Mathematics Study Guide for the TEAS

Measurement

Most people in the US grow up using the customary or household system of measurement, including such units as feet, pounds, and fluid ounces. However, given the prevailing use of metric units in the fields of science and medicine, it is important to become familiar with the principles of measurement pertaining to the use of liters, grams, and meters, as well.

Metric Units of Measurement

The metric system is the most widely used system of measurement in the world, consisting of a coherent series of units compatible with the decimal system of numbers. Its modern form is the International System of Units (SI) and it has become the system of measurement upon which most of science and medicine depend.

The metric system is a system of measurement consisting of basic units, such as liters (L), grams (g), and meters (m), along with their associated units. The basic units are altered with the addition of standard prefixes.

The prefix added to a base informs you of the size of a unit by designating what multiple of 10 it represents. Hence, each prefix always means the same thing, regardless of the base to which it is attached. The relationships between the most commonly used units in health care are primarily found in multiples of 1,000. They are:

kilo = thousand
deci= tenth
centi = hundredth
milli = thousandth
micro = millionth

Additional prefixes include:

hecto = hundred
deka = ten

Converting within the Metric System

Because the metric system is based on the decimal system, conversions within a given denomination can be accomplished simply by relocating the decimal point. For example, 10 milligrams (mg) equals 0.01 grams (g). It is, therefore, critical that healthcare workers not confuse unit names to avoid making medication errors. Likewise, when taking an exam, you need to make sure that the unit names are the same in the given information as they are in the answer choices and make accurate conversions, when necessary.

Converting between Metric and Standard Measuring Units

Healthcare professionals need to be familiar with both the metric and standard household systems. Doctors write prescriptions using the metric system, yet patients need to be able to understand how to measure their prescriptions at home. It is, therefore, important to be able to equate metric to familiar household or customary measurements in order to perform conversions correctly and ensure that doses are properly administered. Common healthcare conversions include:

1 kilogram (kg) = 2.2 pounds (lb)
1 teaspoon (tsp) = 5 milliliters (mL)
½ teaspoon (tsp) = 2.5 milliliters (mL)

Additional conversions:

1 inch = 2.54 centimeters
1 meter = 3.281 feet
1 ounce = 28.35 grams
1 liter = 1.057 quarts

A basic understanding of approximate conversions will also be helpful in checking for accuracy. For example, if you know that 5 mL is approximately equal to 1 teaspoon, you would instantly know that a dosage notation of 50 mL of cough syrup is incorrect.

Geometric Units of Measurement

The term geometric quantities is basically another way of saying physical quantities, and incorporates such concepts as length, width, height, area, perimeter, and other dimensions.

Measuring Non-Curved Shapes

You need to be able to calculate areas and perimeters of fairly simple shapes, such as rectangles, squares, and triangles.

Perimeter

Perimeter is the distance around the outside of an object and can be thought of as a continuous line indicating the boundary of an area. It is typically calculated using a set formula. For example, the formula for finding the perimeter of a rectangle is \(P = 2(l + w)\), where \(P\) stands for perimeter and \(l\) and \(w\) stand for length and width, respectively.

Max is preparing to fence in a portion of his land for cattle. The area he has chosen measures \(100\) feet by \(241\) feet. What is the total length of the fencing material he will need to enclose the entire area?

Using the equation \(P = 2(l + w)\):

\(P = 2 (100 + 241) = 2(341) = 682\) feet

Area

Area is the amount of space filled by the surface of a figure or shape. It, too, is usually determined by a set formula. The standard means of expressing area is in square units. For example, pediatric drug dosages are sometimes based on body surface area, as measured in square centimeters (\(\text{cm}^2\)).

To find the area of a rectangular surface, use the formula \(A = l \times w\), with \(A\) standing for area and \(l\) and \(w\) standing for length and width, respectively.

So, the area of a space that measures \(3\) feet by \(4\) feet would be \(12\) square feet or \(12\; \text{ft}^2\).

Volume

Volume is the amount of space a substance or object occupies in three dimensions. It is most often measured in cubic units, as when expressing density in terms of grams per cubic centimeter (\(\text{cm}^3\)). In the healthcare profession, the liter is the main unit of volume and is routinely applied to drug dosages.

The formula for volume is \(V = l \times w \times h\), with the dimension of height (\(h\)) being added to the formula for area.

So, the equation for the volume of a container that is \(2\) inches wide, \(4\) inches in length, and \(3\) inches tall (height) would be:

\(V = 2 x 4 x 3 = 24\) cubic inches (\(24\;\text{in}^3\))

A Summary of Formulas

Perimeters

• Square: \(P = 4s\), where \(s\) is the side length.
• Rectangle: \(P = 2(l+w)\), where \(l\) and \(w\) are the length and width.
• Triangle: Usually, you just have to figure out the three sides and add them. There is no set formula you need to know.

Areas

• Square: \(A= s^2\), where \(s\) is the length of a side.
• Rectangle: \(A=l \times w\), where \(l\) and \(w\) are the length and width.
• Triangle: \(A= \frac{1}{2} b h\), where \(b\) and \(h\) are the base length and the height.

Volumes of Cubes and Rectangular Prisms (Boxes)

• Cube: \(V=s^3\), where \(s\) is the length of an edge.
• Rectangular prism: \(V= l \times w \times h\), where \(l, w,\) and \(h\) are the length, width, and height.

Converting Squared and Cubed Units

Suppose you need to convert square meters to square feet. Start by comparing square meters to square feet using \(1\) meter = \(3.28\) feet:

\[1 \text{ m} \times 1 \text{ m} = 1 \text{ m}^2\] \[3.28 \text{ ft} \times 3.28 \text{ ft} = 10.76 \text{ ft}^2\]

This tells us that \(1 \text{ m}^2 = 10.76 \text{ ft}^2\), which gives us two conversion factors that we can use to convert \(\text{ m}^2\) to \(\text{ ft}^2\), or vice versa.

\[\frac{10.76 \text{ ft}^2}{1 \text{ m}^2} \text{ or } \frac{1 \text{ m}^2} {10.76 \text{ ft}^2}\]

For example, convert \(4 \text{ m}^2\) to \(\text{ ft}^2\) using a conversion factor:

\[\require{cancel}\] \[4 \require{cancel}\cancel{\text{ m}^2} \times \frac{10.76 \text{ ft}^2} {1 \cancel{\text{ m}^2}} = 43.04 \text{ ft}^2\]

For cubic units, we can generate the same kind of conversion factors:

\[1 \text{ m} \times 1 \text{ m} \times 1 \text{ m} = 1 \text{ m}^3\] \[3.28 \text{ ft} \times 3.28 \text{ ft} \times 3.28 \text{ ft}= 35.29\text{ ft}^3\]

This tells us that \(1 \text{ m}^3 = 35.29 \text{ ft}^3\), which gives us two conversion factors that we can use to convert \(\text{ m}^3\) to \(\text{ ft}^3\), or vice versa.

\[\frac{35.29 \text{ ft}^3}{1 \text{ m}^3} \text{ or } \frac{1 \text{ m}^3} {35.29 \text{ ft}^3}\]

Measuring Circles and Spheres

A circle is defined as the set of all points in a plane that are a given distance (the radius) from a given point (the center). Circles are only two dimensional. The properties of a circle that are generally of interest are the radius, diameter, area, and the circumference.

A sphere is defined as the set of all points in space that are a given distance (the radius) from a given point (the center). Spheres are three dimensional. The properties of a sphere that are generally of interest are the radius, the circumference, the surface area, and the volume.

Circumference

The perimeter of a circle is called the circumference. In a circle, the circumference is a bit more than three times the diameter. More exactly, the ratio of the circumference to the diameter is \(3.14\), called pi. We can write that ratio like this:

\[\pi = \frac{C}{d} = 3.14\]

We can rearrange that to give two different formulas, one to calculate \(C\) and one to calculate \(d\).

\(C = \pi d\) and \(d=\frac{C}{\pi}\)

\(C = 2 \pi r\) also works, because \(d = 2r\).

Area

The area of a circle can be calculated using \(A = \pi r^2\).

For a sphere, there is a formula for the surface area: \(4 \pi r^2\).

Volume

Circles have area, but not volume, since they are only two dimensional. Spheres, being three dimensional, do have volumes, and they can be calculated using \(V=\frac{4}{3} \pi r^3\).

Measuring Irregular Shapes

To make things interesting, we can combine shapes to make new shapes. The following are some examples.

Example 1:

Here you see a rectangle with a half-circle drawn on each end, looking like an ice skating rink. Can you find the area of the overall shape?

First, we can calculate the area of the rectangle part: \(A = 8\text{ m} \times 10\text{ m} = 80\text{ m}^2\).

Second, we can picture combining the two half-circles into a whole circle and find its area:

\[A= \pi r^2 = \pi \times (4\text{ m})^2 = 16 \pi \text{ m}^2\]

Now, we add the two areas together:

\[80\text{ m}^2 + 16 \pi \text{ m}^2\] \[80\text{ m}^2 + 50.27 \text{ m}^2 = 130.27 \text{ m}^2\]

Example 2:

Assuming all angles are right angles, what is the area of the figure?

Mentally separate the rectangle on the right (\(3\times 8\)) and the rectangle on the left (\(7 \times 12\)) and calculate the area of each one. Then add them:

\[A= 3\times 8 +7\times 12\] \[A=24+84= 108\]

Measurement Problems that Omit Some Data

If all the angles in the figure below are right angles, can you calculate its area?

There are different ways to split up the figure into smaller rectangles. Here is a plan that will work. First, imagine that the figure is a single \(9 \times 11\) rectangle. That’s \(99\) square units. Now subtract the areas of the two cutout sections. We aren’t given their height, but it must be the total height, \(9\), minus the height of the bottom section, \(4\). That’s a height of \(5\).

We also need the width of the right-hand cutout. It must be \(11 -3-4-2 = 2\), though it doesn’t look like it (images are not always drawn to scale). The two cutout sections have areas of \(4 \times 5 = 20\) and \(2 \times 5=10\), which is \(20+10=30\).

That makes the area of the figure \(99-30=69\) square units.