Mathematics Study Guide for the HESI Exam

General Information

You’re going into a science-related profession, so why are you being asked to do math? Love it or hate it, nurses use mathematics in nearly every area of their job. From urology to gastroenterology, nurses use math to conduct analysis, read and analyze patient charts, and carry out day-to-day functions.

For this section of the Health Education Systems Incorporated (HESI) exam, you will be given 50 questions and will have 50 minutes to complete them. You will have access to a basic on-screen calculator for the math questions, but you may not use your own calculator, so don’t count on being able to do much beyond the four basic operations (add, subtract, multiply, and divide) with a calculator. This test should not require you to do anything beyond those, anyway.

This guide will help you review the basic math concepts that you should understand before taking the mathematics section of the HESI exam.

Whole Numbers

Computation with whole numbers includes basic operations such as addition, subtraction, multiplication, and division. It also relies on the concept of place value in our number system.

Place Value

A digit is a single symbol used to represent numbers. In the base-ten number system, the digits are $0$ through $9$. When digits are combined, their position, or place, determines their value. The same digit can mean very different things depending on where it is located in a number. You might remember the charts from school showing how each digit in a number, for example $7 \text{,}654\text{,}321$, has a place value. Here is an example of one such chart:

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
7	6	5	4	3	2	1

When reading this number aloud, you would say, “Seven million, six hundred fifty-four thousand, three hundred and twenty-one.” This system also allows us to borrow and carry digits when computing addition and subtraction because our number system relies on powers of ten in the base-ten system.

For example, the number $45$ can be understood to be $4$ tens and $5$ ones, or $45$ ones. Similarly, the number $7\text{,}200$ can be thought of as $72$ hundreds or $7$ thousands and $2$ hundreds.

Note: In this guide, we use the terms “borrow” and “carry.” In modern math courses, both of these are covered under the term “regroup.” The words mean the same.

Addition

Addition is the most basic computation we can do with whole numbers. It is the process of combining two or more numbers into one. The numbers being added are called addends, and the answer is called the sum.

For example, in $4 + 5 = 9$, $4$ and $5$ are the addends and $9$ is the sum.

1 Addition Parts (NEW).png

Traditionally, the easiest way to do this is by writing the two numbers in column format, or vertically, so that each place value is aligned, and first adding the ones values, then the tens values, then the hundreds, and so on until the problem is complete. In other words, when setting up your problem, be sure to align the numbers so that the rightmost digits―the ones values―are on top of each other.

We’ll do an example problem.

Cameron has $\$8\text{,}726\text{,}904$ in his bank account. At the end of this work week, his paycheck is $\$791$ after taxes. How much money is in his bank account after he deposits the check?

Solution

Again, we want to make sure the ones values sit on top of each other when we set up this problem. Also, if a column adds up to $10$ or higher, we will have to carry a $1$ over to the next column.

\[\begin{align} 8,726,904& \\ \underline{+\quad\quad 791}& \\ 8,727,695 \end{align}\]

Therefore, Cameron has $\$8\text{,}727\text{,}695$ in his account after his paycheck is deposited.

Subtraction

Subtraction is the process of finding the difference between two numbers. It tells us how much remains or how far apart two numbers are. The number you start with is called the minuend, the number you subtract is the subtrahend, and the result is the difference.

For example, in $10 -4 = 6$, $10$ is the minuend, $4$ is the subtrahend, and $6$ is the difference (result).

2 Subtraction Parts (NEW).png

Subtraction requires you to be careful with the order of the numbers, because, unlike with addition, changing the order will change the result.

Much like addition, we traditionally solve subtraction problems by aligning the ones position into a column. Then, we take away some number from the top. If the bottom digit is larger than the top digit, we will have to borrow from the next column. Let’s look at an example.

A bag of saline solution originally containing $975$ milliliters has delivered $190$ milliliters to the patient. How much saline is left in the bag?

Solution

The original amount of saline we started with is $975$ milliliters, so that number will be on top. We are subtracting $190$ milliliters, so that will be placed on the bottom as we set up the subtraction problem:

$\require{cancel} \;8\;17$ $\begin{align} \cancel{9}\cancel{7} 5 \\ \underline{- 1\,9\,0}& \\ 7\,8\,5& \\ \end{align}$

Since we cannot take $9$ tens from $7$ tens, we have to borrow (regroup) from the $9$ in the hundreds column. The $9$ hundreds becomes $8$ hundreds and $10$ tens. The $10$ tens combine with our $7$ tens to become $17$ tens. Now we can subtract down each column. The amount of saline left in the bag is $785$ milliliters.

Multiplication

Multiplication is a shortcut for repeated addition. When you multiply two numbers, one of them (the multiplicand) is added to itself repeatedly, as many times as specified by the other number (the multiplier). The result is called the product.

For example, in $4 \times 3 = 12$, we are essentially adding the number $4$ three times:

\[4 + 4 + 4 = 12\]

Here, $4$ is the multiplicand, $3$ is the multiplier, and $12$ is the product.

3 Multiplication Parts (NEW).png

This sort of simple repeated addition works for smaller, familiar multiplication problems. But what happens when you have to multiply large numbers, like $360 \times 21$? Then, a column format similar to what we used for addition and subtraction is helpful. It’s easiest to place the larger number on top for your calculations.

The first step is to multiply the ones digit from the bottom number by each digit of the top number, working from right (the ones column) to left:

\[\begin{align} 360& \\ \underline{\times\quad 21}& \\ 360& \\ \end{align}\]

In this particular multiplication problem, we start by multiplying the $1$ in $21$ by $360$, digit by digit, from right to left.

Our next step is to multiply the tens digit from the bottom number by the top number. Before we do that, we’re going to insert a placeholder zero to show that the $2$ in $21$ indicates we’re multiplying by $2$ tens (i.e., 20):

\[\begin{align} 360& \\ \underline{\times\quad 21}& \\ 360& \\ \underline{+7200}& \\ 7560& \\ \end{align}\]

The final step is to add the two products: $360 + 7\text{,}200 = 7\text{,}560$.

Note that we had to carry a one over to the hundreds column after multiplying $2$ by $6$ to get $12$. This carried number is then added to our answer when we multiply the $3$ from $360$ by the $2$ in $21$.

Before we look at a word problem example, ask yourself, “How does the place value system help us do long multiplication?”

Jada owns a popular mobile phone and technology store. She receives $245$ boxes, each filled with $27$ brand-new phones to be released for sale the next week. How many of these new phones does she have in stock?

Solution

One way to visualize this problem is to picture one box that you open up to see $27$ phones packed inside. As you open up each new box, you have to add another $27$ phones to your inventory. But with $245$ boxes, that can take quite a while. So we’re going to work it out in column format.

We put the larger number, $245$, on top. Step one is to multiply the ones column from the bottom number, $7$, by $245$:

$\;\;33$ $\begin{align} 245& \\ \underline{\times\;27}& \\ 1715& \\ \end{align}$

Again, we have to carry over a number after multiplying $5$ by $7$ and $4$ by $7$.

The next step is to multiply the tens column of the bottom number, $2$, by $245$. Don’t forget your placeholder zero, because the $2$ in $27$ is really $2$ tens:

$\quad\quad 1$ \ $\begin{align} 245& \\ \underline{\times\quad 27}& \\ 1715& \\ \underline{+4900}& \\ 6615& \\ \end{align}$

Our final step was adding $1\text{,}715$ and $4\text{,}900$. Our answer is that Jada has $6\text{,}615$ shiny new phones to sell in her store.

Division

If multiplication is repeated addition, then division is repeated subtraction. We start off with some number and what we want to know is how many times we would have to subtract another number from it until we arrive at zero. The number we start with is the dividend, while the number we repeatedly subtract is the divisor. In division problems, the answer is called the quotient. In other words, dividend $\div$ divisor $=$ quotient.

Again, this is easy to visualize and do with smaller numbers, but when it comes to larger numbers, it’s not so practical. We have a method for this: long division. This process is different from addition, subtraction, and multiplication problems. We’ll go over the steps of long division with an example problem.

Alexis has $4\text{,}630$ milliliters of a chemical. Each time she does an experiment, she uses $125$ milliliters. How many experiments can she perform?

Solution

These are the steps of long division:

Place the divisor (in this case $125$) outside the long division symbol and the dividend ($4630$) underneath it.
Work out the smallest portion of the dividend that the divisor can go into. For instance, $125$ goes into $463$. Why $463$? The number $46$ is too small, while $4\text{,}630$ is too large.
Determine how many times the divisor goes into that dividend portion. In this example, $125$ goes into $463$ three times, so we write $3$ up top, over the last digit we included in the dividend portion (in this case, $3$).
Multiply the number in the quotient ($3$) by the divisor ($125 \times 3 = 375$) and subtract that difference from the portion of the dividend you just used ($463 - 375 = 88$), and write that below the dividend.
Drop the next digit in the dividend down to the end of this new number ($88$ becomes $880$).
Repeat steps three to five until you’ve reached the end of the dividend. Here, $125$ goes into $880$ seven times, so we write $7$ up top, next to the $3$. If the divisor goes into this portion of the dividend evenly, you are done and you have your quotient. If not, you have a remainder.
If there is a remainder (a leftover amount because the divisor does not break the dividend into equal parts), write an “R” and then the remainder number ($5$ in this case, giving us an answer of $37R5$).

This is what these steps looks like using our example problem:

\[\require{enclose} \begin{array}{r} 37R5 \\[-3pt] 125 \enclose{longdiv}{4630} \phantom{00.} \\[-3pt] \underline{-375}\phantom{000.} \\[-3pt] 880 \phantom{00.} \\[-3pt] \underline{-875} \phantom{00.} \\[-3pt] \quad \; \phantom{8}\phantom{7}5 \phantom{00.} \\[-3pt] \end{array}\]

Thus, Alexis can do $37$ of her experiments with $4\text{,}630$ milliliters of the chemical, but she will have $5$ milliliters left over.

Signed Numbers

Signed numbers are numbers that include either a positive ($+$) or negative ($-$) sign. These signs indicate whether a number is positive (above zero) or negative (below zero). On a number line, like the one below, positive numbers are located to the right of $0$, while negative numbers are located to the left of $0$:

7 Signed Number Number Line (NEW).png

Note: If there is no sign in front of a number, it is assumed to be positive.

Doing operations with signed numbers can be slightly more complex than when you are only working with positive numbers. Each operation varies depending on whether the signs are the same or different.

You also need to be familiar with the concept of absolute value, which is the non-negative value of any number, whether it’s positive or negative. It can be found by counting a number’s distance from $0$ on the number line. For instance, the absolute value of both $4$ and $-4$ is $4$ because they are both four spaces from $0$.

Now, let’s look at how each operation with signed numbers works.

Addition

same signs—When the numbers have the same sign, add their absolute values and assign the sum the same sign as the numbers. For example, $(+5) + (+3) = 8$. Since both numbers are positive, we add them directly: $5 + 3 = 8$. For comparison, $(-5) + (-3) = -8$, which we can also get by adding $5$ and $3$ and then placing a negative sign in front of the answer.
different signs—When the numbers have different signs, subtract the smaller absolute value from the larger absolute value, and assign the sign of the number with the larger absolute value. For example, $(+7) + (-4) = 3$. Since the signs are different, we subtract $4$ from $7$ and keep the sign of the larger value, resulting in $+3$.

Subtraction

same signs—When the numbers have the same sign, subtract the smaller absolute value from the larger absolute value, and if the first number is larger in absolute value than the second number, give the result the same sign as the first number. Otherwise, give the result the opposite sign of the first number. For example, $(-6) - (-3) = -3$. Since both numbers are negative and $6$ is larger than $3$, we subtract $3$ (the absolute value of $-3$) from $6$ (the absolute value of $-6$), which gives us $3$. Finally, because $-6$ is larger in absolute value, we add a negative sign to our answer, giving us $-3$.
different signs—When the numbers have different signs, convert the subtraction into addition by changing the sign of the number being subtracted. Then, follow the addition rules for either the same or different signs. For example, $(+5) - (-3) = 8$. When subtracting a negative, it becomes addition, so it can be thought of as $+5 + 3 = 8$. If we had $(-5) - (+3)$, we would change $+3$ into $-3$, which then gives us $-5 - -3 = -5 + 3$ (two negatives become positive), resulting in $-2$.

Multiplication and Division

For multiplication and division with signed numbers, the rules are simple. You simply multiply or divide as usual and then attach the sign as follows:

same signs—If the signs are the same, the result is always positive:

\[(+4) \times (+3) = +12\] \[(-12) \div (-3) = +4\]

different signs—If the signs are different, the result is always negative:

\[(+6) \times (-2) = -12\] \[(+6) \div (-2) = -3\]

Order of Operations

When solving problems with more than one operation, you must follow a specific order to get the correct answer. This order ensures that everyone solves problems the same way and arrives at the same answer. This order is conveyed with the acronym PEMDAS:

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), left to right.
Then do all addition and subtraction (AS), left to right.
A good way to remember: Please Excuse My Dear Aunt Sally.

Let’s look at a couple of problems and solve them using the order of operations.

Solve: $6 + (3^2 - 1) \div 2$

Solution

The first step of PEMDAS is working out the parentheses. For this problem, inside the parentheses, we have:

\[3^2 - 1\]

Exponents come before subtraction in PEMDAS, so let’s do that first:

\[3^2 = 9\]

So, now we have $9 - 1 = 8$, which makes our expression:

\[6 + 8 \div 2\]

According to PEMDAS, we perform division before addition:

\[8 \div 2 = 4\]

Now we do the addition:

\[6 + 4 = 10\]

That’s the correct answer. If we had done this problem in any other order, it’s likely we would have gotten a very different (and incorrect) answer.

Here is another example.

Solve: $(5 + 2) \times 3^2 - 4$

Solution

Again, we start with the parentheses:

\[5 + 2 = 7\]

Next, let’s do the exponents:

\[3^2 = 9\]

Now, we multiply:

\[7 \times 9 = 63\]

Finally, we subtract:

\[63 - 4 = 59\]

Properties of Operations

The properties of operations give us flexibility in solving mathematical problems. Whether we’re adding, subtracting, multiplying, or dividing, following the properties ensure that we work through expressions in a consistent, logical way.

These properties apply across all areas of math, including fractions, decimals, and algebra, and understanding them is essential to solving even basic math questions correctly.

Commutative Properties

The commutative property explains that the order in which you add or multiply numbers does not change the result. In other words, the numbers can switch places and the outcome stays the same.

The commutative property of addition states:

\[a + b = b + a\]

This means that, for example, if you’re adding $3$ and $5$, it doesn’t matter in which order you do it:

\[3 + 5 = 8 \quad \text{and}\quad 5 + 3 = 8\]

The commutative property of multiplication states:

\[a \times b = b \times a\]

For example, if you multiply $7$ and $4$, no matter the order, you’ll get the same answer:

\[4 \times 7 = 28 \quad \text{and}\quad 7 \times 4 = 28\]

This property helps simplify problems, especially when rearranging numbers mentally to make calculations easier.

Associative Properties

The associative property tells us that when adding or multiplying three or more numbers, the way we group them (using parentheses) does not change the result. This is especially helpful when solving problems mentally or rearranging steps for convenience.

The associative property of addition states:

\[(a + b) + c = a + (b + c)\]

Let’s say you’re adding $2$, $3$, and $4$. You could do:

\[(2 + 3) + 4 = 5 + 4 = 9 \quad \text{or}\quad 2 + (3 + 4) = 2 + 7 = 9\]

Either way, the total is $9$.

The associative property of multiplication states:

\[(a \times b) \times c = a \times (b \times c)\]

Try multiplying $2$, $3$, and $5$ using this property:

\[(2 \times 3) \times 5 = 6 \times 5 = 30 \quad \text{and}\quad 2 \times (3 \times 5) = 2 \times 15 = 30\]

Note: This property doesn’t apply to subtraction or division, so always double-check when dealing with those operations.

Distributive Property of Multiplication

The distributive property shows how multiplication interacts with addition or subtraction. It allows you to break a problem into smaller parts to simplify it, especially when dealing with parentheses.

The distributive formula is generally presented as:

\[a \times (b + c) = (a \times b) + (a \times c)\]

This is the same for subtraction:

\[a \times (b - c) = (a \times b) - (a \times c)\]

Using this property, we can evaluate the expression $3 \times (4+5)$. Instead of adding first, you can break it apart: $3 \times 4 + 3 \times 5 = 12 + 15 = 27$. Or, if you do it the standard way (order of operations), you get:

\[3 \times 9 = 27\]

It’s the same result either way! This property is especially important when solving algebra problems and also shows up often in fraction and decimal operations.