Arithmetic Reasoning Study Guide for the ASVAB

Specific Skills to Practice

Knowing how to do a math procedure can be the difference between finding the correct answer on the ASVAB Arithmetic Reasoning Test and staring blankly at the four choices. Be sure you are comfortable with the following procedures, and search for additional paper or online practice if you need it.

Order of Operations

Order of operations tells us the order in which operations are performed in a mathematical expression. An easy way to remember the rule is the mnemonic PEMDAS, which some may recognize as Please Excuse My Dear Aunt Sally. P stands for parentheses. E stands for exponents. M stands for multiplication. D stands for division. A stands for addition. S stands for subtraction.

The rule is applied this way:

When evaluating an expression, first evaluate anything within parentheses.
Next, evaluate anything with an exponent.
After that, working left to right, evaluate anything involving multiplication and division, in order of appearance.
And finally, working left to right again in order of appearance, evaluate anything involving addition or subtraction.

Example:

\(2(x -2x + 3)+ 3^2+ 23 - \frac{14}{2}\cdot 2 - 5 =\)
\(2(-x+3) +9 +23 - 7\cdot 2 - 5 =\)
\(-2x +6 + 9 + 23 -14 - 5 =\)
\(-2x + 38 - 14 - 5 =\)
\(-2x +19\)

Ratios and Proportions

A ratio shows the numerical relationship between two categories of things. If I have two apples and three oranges in a basket, I know that the ratio of apples to oranges is 2 to 3.

We write ratios in four different ways:

Using a colon: 2:3
As fractions: \(\frac{2}{3}\)
As percent: two out of five are apples, so 40% are apples Or as decimals: .4 of the fruit are apples

The term proportion describes whether the ratio of two kinds of things is constant when the total number of things increases. When we talk about proportion we use a colon.

With our apples and oranges example, our ratio is 2:3 (apples to oranges). If we want to have a bigger basket of fruit containing only apples and oranges and we want the fruits to have the same proportion, we have to multiply the starting numbers in our ratio by the same number: 2:3 is proportional to 6:9 because 2 and 3 have both been multiplied by 3.

Definition Relating to Proportions

• Inversely Proportional—Two quantities are inversely proportional if, for example, one is doubled and that causes the other to be cut in half. In short, whatever one is multiplied by, the other is divided by, and vice versa. A good example is car speed vs. time traveled. If it takes you 30 minutes to drive to work at a speed of 25 mph, what if you drove 75 mph? It would take you only 10 minutes. Tripling your speed cuts your time down to one third. Time and speed are inversely proportional.

Rates

The term rate is used in mathematics to describe speed as it relates to time and distance in distance/rate/time problems, where we want to know either how far someone has traveled, what their rate of speed was, and how long it took them to cover the distance. If we know two of the three variables (distance/rate/time) we can find the other with algebra. The distance/rate/time formula is \(d = rt\).

The term rate is also used to describe the interest that accumulates on a principal amount of money over a period of time. For simple interest calculations, the formula is: \(FV = P(1+nr)\), where FV is future value, P is principal, r is interest rate, and n is number of years or periods.

Combined Work/Time

Combined work/time problems generally challenge you to find out how long it will take two people to do a certain task or activity given that person a can do a task in a certain amount of time, and person b can do the same task in a certain amount of time. You are then asked if the two people are working together, how long would it take for them to complete one task.

These kinds of problems are really distance/rate/time problems in disguise, except distance is not distance—it is something else. The trick in solving these problems is to figure out a uniform rate for each person doing their task, add them together to get one rate, then use them in the distance/rate/time formula.

Work through this example problem, then try to make one up yourself:

John and Tom mow their neighbors’ lawns every other Saturday morning to make extra money. John and Tom live on the same street, and all of the lawns are about the same size.
It takes John about two hours to mow one lawn.
Tom, who is a few years younger than John, takes about four hours to mow one lawn.

If the two boys work together, how long does it take them to mow one lawn?

Explanation: In this problem, it is lawns mowed, not distance we are interested in, and in this case we want to know how long it takes for the two boys working together to mow one lawn. Set the distance/rate/time formula up:

\[d = rt\]

Substitute 1 for d to represent one lawn:

\[1 = rt\]

Next, we need to figure the rate for the two boys working together. John takes two hours to mow one lawn so his per-hour rate is \(1\over 2\) (half of a lawn in one hour). Tom takes four hours to mow one lawn so his per-hour rate is \(1\over 4\) (one-fourth of a lawn in one hour).

Finally, add the two rates together to get an hourly rate of lawn mowing for the two boys working together:

\[\frac{1}{ 2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}\]

Substitute \(3\over 4\) into the equation:

\[1 = \frac{3}{4}t\]

Solve for t:

\[\frac{4}{3} = t\]

The time it takes for the two boys working together to mow one lawn is \(\frac{4}{3}\) or \(1\frac{1}{3}\) hours.

Percent

Percent means parts out of one hundred. If you have 100 of something, 30 of those are one kind of thing, and 70 are another kind of thing, we say we have 30% of the first thing. Using fruits as an example again, if we have 100 pieces of fruit, and 30 of those are apples and 70 are oranges, we say 30% of our fruit are apples.

Finding the Percent from a Fraction

If our total number of things is not equal to 100, we want to convert our ratio so that the denominator is 100, the same way an equivalent fraction is made. If we have only 50 pieces of fruit, and 20 of those are apples and 30 are oranges, we want to make an equivalent fraction for 20/50.

First, divide 50 into 100 to get 2, then multiply 2 by the numerator 20 to get 40. Our parts out of 100 fraction is \(\frac{40}{100}\).

We can now express the number of apples as a percent (parts out of 100):
40% of the fruit in this example are apples.

The Proportion Method

Many questions in the ASVAB Mathematics Knowledge section can be solved using the proportion method of solving percentages, which may be summed up as follows:

\[\frac{\text{percent}}{100} = \frac{\text{part}}{\text{whole}}\]

Take this question, for instance:

What percentage of 20 is 5?

In this case, the percentage is unknown and may be represented by x.
The number 5 is the part and 20 is the whole.
Plugged into the relationship above, the equation would look like this:

\[\frac{x}{100} = \frac{5}{20}\]

It will then be an easy matter of solving for x, the percentage.

Cross-multiplying would yield:

\[20x = 500\]

$x = 25$$

Calculating Percent Change

The proportion method described above comes in handy when solving more difficult percentage questions. Try using it in the following question.

A laptop now sells at $499 but its original price was $699. By what percent was the laptop discounted?

Solve first for the decrease in price:

\[699 – 499 = 200\]

This is the part of the whole price that was discounted from the original price. The percent discount x, then, is:

\[\frac{x}{100} = \frac{200}{699}\]

Solving for x, the percent representing the discount is 28.61%.

Converting Fractions to Percentage or Decimal

Fractions, percentages, and decimals are actually different ways of presenting the same concept of parts of a whole. For instance:

The fraction \(\frac{1}{100}\) is the same as the percentage 1% and the decimal [0.01].

The fraction \(\frac{1}{2}\) is the same as 50% and [0.5].

ASVAB questions often involve converting fractions to percentages or decimals, or the other way around. The methods are quite straightforward, actually, and are as follows:

To convert a fraction to its percentage form: divide the numerator by the denominator, multiply by 100, and then add the % sign.

Thus, converting \(\frac{1}{2}\) to a percentage involves the following steps:

Divide the numerator by denominator:

\[\frac{1}{2} = 0.5\]

(Note: This is the decimal form of \(\frac{1}{2}\).)

Multiply by 100: 0.5 x 100 = 50

Add the percentage sign: 50%

To revert 50% to fraction form:

Write 50% as \(\frac{50}{100}\).

Reduce \(\frac{50}{100}\) to its lowest form, which is \(\frac{1}{2}\).

Sequences

A sequence is a series of numbers. Each number follows the one that preceded it based on a certain pattern or rule. The set of even numbers is a sequence that starts with 2, with each new number increasing by 2: {2,4,6,8…}

Another example is multiplying the previous number by 2, then adding one. The general form of this rule is:

\(2n+1\) for a sequence that starts with 1, produces: {1,3,7,15…}

Definitions Relating to Sequences

• Arithmetic growth—Arithmetic growth gives a sequence of numbers such that each number differs from the previous neighbor by the same amount. The sequence is called an arithmetic sequence. An example would be 2, 5, 8, 11, 14, 17, 20. In other words, to form an arithmetic sequence, you start with any number and pick some number and add that to your starting number and then add it again, and again, and again as far as you please.
  Note: In this context, arithmetic is pronounced air-ith-MET-ic.

• Common difference—In an arithmetic sequence, as noted above, each number differs from the previous one by a fixed amount. That fixed amount is called the common difference. In the example above, the common difference is 3.

• Recursion—Recursion produces a sequence in which each number depends on a previous number according to some rule. For example, suppose we start with the numbers 0 and 1 and make a rule that to get the next number in the sequence, we add the two numbers before it.

We could build the sequence like this.
Start with 0, 1, __. To get the third number, we add the 1 to the 0.
Now we have 0, 1, 1, __. To get the fourth number, we now add 1 and 1.
Now we have 0, 1, 1, 2, __. To get the fifth number, we add 1 and 2.
Now we have 0, 1, 1, 2, 3, __. Now we add 2 and 3.

Keep this up and you will have 0, 1, 1, 2, 3, 5, 8, 13, 21…