Arithmetic Reasoning Study Guide for the ASVAB

Geometry Basics

There will probably not be much geometry on the ASVAB Arithmetic Reasoning Test, but it couldn’t hurt to know these basic concepts and how to calculate them.

Perimeter

The distance around an object or figure is the perimeter of that object. It’s not hard to calculate; you just add all the side lengths together, or, in the case of a circle, use the formula \(C=\pi \cdot d\), where C is the circumference, \(\pi\) is 3.14, and d is the diameter.

Area

Area is the amount of space enclosed in a two-dimensional (flat) object or figure. For example, a rug covers a certain area on the floor. To paint a wall, it would be a good idea to know the area of the wall, so you can buy enough paint.

Every shape has its own formula to calculate area. These four are very commonly used:

Square: \(A=s^2\), where \(s\) is the length of a side.

Rectangle: \(A = l \cdot w\), where \(l\) is the length and \(w\) is the width.

Triangle: \(A = \frac{1}{2} b \cdot h\), where \(b\) is the base and \(h\) is the height.

Circle: \(A = \pi \cdot r^2\), where \(\pi\) is approximately \(3.14\) and \(r\) is the radius.

Volume

Volume is the amount of space in a three-dimensional object or figure. Examples are the volume of coffee in a cup or the volume of concrete in a basement floor. Just as with area, each shape has its own formula to calculate volume.

Rectangular prism (box shaped): \(V= l \cdot w \cdot h\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height.

Sphere (ball shaped): \(V=\frac{4}{3} \pi \cdot r^3\), where, again, \(\pi\) is approximately \(3.14\), and \(r\) is the radius.

Working with Data

Average, Mean, Median, and Mode

Average, mean, median and mode are measures of central tendency that describe features of data in a data set.

Suppose we have a class of 10 students who took a math exam and received the following scores:

\[59\quad 67\quad 82\quad 82\quad 82\quad 85\quad 89\quad 91\quad 92\quad 94\]

Average is the total of the scores spread among all 10 students. Average is the sum of all the numbers in a data set, divided by the number of items in our data set.

In our test example: the average score is:

\[(59 + 67 + 82 + 82 +82 +85 + 89 + 91 + 92 + 94)\div10 = 82.3\]

Average and mean have the same meaning, so using the word mean is just a different way to say average.

The median of a data set is that data item that lies exactly in the middle of the data set. To find the median, first arrange the data items in ascending order. If you have an odd number of data items, subtract one from the number of items you have, divide by 2, then count that many items into the data set from both the left and right. The one data item left in the center is the median.

If you have an even number of data items in your data set, divide the number of items by 2, subtract 1, and then count that many items into the data set by both the left and right. You now have two items left exactly in the center. Add these two numbers and divide by 2 to find the median.

With our 10 test scores above, divide 10 by 2 to get 5. Subtract 1 from 5 to get 4. Count four data items in from both the left and right, leaving 82 and 85 in the middle. Add these two numbers to get 167. Divide this by 2 to get a median of 83.5.

Mode means the data item in the data set that occurs most frequently. Again, the data set must be arranged in ascending order so we can identify the mode. By visual inspection with the 10 test scores, we see that 82 occurs most often (three times), so 82 is the mode.

Outlier—If you have a data set in which most of the values are somewhat close to each other, but there is one value that is far away from the rest, that value is an outlier. Depending on how far out the outlier is, it may be ignored when doing calculations with the data set.
Spread/Dispersion—Spread and dispersion are synonyms referring to how spread out the data points are. There are several ways to come up with a value for spread: range, variance, and standard deviation. When these values are low, it means that the data points are clustered very close to the mean. That’s a good thing.
Range—Range is a measure of how spread out the data elements are. It is calculated by subtracting the lowest data point from the highest.
Variance—Variance is a different way to measure the spread of the data. It is based on the average of the differences between each data point and the mean, and gives you an idea how far the data points stray from the mean. The higher the variance, the more the data points are spread out. The variance calculation is fairly involved.
Standard deviation—Standard deviation is another way to measure data spread. Again, the higher the standard deviation, the more spread out the data points are. Calculating standard deviation is pretty simple. You just take the square root of the variance.
Quartiles—Quartiles are a way of dividing a data set into four equal groups of elements. The first quartile is the value that has 25% of all the elements equal to or below it. The second quartile is the value where 50% of elements are equal to or below it, and the third quartile is the value where 75% of elements are equal to or below it. The second quartile is the median.
Interquartile range—Interquartile range is the range of the middle percentiles. It’s calculated by subtracting the 25th percentile from the 75th percentile.

Probability

Probability is describing the chance that something will occur based on the set of all possible outcomes in an event, called the sample space. If we conduct an experiment in which a coin is tossed two times, the set of all possible outcomes (the sample space) is: HH, HT, TH, and TT. The probability the coin will land tails up two times is one out of four:

\(\frac{1}{4}\) or 25%

The probability of an event happening can be independent as with the coin flip experiment above. Or probability can be dependent. If you have two yellow balls, one red ball, and one blue ball, then take one ball out of the bag, note the color, and do not put the ball back in the bag, the probability that you take out a certain colored ball on the second selection depends on the color of the ball you took out on the first draw. If you took a yellow ball out on the first draw, the probability of taking a yellow out on the second draw is:

\(\frac{1}{3}\;\text{or}\;33 \%\).

Definitions Relating to Probability

Single event—An event refers to one or more outcomes. For example, suppose you want to roll a die and get an even number. There are three ways you can roll an even number: 2, 4, or 6. The 2, 4, and 6 are three different outcomes that will produce the desired event, i.e. rolling an even number.
Favorable outcome—A favorable outcome is one that satisfies a condition that has been specified. For example, in the passage above, the specified condition is to roll an even number, so rolling a 2, 4, or 6 would be a favorable outcome. Rolling a 1, 3, or 5 would be an unfavorable outcome.
Complement—The complement of an event is made of all outcomes that are not favorable. The complement of rolling an even number consists of the outcomes 1, 3, and 5.
Multiple events—Sometimes we want to find the probability of two or more different events. For example: when drawing two cards from a normal deck of cards, what is the probability of drawing an ace and a king? Or, if you’re drawing one card, what is the probability of drawing an ace or a king?
Conditional Probability—This is a probability determination where a second event happens that depends on a first event. For example, suppose you have two red socks and two blue socks. What is the probability of drawing a blue sock if you have already drawn a blue sock? Drawing the blue sock first changes the number you choose from for the second sock.
Mutually exclusive events—Mutually exclusive events are those that cannot possibly happen at the same time. For example, when drawing a card, it can be a king or a queen, but it can’t be a king and a queen. The events drawing a king and drawing a queen are mutually exclusive. They can’t occur at the same time.
Mutually inclusive events—Mutually inclusive events are those that can happen at the same time. For example, sticking with cards, it’s possible to draw a jack and a diamond at the same time. The events drawing a jack and drawing a diamond are mutually inclusive. They can occur at the same time.