Mathematics Study Guide for the HESI Exam

Fractions

Fractions are parts of numbers, or portions of a whole. You might remember seeing models of fractions in school, and creating a fraction out of the number of shaded parts over how many parts there were in total. For example, below is a rectangle divided into \(12\) equal parts, with six of these pieces shaded:

To write a fraction to represent this rectangle, we put the number of shaded pieces on top of the fraction bar and the total number of sections on the bottom:

\[\frac{6}{12}\]

Fraction Terms

Fractions are a vital part of math, especially in nursing and healthcare, where you’ll often work with medication dosages, IV rates, and lab values that involve parts of a whole.

Understanding the language and structure of fractions helps you confidently interpret and solve real-life clinical problems. Before diving into how to add, subtract, or reduce them, it’s important to understand what each part of a fraction means and the types of fractions you’ll encounter.

Parts of a Fraction

Fractions consist of two main parts:

• The numerator (top number) shows how many parts you have.
• The denominator (bottom number) shows how many total equal parts make up the whole.

The horizontal bar separating the numerator from the denominator is called the fraction bar.

For example, in the fraction \(\frac{3}{4}\), \(3\) is the numerator (it tells you that you have three parts) and \(4\) is the denominator (it tells you the whole is divided into four equal parts). In nursing, this might represent three-quarters of a pill, three-fourths of a liter, or any similar division of a whole.

Types of Fractions

Before you learn how to work with fractions, it’s helpful to understand the different types you’ll see. Each kind of fraction represents numbers differently and plays a unique role in calculations. Let’s look at the four main types:

• proper fraction—A proper fraction is one where the numerator is smaller than the denominator. This means the value of the fraction is less than one. For example, \(\frac{2}{5}\) means you have two out of five parts. Since \(2\) is less than \(5\), it’s a proper fraction.

• improper fraction—An improper fraction has a numerator that is equal to or greater than the denominator, meaning the value is \(1\) or more. For example, \(\frac{7}{4}\) means there are seven parts out of four, which is more than one whole. Improper fractions are often converted into mixed numbers.

• mixed number—A mixed number (or mixed fraction) is made up of a whole number and a proper fraction. For example, \(2\frac{1}{3}\) means there are two wholes and one part out of three of another whole. Mixed numbers are useful in real-world settings like cooking and dosage calculations.

• reciprocal fraction—A reciprocal is what you get when you flip a fraction. You switch the numerator and the denominator. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). When you multiply a fraction by its reciprocal, it equals \(1\). Reciprocals are especially important when dividing fractions.

Reducing Fractions

Sometimes, a fraction can be written in a simpler form without changing its value. This is called reducing (or simplifying) the fraction. You simplify a fraction by dividing both the numerator and the denominator by the same number, called a common factor. It’s helpful to simplify fractions so that they are easier to compare, calculate with, and interpret.

Factors

Think about the rectangle from the beginning of this section and the first fraction we saw, \(\frac{6}{12}\). While that fraction accurately represents the rectangle, it can be reduced so that both the numerator and denominator are smaller. We can divide \(6\) and \(12\) by \(2\), which gives us \(\frac{3}{6}\). Both \(\frac{6}{12}\) and \(\frac{3}{6}\) represent how much of the rectangle is shaded.

We don’t need to stop there, though. We can divide \(\frac{3}{6}\) by \(\frac{3}{3}\), which gives us \(\frac{1}{2}\). Again, \(\frac{6}{12}\), \(\frac{3}{6}\), and \(\frac{1}{2}\) all represent the same amount of the shaded rectangle. These are equivalent fractions.

Additionally, since \(\frac{1}{2}\) cannot be reduced anymore, it is called the simplified form of \(\frac{6}{12}\).

Let’s do an example problem.

Simplify \(\frac{42}{56}\)

Solution

The first thing you should notice is that both the numerator and the denominator have a common factor of \(2\), since they are even numbers.

Dividing the top and bottom by \(2\) gives the equivalent fraction \(\frac{21}{28}\). We know \(21\) and \(28\) share a common factor of \(7\), so dividing the numerator and denominator by \(7\) gives us the equivalent fraction \(\frac{3}{4}\).

There are no more common factors for \(3\) and \(4\), so this is the simplified form of \(\frac{42}{56}\).

Greatest Common Factor (GCF)

The GCF is the largest number that divides evenly into both parts. Using the GCF is the fastest way to reduce a fraction completely. You can find the GCF of the numerator and the denominator by doing the following:

1. List the factors of the numerator and the denominator.
2. Identify the common factors.
3. Choose the greatest of them.

Once you determine the GCF, divide both the numerator and the denominator by it to simplify a fraction to its lowest terms.

Here’s an example.

Simplify the fraction \(\frac{40}{60}\) by using the GCF method.

Solution

The numerator is \(40\) and the denominator is \(60\). Let’s list all the factors of \(40\) and \(60\):

\[40: 1, \, 2, \, 4, \, 5, \, 8, \, 10, \, 20, \, 40\] \[60: 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 10, \, 12, \, 15, \, 20, \, 30, \, 60\]

As we can see, the common factors are \(1, \, 2, \, 4, \, 5, \, 10, \, 20\). Out of these, which one is the greatest? It is \(20\). So, the GCF of \(40\) and \(60\) is \(20\). Therefore, to simplify the fraction to its lowest terms, we divide both the top and bottom by \(20\):

\[\frac{40 \div 20}{60 \div 20} = \frac{2}{3}\]

Operations with Fractions

Fractions can be added, subtracted, multiplied, and divided. In this section, we’ll go over each of these operations in detail, with clear examples to show you how it’s done. You’ll often use these skills in calculating dosages, IV flow rates, and other clinical tasks.

Adding and Subtracting

When fractions have the same denominator, adding or subtracting them is simple, you just add or subtract the numerators. For example, say you needed to add these two fractions:

\[\frac{3}{7} + \frac{2}{7}\]

Since the denominators are the same (both \(7\)), you do nothing with the denominator and simply add the numerators:

\[\frac{3+2}{7} = \frac{5}{7}\]

That’s it. There is no need to change anything when the denominators are the same.

Subtracting fractions with the same denominator is just as straightforward:

\[\frac{3-2}{7} = \frac{1}{7}\]

Common Denominator

If fractions have different denominators, you must first find a common denominator, specifically the least common denominator (LCD) of both fractions. The LCD is the least common multiple (LCM) of the numbers in the denominators, meaning the smallest number they both can divide into. For reference, the multiples of \(2\) are \(2, \, 4, \, 6,\) etc.

To find the LCM of two numbers, follow these steps:

1. List the multiples of each denominator separately in a list.
2. Choose the first common number in both the lists. This is the LCM.

Let’s do an example problem to show how this would work.

Add \(\frac{1}{4} + \frac{2}{3}\).

Solution

To find the LCM of the two denominators, we first list several multiples of both numbers:

\[4: 4, \, 8, \, 12, \, 16, \, 20, \, 24…\] \[3: 3, \, 6, \, 9, \, 12, \, 15, \, 18…\]

The first (least) common number in both lists is \(12\). Thus, the LCM of both denominators, \(4\) and \(3\), is \(12\). We need to convert both fractions so they each have this common denominator.:

\[\frac{1}{4} \times \frac{3}{3} = \frac{3}{12}\quad\text{and} \quad \frac{2}{3} \times \frac{4}{4} = \frac{8}{12}\]

Now, we can add as usual:

\[\frac{3}{12} + \frac{8}{12} = \frac{11}{12}\]

Note: Whenever possible, you will want to reduce the answer. In this case, though, the fraction can’t be reduced further, so that is your answer.

Adding Mixed Numbers and Improper Fractions

Think of quarters as parts of a dollar. It takes four quarters to make a dollar, so \(4\) would be the denominator of any fraction that involves quarters and dollars. The number of quarters it takes to equal a dollar is not going to change, but the number of quarters you have can change. You can have three quarters, for example, or five. If you have five quarters, you would have more than a dollar (the whole). This would be represented by the mixed number \(1\frac{1}{4}\) or the improper fraction \(\frac{5}{4}\).

In the real world, you will often need to add these types of fractions. Let’s try an example problem.

Denali owns a party space that hosts rentals for birthdays, weddings, and other celebrations. One day, while cleaning up after two birthday parties, he finds leftover pizzas, which are served as eight slices per pie. One party left behind \(2\frac{3}{8}\) pizzas. The other party left behind \(\frac{9}{8}\) pizzas. How much pizza was left behind in total?

Solution

This is a fraction addition problem with one mixed number and one improper fraction. To add these fractions together, it is best to get them in the same form. It’s generally easier to work with improper fractions, so let’s convert the mixed number, 2\(\frac{3}{8}\).

If each pizza has \(8\) slices and there are \(2\) whole pizzas left, then that means the whole number \(2 = \frac{16}{8}\). Then \(2\frac{3}{8} = \frac{16}{8}\) + \(\frac{3}{8} = \frac{19}{8}\).

So, the first party left behind \(\frac{19}{8}\) pizzas, while the second party left behind \(\frac{9}{8}\) pizzas. All together, there were \(\frac{19}{8} + \frac{9}{8} = \frac{28}{8}\) pizzas left over.

But wait! \(\frac{28}{8}\) is an improper fraction, and they are called that for a reason. The proper way to write a fraction when the numerator is larger than the denominator is as a mixed number.

To get there, we divide the numerator by the denominator. The number of times it goes in evenly is the whole number, while the remainder is our numerator in the fraction part.

For this example, \(28 \div 8 = 3 \text{ R }4\).

Therefore, as a mixed number, there are \(3\frac{4}{8}= 3\frac{1}{2}\) pizzas left over.

Subtracting Mixed Numbers and Improper Fractions

Subtraction with mixed numbers and improper fractions is very similar to addition. The denominator must be the same, and it is easiest if any mixed numbers are converted to improper fractions first. We’ll do an example problem.

Subtract and simplify: \(2\frac{14}{15}-\frac{3}{5}\)

Solution

First, we’ll convert \(2\frac{14}{15}\) to an improper fraction. Since our denominator is \(15\), we multiply \(15\) by \(2\) to get \(30\). So:

\[2 \frac{14}{15} = \frac{30}{15} + \frac{14}{15} = \frac{44}{15}\]

Next, we want to make it so that \(\frac{3}{5}\) has the same denominator as \(\frac{44}{15}\) by finding the least common multiple of the two numbers. Notice that \(5 \times 3 = 15\), so we multiply the top and bottom of the fraction by \(3\), like so:

\[\frac{3}{5} \times \frac{3}{3} = \frac{9}{15}\]

Now we can set up our subtraction problem and subtract the numerators straight across while keeping the same denominator:

\[2 \frac{14}{15} – \frac{3}{5} = \frac{44}{15} –\frac{9}{15}= \frac{35}{15}\]

The final step is to convert from an improper fraction back to a mixed number. We do this by dividing the numerator by the denominator. The whole number is the number of times it goes in evenly while the remainder is the numerator of the fraction. The denominator remains the same. Since \(35 \div 15 = 2 \text{ R }5\), we simplify:

\[\frac{35}{15} = 2\frac{5}{15}= 2\frac{1}{3}\]

Multiplying

When fractions need to be multiplied, you work straight across, multiplying one numerator by the other numerator and one denominator by the other denominator. You’ve already seen this in practice when we’ve multiplied fractions to get common denominators (or divided to simplify them). When converting to a common denominator, the numerator and the denominator must both be multiplied by a fraction that has the same number on top and on bottom (which is equal to \(1\)).

Let’s practice multiplying fractions with an example problem.

Harriet and Samantha are sisters. In the spring, their father tells them they can each have \(\frac{1}{3}\) of the garden to grow whatever they want during the summer, but they have to plan it out first. Harriet decides she wants to grow \(\frac{3}{4}\) of her land with corn. How much of the total garden will have her corn growing on it?

Solution

To find the answer to this problem, we must multiply \(\frac{1}{3}\) by \(\frac{3}{4}\). We will multiply straight across the top and bottom numbers to find the answer, then simplify:

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

So, Harriet’s corn will take up \(\frac{1}{4}\) of the total garden.

Dividing

You may have noticed that a fraction is just another way to write “division.” We start with a whole piece and divide it into parts. So, when we divide fractions by each other, we’re really just taking a portion of a portion. With the garden question above, we found a portion of Harriet’s part of the garden.

In fact, multiplying and dividing fractions are very similar processes. The only difference is that, with division, you need to work with reciprocal fractions. An example problem will help us explain this.

Divide \(\frac{2}{5}\) by \(\frac{1}{3}\).

Solution

As said above, dividing fractions actually involves multiplying fractions. However, instead of multiplying the two given fractions by each other, you multiply the first fraction by the reciprocal of the second fraction, meaning you flip the second fraction upside down. Recall that the reciprocal fraction is the fraction that, when multiplied by the original fraction, equals \(1\).

So, the reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\):

\[\frac{1}{3} \times \frac{3}{1}= \frac{3}{3}=1\]

Now that we have the reciprocal of the second fraction, it’s just a simple multiplication problem:

\[\frac{2}{5} \div \frac{1}{3} = \frac{2}{5} \times \frac{3}{1}=\frac{6}{5}\]

From here, we can simplify by converting to a mixed number:

\[\frac{6}{5} = 1 \frac{1}{5}\]

Changing Fraction Types

In nursing and healthcare calculations, you’ll often need to switch between different types of fractions—specifically improper fractions and mixed numbers—depending on the context. Knowing how to convert between these forms helps ensure clarity and accuracy in drug dosages, IV calculations, and chart values.

Improper Fraction to Mixed Number

An improper fraction has a numerator that is equal to or larger than the denominator, such as \(\frac{11}{4}\). To make it easier to understand or apply in clinical contexts, we often convert improper fractions into mixed numbers, which show both the whole number part and the leftover fraction.

Here are the steps to convert an improper fraction to a mixed number:

1. Divide the numerator by the denominator.
2. Write down the whole number part of the answer.
3. Take the remainder and write it over the original denominator.
4. Combine the whole number and the new fraction.

Let’s do an example.

Convert \(\frac{11}{4}\) to a mixed number.

Solution

First, we divide \(11\) by \(4\) to get \(2\) and a remainder of \(3\).

Thus, the whole number portion of the mixed number is \(2\) and the fractional part will be \(\frac{3}{4}\). We combine them and get the final result:

\[\frac{11}{4} = 2\frac{3}{4}\]

Mixed Number to Improper Fraction

The process for converting from a mixed number to an improper fraction essentially reverses the steps you just learned:

1. Multiply the whole number portion of the mixed number by the fraction’s denominator.
2. Add that product to the numerator.
3. Place that result above the original denominator.

Here is an example.

Convert \(1\frac{1}{4}\) to an improper fraction.

Solution

First, multiply \(1\) (the whole number) by \(4\), which gives you \(4\). Next, add this \(4\) to the numerator, \(1\), which gives you \(5\). Finally, put this total atop the denominator:

\[1\frac{1}{4} = \frac{5}{4}\]