Next Generation Arithmetic Study Guide for the ACCUPLACER Test

Comparisons and Equivalents

Some of the questions on this test ask you to determine if two numbers, presented in different formats (fractions, whole numbers, etc.) are equal. To answer other questions, you will need to put a series of numbers in order, according to their relative value or tell the first or last number in the given series, again, according to value. You may even be asked where a given number belongs in a sample series of numbers, according to its value. To accomplish these tasks, there are several strategies that will help you and certain symbols you must understand.

Changing Numbers to a Common Format

The only way to compare the value of numbers is to have them in the same format. If they are given in different formats you will need to convert some of them so that the format is all the same.

Suppose you are asked to compare the numbers \(0.37\), \(\frac{2}{5}\), and \(\frac{1}{3}\). If you are familiar with the decimal equivalents given above, it is easier to convert the fractions to decimals. Now we have \(0.37, 0.4,\) and \(0.33\) and we can rearrange them from greatest to least: \(0.4, \;0.37, \;0.33\).

Using a Number Line

A number line is a great tool for comparing numbers when some or all of them are negative. Numbers always increase in value from left to right on a number line and decrease in value from right to left. You don’t always need to place the number exactly on the number line if it is in between the markings.

For example, to compare -6.5 and -3.2 using the number line below, It is enough to know that -6.5 is about halfway between -6 and -7 and -3.2 is between -3 and -4 but closer to -3. Even with these approximate placements, it is clear that -6.5 is well to the left of -3.2 and is less than -3.2 in value.

Symbols to Know

There are a few symbols that are used when comparing numbers:

\(=\) is equal to
\(\lt\) is less than
\(\gt\) is greater than
\(\le\) is less than or equal to
\(\gt\) is greater than or equal to

A slash through the sign means “is not.” The most common example, \(\neq\), means “not equal to.”

An expression like \(-5 \lt -0.5 \lt 5\) means “\(-5\) is less than \(-0.5\) which, in turn, is less than \(5\).”

Some math skills are used in a variety of problem types, so you need to be familiar with the process of each. Here are some common ones you’ll need for this test.

Estimation and Rounding

Mathematics is generally a pretty exact area, but there are times when you’ll need to just get an idea of what the answer should be. This is true whether you are working with whole numbers, decimals, or fractions. You may want to shorten your calculating time or evaluate your answer choice to see if it makes sense. Some questions even ask you to do these things before choosing an answer.

Estimation

Estimation is the use of rounding or using a close but easy number to work with to find the approximate value of the answer. For example, when multiplying \(43.5 \times 2.13\), it is easy to estimate with \(43 \times 2\), or \(86\). Since the actual numbers are a bit larger, you should expect an answer to be a little more than \(86\). The exact answer is \(92.655\).

Estimation is a good way to quickly check the results of a calculation. If the question is multiple-choice, an estimate may be all that is needed to identify the correct response.

Rounding

Rounding to a given place means to eliminate any digits to the right of that place, replacing them with zero, and then adjusting the digit in that place depending on the value of the digit to the immediate right.

If the digit to the right is 5 or greater, round up to the next digit.
If it is 4 or less, do not change the value.

For example, to round 92.655 to the nearest tenth, look at the digit in the hundredths place. Since that is 5 we will round the 6 up to a 7 and the result is 92.7. Rounding 92.655 to the nearest ten, we look at the ones place and since that is 2 we do not change the 9, and the result is 90. Rounding to the nearest integer means rounding to the ones place.

Order of Operations

When a problem requires a series of arithmetic calculations, they must be done in the correct order, remembered by the acronym PEMDAS, which can be remembered with the phrase “Please Excuse My Dear Aunt Sally.”

Parentheses: first do any operations contained in parentheses
Exponents: next, evaluate any exponents
Multiplication and Division: Do all multiplications and divisions in order from left to right
Addition and Subtraction: Do all additions and subtractions in order from left to right

For example, for the sequence \(3 - 6 \div 2 + 7 \times 3\) the correct order is:

\[3 - 6 \div 2 + 7 \times 3\] \[3 - 3 + 21\] \[0 + 21\] \[21\]

This is incorrect:

\[3 - 6 \div 2 + 7 \times 3\] \[-6 \div2 + 7 \times 3\] \[-3 + 7 \times 3\] \[4 \times 3\] \[12\]

If the problem was written with parentheses \((3-6) \div 2 + (7 \times 3)\) then this is correct:

\[(3-6) \div 2 + (7 \times 3)\] \[-3 \div 2 + 21\] \[-1.5 + 21\] \[19.5\]

Solving “Word Problems”

When the question contains only numbers and a brief question, you just have to deal with the numbers and operations shown. But, if the question is presented in “story form,” you’ll need to turn the words and numbers into number sentences before you can solve anything. Here are some strategies to use when doing this:

Read the Entire Question

Read the problem through completely to the end, making notes or highlighting what looks like important information. Some language may indicate operations to be performed, for example:

More than: addition
Less than: subtraction
Diminished by: subtraction
Times: multiplication
Rate: division
Per: division

Establish the Goal

After reading the question, determine what you are being asked for. What does a correct answer contain? An amount? If so, an amount of what? Will the answer be a rate such as a speed in miles per hour? Be sure that you understand what is needed to actually answer the question.

Decide on a Method

Now that you know what you are looking for, what do you need to do to get there? Will you need to do several steps, calculating some of the information that will go into the final result?

For example, if you are being asked for a speed in miles per hour, you will need an amount of miles and the length of time in hours. You may have to convert minutes into hours or do some other calculations to come up with the total miles.

Gather Information

Go through the problem and look for the information that will help you. You might be able to compare this list to the items you decided that you need from the previous step.

Keep in mind that some information may be included that you don’t really need. Look at every piece of information in the problem and decide whether you will use it. If you leave anything out, it should be because you decided you didn’t need it, not because you overlooked it.

Combine and Structure Information

Now you are ready to perform calculations with the information. Which operations will you need, and what will you need to do first? Some calculations may require that you complete others first. For example, you may need to subtract a beginning mileage from an ending mileage to get total miles before you can compute a speed.

Do the Problem and Check Your Answer

Finish the calculations. Use estimation to see if the answers seem correct. You might be able to take the answer and put it back in an initial equation to see that it comes out right. Be sure that the answer makes sense for the question and matches what you decided the answer should contain.

For example, if the question asked for an amount and you got a negative number, will that make sense? Negative gallons of water will not make sense, but negative dollars might mean that money is owed instead of received. If you got a fraction, does that make sense? For a number of people, you may need to round to the nearest integer. If you are asked for ages of a mother and daughter, is the mother older than the daughter by more than just a few years?