -8\]

Note: The absolute value of a number is simply the distance of that number from \(0\) and is always positive. For instance, the absolute value of both \(3\) and \(-3\) is \(3\). This topic will be discussed at greater length below.

• different signs—When adding integers with different signs, subtract the smaller absolute value from the larger one, and use the sign of the number with the larger absolute value for the result. For example:

\[(-7) + (+4) = -(\vert -7 \vert - \vert +4 \vert) = -(7 - 4) = -3\]

Subtracting Integers

Subtracting integers is a similar process to adding integers. In fact, you can transform subtraction into addition by changing the sign of the number being subtracted. Follow these steps:

• Change subtraction to addition by changing the sign of the number being subtracted. For example:

\[(+8) - (-3) = (+8) + (+3) = 11\]

• Once you have converted the problem to addition, follow the rules for adding integers.

Multiplying Integers

The process for multiplying integers varies based on the signs of the numbers:

• same signs—When multiplying two integers with the same sign, the result is always positive. For example:

\[(+4) \times (+2) = +(4 \times 2) = +8\] \[(-3) \times (-5) = +(3 \times 5) = +15\]

• different signs—When multiplying two integers with different signs, the result is always negative. For example:

\[(-4) \times (+6) = -(4 \times 6) = -24\]

Dividing Integers

Dividing integers follows similar rules to multiplication. Once again, the signs of the numbers determine how you complete the process:

• same signs—When dividing two integers with the same sign, the quotient is always positive. For example:

\[(+4) \div (+2) = +(4 \div 2) = +2\] \[(-15) \div (-5) = +(15 \div 5) = +3\]

• different signs—When dividing two integers with different signs, the result is always negative. For example:

\[(-24) \div (+6) = -(24 \div 6) = -4\]

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 \(9\).

Rational and Irrational Numbers

Rational and irrational numbers are two key categories of numbers that help us understand and categorize numerical values. Rational numbers can be expressed as fractions or as decimals that either terminate or repeat, while irrational numbers cannot be represented as simple fractions and have non-terminating, non-repeating decimal expansions.

Rational Numbers

Rational numbers include all integers along with fractions and decimals that can be expressed as a quotient or a fraction in the form \(\frac{a}{b}\), where \(a\) is an integer and \(b\) is a non-zero integer. Rational numbers provide a comprehensive representation of numerical values, including both integers and fractional parts. For example, \(6, 1.4,\) and \(\frac{2}{5}\) are all rational numbers.

There are properties of rational numbers you need to know:

• Rational numbers can be expressed as a fraction in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b \neq 0\).

• Decimal expansions are either terminating (e.g., \(0.75\)) or repeating (e.g., \(0.333…\)).

• Rational numbers include all integers (e.g., \(7, -3\)), as they can be written as \(\frac{a}{1}\).

Here are some rules of rational numbers:

• adding or subtracting rational numbers—The sum or difference of two rational numbers is always rational. For example, adding two fractions results in another rational number (e.g., \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\)). Similarly, subtracting rational numbers also yields a rational result (e.g., \(7 - 2.5 = 4.5\)).

• multiplying rational numbers—The product of two rational numbers is always rational. For instance, \(3 \times \frac{2}{5} = \frac{6}{5}\) and \((-2) \times 4 = -8\) are two examples where the result remains rational.

• dividing rational numbers—The quotient of two rational numbers is always rational, provided the divisor is not zero. For example, with \(10 \div 2 = 5\) and \(\frac{7}{3} \div \frac{2}{3} = \frac{7}{2}\), both results are a rational number.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). They have decimal expansions that neither terminate nor repeat. Examples of irrational numbers include \(\sqrt{2}\), \(\pi\), and \(e\). For instance, \(\pi\), a mathematical constant, is approximately \(3.14159\), but it continues infinitely without repeating.

There are properties of irrational numbers you need to know:

• Irrational numbers cannot be expressed as a fraction in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\),

• Decimal expansions are infinite and non-repeating.

• Irrational numbers often arise from roots of non-perfect squares or transcendental numbers (e.g., \(\sqrt{2}\), \(\pi\)).

Here are some rules of irrational numbers:

• adding or subtracting irrational numbers—When you add or subtract two irrationals, the result may be rational or irrational. For example, \(\sqrt{2} + (-\sqrt{2}) = 0\), a rational result, while \(\sqrt{2} + \pi\) produces an irrational result.

• multiplying irrational numbers—When you multiply two irrationals, the product may be rational or irrational. For example, \(\sqrt{2} \times \sqrt{2} = 2\), a rational result, while \(\sqrt{2} \times \pi\) produces an irrational result.

• dividing rational numbers—When dividing two irrationals, the quotient may be rational or irrational. For example, \(\pi \div \pi = 1\), a rational result, while \(\sqrt{2} \div \pi\) produces an irrational result.

Exponents

Exponents represent the number of times a base is multiplied by itself. This is useful notation when multiplying a number by itself multiple times. 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 once. 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.

Other Number Skills

Using a Number Line

A number line is a visual representation of numbers on a straight line, typically drawn horizontally. The line extends infinitely in both directions, with numbers increasing to the right and decreasing to the left. The central point on a number line is usually labeled as zero, with positive numbers to the right and negative numbers to the left.

The main purpose of a number line is to help visualize numerical relationships, such as addition and subtraction, and to understand the order of numbers. It’s a simple yet powerful tool that can be used to demonstrate a wide range of mathematical concepts. Take the case of adding and subtracting numbers using a number line.

To add two numbers using a number line, start at the first number and move to the right by the number of steps equal to the second number. For example, if you want to illustrate \(2 + 3\) on a number line:

• Start at \(2\) on the number line.
• Move \(3\) steps to the right.

You will land on \(5\), which is the sum of \(2 + 3\).

To subtract a number, start at the first number and move to the left by the number of steps equal to the second number. For example, if you want to illustrate \(4 - 2\) on a number line:

• Start at \(4\) on the number line.
• Move \(2\) steps to the left.

You will land on \(2\), the result of \(4 - 2\).

A number line can also be used to compare the size of numbers, understand the concept of negative numbers, and visualize even more complex operations like finding the absolute value of a number.

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:

• Determine to which digit (place value) you want to round a number. We 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.

We look at the thousands place and see that the rounding digit is \(1\).

The digit to the right of the rounding digit is \(7\), so the \(1\) is rounded up.

The rounded number is \(52\text{,}000\).

Absolute Value

Absolute value is a mathematical concept representing the distance of a number from zero on the number line, regardless of direction. In other words, absolute value tells us how far a number is from zero, but it doesn’t matter whether the number is positive or negative.

The absolute value of a number is always non-negative. It’s denoted by placing the number inside vertical bars like this:

\[\vert x \vert\]

For example, the absolute value of both \(-3\) and \(3\) is \(3\), because both are three units away from zero on the number line.

A number line can be very helpful in visualizing absolute value. Let’s look at the number \(-4\) and \(4\) on a number line:

Both \(-4\) and \(4\) are four units away from zero. Therefore, the absolute value of \(-4\) is \(\vert -4 \vert = 4\), and the absolute value of \(4\) is \(\vert 4 \vert = 4\).

Absolute value is particularly useful in real-life scenarios where the direction doesn’t matter, only the magnitude. For example, when calculating distances or dealing with quantities that cannot be negative, such as speed or time.

Fractions

Fractions allow us to express parts of a whole. For example, \(\frac{2}{3}\) (two-thirds) and \(\frac{4}{5}\) (four-fifths) are fractions. In this section, we will learn basic fraction concepts, the processes for converting different fraction types and reducing fractions, and operations with fractions.

Basic Fraction Concepts

Fractions are crucial for various math tasks and real-life situations. Whether you’re measuring ingredients for a recipe or shopping, knowing how fractions work allows you to get an accurate count.

Parts of a Fraction

The two important parts of a fraction are the denominator, which reveals how many equal parts make up the whole, and the numerator, which lets you know how many parts of the whole you have. The denominator is the bottom of the fraction, and the numerator is the top, and they’re separated by the fraction bar.

Note: When the denominator is \(1\), you have a whole number. For example, the whole number \(5\) can be written as the fraction \(\frac{5}{1}\).

Types of Fractions

There are different types of fractions that represent different situations. Let’s explore them briefly:

• proper fraction—This is a type of fraction where the numerator is less than the denominator. For example, \(\frac{1}{2}\) and \(\frac{4}{11}\) are proper fractions.

• improper fraction—This is a type of fraction where the numerator is greater than the denominator. For example, \(\frac{4}{3}\) and \(\frac{12}{5}\) are improper fractions.

• mixed number—This is a combination of a whole number and a proper fraction. For example, \(4 \frac{1}{2}\) is a mixed number. Mixed numbers are also known as mixed fractions.

• unit fraction—This is a type of fraction where the numerator is \(1\). For example, \(\frac{1}{10}\) and \(\frac{1}{7}\) are unit fractions.

• complex fraction—This is a type of fraction where the numerator, the denominator, or both are fractions. Examples of complex fractions are shown below:

\[\frac{\left( \frac{1}{2} \right)}{5}, \frac{6}{\left( \frac{2}{3} \right)}, \frac{\left( \frac{1}{4} \right)}{\left( \frac{3}{8} \right)}\]

• equivalent fraction—When two or more fractions represent the same amount, they are known as equivalent fractions. For example, \(\frac{1}{2}\) and \(\frac{3}{6}\) are equivalent since both represent half of a whole.

Other Fraction Terms

There are specific terms that can help you better understand and use fractions. This section will review the most important terms you should know.

• greatest common factor (GCF)—The GCF is the largest number that divides two or more numbers without leaving a remainder. It’s a fundamental concept in simplifying fractions to their lowest terms.

• least common multiple (LCM)—The LCM is the smallest multiple that two or more numbers share. It’s particularly useful when finding a common denominator for two or more fractions.

• least common denominator (LCD)—The LCD is the LCM of the denominators of two or more fractions. This is important when adding or subtracting fractions with different denominators.

• relatively prime—When numbers are relatively prime, it means that they don’t share any common factors other than \(1\), meaning there’s no number other than \(1\) that can divide both of them evenly. For instance, \(7\) and \(20\) are relatively prime numbers since the only common factor between them is \(1\). Identifying relatively prime numbers can simplify the process of reducing to lowest terms.

• lowest terms—Reducing a fraction to its lowest terms involves simplifying it as much as possible. This is achieved by dividing both the numerator and denominator by their GCF. The fraction \(\frac{9}{12}\) reduced to its lowest terms is \(\frac{3}{4}\), because the GCF of \(9\) and \(12\) is \(3\). If you recognize that the numerator and denominator of a fraction are relatively prime, you know immediately that it can’t be reduced further.

• reciprocal—The reciprocal of a fraction is obtained by swapping its numerator and denominator. The reciprocal of the fraction \(\frac{6}{13}\) is \(\frac{13}{6}\). Multiplying a fraction by its reciprocal results in a product of \(1\). That is a useful operation in various mathematical scenarios.

Converting Fractions

The various forms of fractions provide versatile ways of expressing and manipulating quantities. Let’s explore the conversion between improper fractions and mixed numbers.

From Improper Fraction to Mixed Number

To convert an improper fraction to a mixed number, we first find how many times the denominator goes into the numerator using division. That is the whole number part of the mixed number. Then, we create a fraction with the remainder as the numerator and the original denominator as the denominator. This is the fractional part of the mixed number.

Let’s convert the improper fraction \(\frac{11}{3}\) into a mixed number. First, we determine how many times \(3\) goes into \(11\), which is three times. There is also a remainder of \(2\) left over. So, writing the improper fraction as a mixed number gives us:

\[\frac{11}{3} = 3 \frac{2}{3}\]

From Mixed Number to Improper Fraction

To convert a mixed number into an improper fraction, we first multiply the whole part of the mixed number by the denominator of the fractional part. Then, we add this product to the numerator of the fractional part. Last, we put this sum over the original denominator, giving us an improper fraction.

Let’s convert \(3\frac{1}{2}\) into an improper fraction:

\[3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}\]

Reducing Fractions

Reducing fractions means converting them into their simplest form. It involves dividing both the numerator and the denominator by their greatest common factor. This ensures that the fraction retains its value but is expressed in its most compact representation.

Let’s reduce the fraction \(\frac{6}{24}\).

The numerator and denominator of \(\frac{6}{24}\) have multiple factors in common, but their GCF is \(6\), because both \(6\) and \(24\) are divisible by that number. As such, we can simplify our original fraction a great deal:

\[\frac{6}{24} = \frac{1}{4}\]

Operations with Fractions

Whether you’re working with measurements, following recipes, or taking medicine, learning how to add, subtract, multiply, and divide fractions can help you get an accurate result.

Adding and Subtracting Fractions

When adding or subtracting fractions, the process is fairly easy if the denominators are the same. You simply add or subtract the numerators while keeping the denominator unchanged:

\[\frac{1}{4} + \frac{5}{4} = \frac{1+5}{4} = \frac{6}{4} = \frac{3}{2}\] \[\frac{7}{2} - \frac{4}{2} = \frac{7 - 4}{2} = \frac{3}{2}\]

Note: In the first example, we went ahead and reduced the fraction to its simplest form. You will be expected to do that on most questions.

If the denominators are not the same, you first have to find a common denominator before performing the operation. An easy way to do this is to multiply both fractions by a fraction that has the denominator of the other fraction and is equal to \(1\). Here’s an example:

\[\frac{5}{3} + \frac{13}{8}\] \[\left( \frac{8}{8} \right) \frac{5}{3} + \left( \frac{3}{3} \right) \frac{13}{8}\] \[\frac{8 \times 5}{8 \times 3} + \frac{3 \times 13}{3 \times 8}\] \[\frac{40}{24} + \frac{39}{24}\]

Now, you can add (or subtract) the numerators while keeping the denominator unchanged:

\[\frac{40}{24} + \frac{39}{24} = \frac{79}{24}\]

Note: Although both numbers are large, this fraction cannot be reduced any further, so this is your final answer.

Multiplying Fractions

To multiply fractions, simply multiply the numerators to get the new numerator, and do the same for the denominators. If possible, simplify the result. For example:

\[\frac{1}{5} \times \frac{2}{3} = \frac{1 \times 2}{5 \times 3} = \frac{2}{15}\]

Here, the fraction can’t be reduced any further, so \(\frac{2}{15}\) is your final result.

Dividing Fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. Flip the divisor (the numerator becomes the denominator, and vice versa) and proceed with multiplication as usual:

\[\frac{1}{4} \div \frac{3}{8} = \frac{1}{4} \times \frac{8}{3} = \frac{1 \times 8}{4 \times 3} = \frac{8}{12} = \frac{2}{3}\]

Decimals

Decimals provide a way to express numbers that fall between whole numbers, allowing for more precise representation. The decimal point, a small but mighty dot, is the key player here. It separates the whole part of a number from the fractional or decimal part. Decimals are fundamental in mathematics, especially when dealing with quantities that involve parts of a whole, such as precise measurements or currency.

Basic Decimal Concepts

To understand decimals, we need to know where each digit stands, and that’s where a decimal place value chart comes in handy. It shows the value of each digit. Below is an example of a decimal place value chart:

Note that the digits to the left of the decimal point represent the whole part of the decimal number, and the digits to the right of the decimal point represent the fractional part of the decimal number.

When you read a decimal aloud, the whole number part is said first, and then the word and is included before expressing the fractional part. For example, with the decimal \(3.14\), you should say, “Three and fourteen hundredths.” A decimal place value chart can help make each digit’s significance clearer, especially when dealing with large numbers.

Converting Between Fractions and Decimals

Converting between fractions and decimals is crucial for seamlessly switching between these two numerical forms. It enables you to make accurate calculations in a variety of scenarios. Let’s look at how to convert between the two forms.

Fraction to Decimal

When converting a fraction to a decimal, remember that the fraction bar means division. Simply divide the numerator by the denominator. For example, to convert \(\frac{3}{4}\) to a decimal, you divide \(3\) by \(4\), resulting in \(0.75\). If you have a mixed fraction, first convert it to an improper fraction, then do the division.

Decimal to Fraction

Converting decimals to fractions involves recognizing the place value of each digit. Identify the decimal place and use it to create the fraction. For instance, in the decimal \(0.6\), the \(6\) is in the tenths place. Therefore, it becomes \(\frac{6}{10}\). When possible, simplify the fraction. In this case it simplifies to \(\frac{6}{10}=\frac{3}{5}\). If you had two decimal places, you would divide by \(100\), and so on. Understanding this conversion process is key to fluidly working with fractions and decimals in various mathematical scenarios.

Operations with Decimals

Navigating decimal operations opens up avenues for precise mathematical calculations and practical applications. These basic decimal operations make it possible to tackle a variety of problems, from everyday situations to more complex mathematical challenges.

Adding and Subtracting Decimals

Adding and subtracting decimals involves aligning the decimal points of the numbers. This alignment ensures that corresponding place values match up. Once aligned, you can add or subtract as if they were whole numbers, and then you just place the decimal point in the result. If the two numbers involved in the operation are not of the same length (the same number of decimal places), make sure to add trailing zeros as needed until they are. Carrying and borrowing (often called regrouping) remain the same as with whole number addition/subtraction.

Note: Trailing zeros don’t change the value of a decimal number, so you can add as many zeros after a decimal number as you need to make sure the numbers align.

Multiplying Decimals

When multiplying decimals, you can multiply the numbers as if they were whole numbers. Afterward, count the total number of digits to the right of the decimal point in both factors. The product should then have the same number of digits to the right of the decimal point.

Let’s look at an example.

\[2.7 \times 1.3 = ?\]

Solution

First, you need to convert the decimal numbers (the multiplicand and multiplier) to whole numbers by multiplying them by the power of \(10\) that is equal to the number of decimal places (for instance, one decimal place will require multiplying by \(10\), or \(10^1\)). Once that is done, perform the multiplication:

To get your final answer, you need to divide this modified product by the total power of \(10\) that was needed to convert both the multiplicand and multiplier to whole numbers. In this case, that is \(100\) because both numbers had one decimal place (\(10^1 \times 10^1\)). That will give you the product:

\[351 \div 100 = 3.51\] \[2.7 \times 1.3 = 3.51\]

Multiplying Decimals by Powers of 10

Multiplying decimals by powers of \(10\) involves shifting the decimal point to the right by a number of places equal to the exponent of \(10\). The number \(10\) can be written as \(10^1\), \(100\) can be written as \(10^2\), and so on. This operation is essential for scaling quantities up or down, as each shift in the decimal point represents a multiplication or division by \(10\).

Dividing Decimals

Division of decimals requires aligning the divisor and dividend in a similar manner as you do with long division of whole numbers. Once aligned, you perform the division as you would with whole numbers and then adjust the decimal point in the quotient accordingly.

Let’s look at an example.

\(7.8 \div 1.2 =\)____

Solution

We move the decimal points of the divisor and dividend one place to the right to obtain a division operation with a whole divisor. Moving the decimal points in \(7.8\) and \(1.2\) by one place results in \(78\) and \(12\), respectively. We can do this because \(78 \div 12\) gives the same result as \(7.8 \div 1.2\).

Dividing Decimals by Powers of 10

Dividing decimals by powers of \(10\) involves shifting the decimal point to the left by a number of places equal to the exponent of \(10\). As with multiplication, this operation is essential for scaling quantities up or down.

Percentages

A percentage is a way to express a part of a whole in relation to \(100\). For example, \(50\%\) means \(50\) out of \(100\), which represents half of a whole. Similarly, \(25\%\) means \(25\) out of \(100\), which represents one-fourth of a whole.

Percentages establish relationships where proportions matter. By using the standard of \(100\) and the percent symbol (\(\%\)), we can get a clearer picture of the relationship between one number and another. For example, suppose you were given this data:

”Only \(116\) of the \(580\) college freshmen are expected to graduate in four years.”

While it’s obvious that the first number is much less than the second, it would be helpful to know how much less. If we add, “This means that only \(20\%\) will graduate on time,” the original numbers are much easier to compare.

Whether you’re analyzing data, calculating discounts, or interpreting trends, percentages provide a standardized way to express values in relation to the entire set.

Conversions with Percentages

Converting decimals and fractions to percentages involves recognizing that one whole (of anything) is equivalent to \(100\%\). This section will show you how to do both types of conversion.

Between Decimals and Percentages

To convert a decimal number to a percentage, you multiply it by \(100\) and add the percent symbol at the end. Remember, multiplication by \(100\) simply means moving the decimal place two places to the right.

For example, to convert \(0.35\) to a percentage, we must first move the decimal place two places to the right, which gives us \(35\). Then, we add the percent symbol:

\[0.35 = 35\%\]

To convert a percentage to a decimal, you remove the percent symbol and divide the number by \(100\). Remember, dividing a decimal number by \(100\) means moving the decimal point two places to the left.

For example, when converting \(72\%\) to a decimal, first, we remove the percent symbol to get \(72\). Then, we move the decimal point (there is an invisible decimal point at the end of every whole number) two places to the left to get \(0.72\):

\[72\%=0.72\]

Between Fractions and Percentages

Converting fractions to percentages, and vice versa, involves knowing how to convert both to a decimal. For example, to convert the fraction \(\frac{2}{5}\) to a percentage, we first have to convert it into a decimal:

\[\frac{2}{5} = 2 \div 5 = 0.4\]

Now, we convert it to a percentage using the process we learned in the previous section. We move the decimal place two places to the right (make sure to put placeholder zeros where required) and add the percent symbol at the end:

\[\frac{2}{5} = 0.4 \times 100 = 40\%\]

To go from a percentage to a fraction, we first convert the percentage to a decimal (by dividing by \(100\)) and then convert the decimal to a fraction. For example, \(65\%\) converted to a decimal is:

\[65\% = 65 \div 100 = 0.65\]

Sixty-five hundredths written as a fraction is \(\frac{65}{100}\). We can reduce this fraction:

\[\frac{65}{100} = \frac{13}{20}\]

Common Percentage/Fraction/Decimal Equivalencies

Knowing the common equivalencies between percentages, fractions, and decimals can help you answer questions more quickly. Here are some common conversions that you should know:

Note: The line over the number in a decimal means that the digit repeats forever.

Solving Percentage Problems

Whether you’re figuring out sales discounts, calculating tax amounts, or determining percentage increases or decreases, understanding how to work with percentages is crucial for accurate and informed decision-making. This section will teach you how to solve percentage problems.

Basic Percentage Problems

The most common scenarios in which you will encounter percentage problems are those involving sales and taxes. When determining the final price after a discount, you are calculating a percentage reduction from the original price. Likewise, calculating the tax on an item requires being able to add a percentage of the retail price.

For instance, if an item is marked down by \(20\%\), you subtract this percentage from the initial retail price to find the reduced price. On the other hand, if there is a tax of \(8\%\), you find \(8\%\) of the retail price and add that amount to the initial cost to get the total price after tax.

Percentage Increase and Decrease

Being able to calculate percentage increases and decreases is essential for evaluating changes in quantities or values. When prices increase or decrease by a certain percentage, it directly impacts the final cost. For example, if the price of a product is \(\$100\) and it increases by \(15\%\), you calculate this percentage of the initial cost and add it to find the new price. The increased price will be \(100 + (0.15)(100) = 100 + 15 = \$115\).

Conversely, if there’s a \(10\%\) decrease, you deduct this percentage from the original price to determine the reduced cost. If the price of a product is \(\$100\), after the \(10\%\) decrease, the final price will be \(100 - 0.1(100) = 100 - 10 = \$90\).