# Page 2 Arithmetic Study Guide for the ACCUPLACER® test

#### Estimation

Estimation is a useful method for approximating the value of a calculation without working through all of the precise details. Proper rounding is essential to performing effective estimations. Knowledge of place value is necessary to properly round.

To round to a particular value, examine the value of the digit to the “particular value’s” right.

If the digit to the right is smaller than 5, the particular value remains the same and every digit to its right becomes 0.

If the digit to the right is 5 or greater, the particular value is rounded up one digit and every digit to its right becomes 0.

For example:

$398,040$ rounded to the nearest $100,000$ is $400,000$ because the digit to the right of the hundred thousand place value is $9$, which is greater than $5$, so the number rounds up.

$398,040$ rounded to the nearest thousand; however, is $398,000$ because $0$ is less than $5$.

Consider the expression:

To estimate the result, first examine the terms involved. Every term in this expression is greater than $100$, but less than $5,000$, so it is wise to round every term to the nearest hundred and then perform the calculation:

Consider the expression:

We can round to the nearest whole number without changing the outcome too much:

which can then be changed to the easy calculation:

One last thing to consider when rounding or estimating the value of an expression. When one term is rounded up, it is wise to round another term down (if possible) as a means of balancing the rounding. If every term in an expression is rounded up, then the final result will be an overestimation. Likewise, an underestimation results from rounding every term down.

### Operations with Decimals and Percents

Like the section listed above, this portion of the test focuses on mathematical operations. Unlike that portion, however, this section focuses on decimals and percentages. Decimals have special rules regarding addition, subtraction, multiplication, and division, and these rules should be reviewed thoroughly to prepare for the test. Meanwhile, problems involving percentages range from as simple as identifying percentages using a series of numbers to as complex as identifying equivalences among percentages and fractions. Estimation also plays a role in this portion of the test. Also, review and understand the basic rules regarding decimals and have a firm grasp on the formula required to correctly construct percentages.

Being comfortable using decimals and percents as part of mathematical operations is a key skill for solving problems on the test. Be sure you understand the rules and procedures for doing these things:

#### Addition and Subtraction of Decimals

To add or subtract decimals, vertically align the numbers in columns, according to the place value of each. Use the decimal point as a reference. Add or subtract as usual, carrying or borrowing where necessary. Examine the final result to ensure that the answer makes sense. For example:

which can be rewritten as:

Notice the addition of the 0 in the second value. Adding this 0 does not change the value of the number and is used only as a place marker.

#### Multiplication of Decimals

Multiplication of decimals is very similar to multiplying integers, with the key difference being the consideration of the decimal point. To multiply decimals, begin by counting the number of digits that are to the right of the decimal point in both numbers in the question. The answer will have the same number of digits to the right of the decimal point. For example:

There are a total of $3 + 4$ digits to the right of the decimal point in the two terms. Consequently, the final answer will have $7$ total digits to the right of the decimal point.

$32364 \cdot 102 = 3301128$, so $32.364 \cdot 0.0102 = 0.3301128$

#### Division Using Decimals

Division of decimals also involves the alignment of a decimal point. If the divisor (number of pieces the dividend is being divided into) contains a decimal point, move the decimal point to the right as many times as is necessary to form a whole number. Be sure to also move the decimal point in the dividend to the right the same number of times. For example:

This problem can be rewritten, without changing the value of the answer as:

Because we moved the decimal point three places to the right in the divisor, we need to also move the decimal point three places to the right in the dividend.

After the decimal point has been moved in problems involving division by a decimal (and when using long division symbol notation), place a decimal point above the long division symbol directly above the decimal point’s location in the dividend (the number inside of the long division symbol).

Once the divisor has been written as a whole number, the decimal point has been moved in the dividend, and the decimal point has been placed above the long division symbol, the division problem can be solved as usual.