Mathematics Study Guide for the HESI Exam

Decimals

Decimals are numbers that represent values less than one (similar to proper fractions). They are the values to the right of a decimal point. They are commonly used in medical measurements, medication dosages, temperature readings, and financial data.

Decimal Place Value

Remember the place value chart for whole numbers? There’s a continuation of the chart that represents numbers smaller than one in the form of decimals. Here’s a sample of the chart for the number \(1.234567\):

Ones	Decimal Point	Tenths	Hundredths	Thousandths	Ten-Thousandths	Hundred-Thousandths	Millionths
1	.	2	3	4	5	6	7

Note how the terms differ from the numbers in the regular place value chart. For instance, instead of tens, the decimal is tenths.

Terminating Decimals

A terminating decimal is a decimal number that comes to an end (it has a finite number of digits after the decimal point). These decimals result from fractions whose denominators have only the prime factors \(2\) or \(5\). For example, \(\frac{1}{4}\) can be written as \(0.25\) (this decimal terminates after two digits) and \(\frac{3}{5}\) can be written as \(0.6\) (this decimal ends after just one digit). Terminating decimals are the easiest to work with because they don’t go on forever like repeating decimals do.

Adding and Subtracting Decimals

When adding or subtracting decimals, it is the same process as adding or subtracting with whole numbers. Just remember to align the ones position or the decimal point. When working down, the decimal point must be carried down into the answer.

Here is an example:

\[\begin{align} $12.56& \\ \underline{-\; 2.24}& \\ $10.32& \\ \end{align}\]

Multiplying Decimals

Multiplying is similar to adding and subtracting decimals, except you ignore the decimal point until the very end. Then, the total number of decimal places in the two numbers (the multiplicand and multiplier) should also be the number of decimal places in your answer.

Note: Sometimes your answer will involve trailing zeroes that can be dropped. For example, \(1.80\) is the same as \(1.8\), so you don’t need to include the \(0\).

An example will make this process clearer.

Multiply: \(15.3 \times 0.25\)

Solution

Let’s set this problem up like any other whole number multiplication problem:

\[\begin{align} 153& \\ \underline{\times\; 025}& \\ 765& \\ \underline {+\;3060}& \\ 3825& \\ \end{align}\]

Now, there are a total of three decimal places in the two numbers (one in the multiplicand and two in the multiplier). As such, our answer also must have three decimal places: \(3.825\).

Dividing Decimals

To divide decimals, you must move the decimal place in the divisor to the right until it is a whole number. However many places you moved it, the decimal place in the dividend must also be moved to the right. You may have to add zeros to the dividend during this process.

Let’s do an example problem.

Divide: \(50.125\) by \(0.05\)

Solution

To start, be sure to move both decimal points two places to the right before dividing. This means \(0.05\) becomes \(5\) and \(50.125\) becomes \(5012.5\):

\[\require{enclose} \begin{array}{r} 1002.5 \\[-3pt] 5 \enclose{longdiv}{5012.5} \\[-3pt] \underline{-5}\phantom{0} \phantom{1}\phantom{2}\phantom{.}\phantom{5} \\[-3pt] 0 \phantom{0} \phantom{1}\phantom{2}\phantom{.}\phantom{5} \\[-3pt] \underline{-0} \phantom{0} \phantom{1}\phantom{2}\phantom{.}\phantom{5} \\[-3pt] 1 \phantom{1}\phantom{2}\phantom{.}\phantom{5} \\[-3pt] \underline{-0}\phantom{1}\phantom{2}\phantom{.}\phantom{5}\\[-3pt] 12\phantom{2}\phantom{.}\phantom{5} \\[-3pt] \underline{-10} \phantom{2}\phantom{.}\phantom{5}\\[-3pt] 25 \phantom{.}\phantom{5}\\[-3pt] \underline{-25} \phantom{.}\phantom{5}\\[-3pt] 0 \phantom{.}\phantom{5} \\[-3pt] \end{array}\]

The answer is \(1002.5\).

Changing Fractions to Decimals

Another facet of decimals is that we can convert them into fractions, and vice versa. Remember how we said earlier that fractions are just a form of division? This is evident when converting from a fraction to decimal. To do so, we divide the numerator by the denominator and work it out with division.

Here is an example.

Convert \(\frac{3}{8}\) to a decimal.

Solution

\[\frac{3}{8} = 0.375\]

Changing Decimals to Fractions

What about when we want to convert from a decimal to a fraction? The nice thing about the place value system is that when we want to convert from a decimal to a fraction, the answer is right in the name of the decimal.

For example, \(0.75\) is read aloud as “seventy-five hundredths” or “seventy-five over one hundred,” which translates as a fraction to this:

\[\frac{75}{100}\]

From here, all we have to do is simplify:

\[\frac{75}{100} \div \frac{25}{25} = \frac{3}{4}\]

Ratios and Proportions

Ratios and proportions are key concepts for comparing quantities, scaling data, and understanding relationships between numbers. In healthcare, they’re used for dosage measurements, blood pressure comparisons, fluid intake and output, and much more. Let’s begin by understanding what ratios are and how to work with them.

Ratios

Ratios express a relationship between two quantities. You might see them when comparing the number of nurses to patients, pills to doses, or ingredient amounts in a solution. Ratios tell us how many times one quantity fits into another.

The Meaning of Ratios

Ratios help you understand how two values compare. For example, suppose there is a classroom in which there are four men and \(12\) women. The ratio of men to women would be \(4\text{:}12\).

This tells us that for every four men, there are \(12\) women. When simplified, that ratio is \(1\text{:}3\), meaning for every one man, there are three women.

Writing Ratios

There are three common ways to write a ratio: colon notation, fractional notation, and in words. For a ratio that expresses three parts out of five, these are the three different ways to write it:

Suppose, there are six nurses and three doctors on shift, the ratio of nurses to doctors can be written as:

\[6\text{:}3 \quad\text{or}\quad \frac{6}{3}\quad\text{or}\quad6 \text{ to } 3\]

Always simplify ratios when possible. In this example, \(\frac{6}{3} = 2\text{:}1\), meaning there are two nurses for every one doctor.

Proportions

Proportions are used to show equivalent ratios and are essential in calculating medical dosages, scaling recipes, and interpreting charts. A proportion tells you that two ratios are equal.

The Meaning of a Proportion

A proportion compares two equal ratios. When we know two ratios are in proportion to each other, we can find unknown values within those ratios. For example, if we know two out of every five patients in a clinic are over \(60\) years old, we can figure out how many patients over \(60\) there would be if the clinic had \(20\) patients total.

To do this, we set up the following proportion:

\[\frac{2}{5} = \frac{x}{20}\]

Here, we’re solving for \(x\), the number of over-\(60\) patients in a group of \(20\) people.

Using Proportions to Problem-Solve

Proportions help solve for missing values in real-life situations by setting up two equal ratios. Above, we set up a proportion, but we didn’t solve it. Now, we’ll solve a proportion problem.

A medication label says “\(2\) tablets per \(8\) hours.” How many tablets are needed for \(24\) hours?

Solution

We let \(x\) be the unknown variable, the number of tablets taken in \(24\) hours, and set up the proportion as follows:

\[\frac{2}{8} = \frac{x}{24}\]

This is where we stopped in the previous section. The next step involves cross-multiplying the two fractions, meaning we multiply the numerator of the first by the denominator of the second, and vice versa:

\[2 \times 24 = 8 \times x\] \[48 = 8x\]

Now, we do simple algebra to solve for \(x\):

\[\frac{48}{8} = \frac{8x}{8}\] \[6 = x\]

Thus, the patient will need \(6\) tablets in the course of a day.

Percentages

Percentages are a way to express a number out of \(\bf{100}\). In fact, the root of the word percentage is “per cent” or “per \(100\)”. Therefore, a percentage such as \(42\%\) means \(42\) out of \(100\). As a fraction, it means \(\frac{42}{100}\).

In this section, we will learn how to convert to and from percentages using fractions and decimals.

Percent to Fraction

Converting from a percentage to a fraction is quite straightforward. You simply put the number over one hundred and drop the percentage symbol (\(\%\)). If the fraction can be reduced, do that. Let’s try a quick example.

Convert \(75\%\) to a fraction.

Solution

We start by getting rid of the percent symbol, then we put the number over \(100\) to get:

\[\frac{75}{100}\]

Always simplify the fraction when possible. So, for the above, you could also write it as:

\[\frac{75}{100} \div \frac{25}{25} = \frac{3}{4}\]

We got our final result by dividing both the top and bottom by their greatest common factor.

Decimal to Percent

To convert a decimal to a percent, simply multiply by \(100\) and add the percent symbol.

Convert \(0.63\) to a percent.

Solution

We multiply by \(100\):

\[0.63 \times 100 = 63\]

Now, put the percent symbol at the end and you have your answer:

\[63\%\]

Note: Multiplying by \(100\) means to move the decimal point two places to the right.

Percent to Decimal

If we want to convert a percentage to a decimal, the first step is converting it to a fraction. Then, it’s simply a matter of using our conversion method for fractions to decimals. Let’s try an example.

Convert \(42\%\) to a decimal.

We’ve just learned how to convert a percent to a fraction. Just put the number over \(100\):

\[\frac{42}{100}\]

If we wanted a fraction, we would reduce, but that’s not necessary here because we want to convert to a decimal. If you forgot that process, reading the fraction aloud gives you a clue how to do it: “forty-two over one hundred.” We simply have to bump the decimal place of the numerator over so that the digits reach the hundredths place:

\[\frac{42}{100} = 0.42\]

Note: You can also do this process by dividing the original percentage by \(100\).

Solving Percentage Problems

We use percentages in many kinds of word problems. They can answer questions like how much of something has been used, how much remains, how much of a goal has been reached, and more. The key to solving these types of questions is knowing what part is the “whole” and what part is the “portion.” Let’s look at different types of percentage calculation problems, and then we’ll practice them with a couple word problems.

Comparing Numbers as a Percentage

When we want to know what percent a number is of another number, we use the following formula:

\[\% = \frac{p}{w} \times 100\]

where \(p\) is the portion and \(w\) is the whole.

Let’s do an example.

What percent of \(50\) is \(30\)?

Solution

We will use the formula from above:

\[\% = \frac{p}{w} \times 100\]

We can substitute the values from the problem to find the answer:

\[\frac{30}{50} = 0.6\] \[0.6 \times 100 = 60\]

So, \(30\) is \(60\%\) of \(50\).

Finding a Percentage of a Number

When we want to find the percentage of a given number, we divide the percent by \(100\), which will give us a decimal number, and then we multiply that decimal number by the original given number. Let’s do an example problem to practice this process.

What is \(25\%\) of \(80\)?

Solution

First, convert the percent to a decimal by dividing by \(100\):

\[25 \div 100 = 0.25\]

Now, multiply to get the answer:

\[0.25 \times 80 = 20\]

Thus, \(25\%\) of \(80\) is \(20\).

Finding the Whole Given a Number and Percentage

When you are given a number and asked what percentage of a whole it is, you will need to do a bit of algebra to find the unknown value (the whole). We’ll show you how this is done with an example problem.

\(12\) is \(30\%\) of what number?

Solution

We can translate the problem given into an equation as follows:

\[12 = 0.3 \times n\]

where the “is” in the question translates to an equal sign and the “of” translates to multiplication. Note that we converted the percentage to a decimal (division by \(100\)) and \(n\) is the unknown.

Now, we use algebra to solve for \(n\):

\[n = \frac{12}{0.3}\] \[n = 40\]

Percentage Word Problems

Sharon is selling tickets for her college’s theater production. There are \(540\) seats in the theater. If she sells \(66.7\%\) of the available tickets, how many seats is that? Round to the nearest whole number.

Solution

To solve this problem, you must first understand what type of problem this is. The question is asking you to find a percentage (\(66.7\%\)) of a whole (\(540\)). Recall that the first step in this type of question is converting the percentage to a decimal:

\[66.7\% \div 100 = 0.667\]

Now, to find the number of seats, we multiply \(0.667\) by \(540\):

\[540 \times 0.667 = 360.18\]

The problem tells us to round to the nearest whole number, which is \(360\).

Let’s try a different type of problem.

Xavier and his friend go out for lunch together. At the end of the meal, Xavier offers to pay. If he leaves a \(\$6.16\) tip and the bill was \(\$34.24\), what percentage of the bill was Xavier’s tip? (Round the percent to the nearest whole number.)

Solution

Though the details in the question may make this seem complicated, this is just a situation where we’re given a number and asked to determine what percent it is of another number. Remember this formula:

\[\% = \frac{p}{w} \times 100\]

To figure out what percentage tip Xavier left, we divide the tip amount (\(p\)) by the bill amount (\(w\)):

\[\$6.16 \div \$34.24 \approx 0.1799\]

The second step is converting to a percent, which we do by multiplying this result by \(100\) (or moving the decimal two places to the right) and adding the percent sign:

\[0.1799 = 17.99\%\]

Since the problem asks us to round to the nearest whole number, we round up to \(18\%\). Thus, Xavier left an \(18\%\) tip on the meal bill.