# ACT Math Skill Review: Arithmetic

Many of the common mistakes made by students on the ACT math sections are due to simple errors in operations: addition, subtraction, multiplication, and division. More often than not, this is the result of rushing through an otherwise easy problem and arriving at one of the wrong answers. The test writers know what mistakes students make repeatedly and they use this information when coming up with the wrong answer choices. Keep in mind this pacing advice:

Take enough time on straightforward questions to avoid careless errors!

Your ability to perform basic mathematical operations will be extremely important to your success on the test. You will be able to use your calculator on this part of the test, but your understanding of rudimentary mathematical operations will help prevent careless mistakes. We’ll review some of the fundamentals (addition, subtraction, multiplication, and division) here. You’ll also want to make sure you understand the full capabilities of your calculator and become adept at using them before test day. Consult the manual for your calculator and practice!

#### Addition

You add two (or more) numbers together to arrive at the sum. When adding integers, always be careful to keep the columns straight and line everything up with the units column.

Here are a few things to keep in mind:

- You can add the numbers in any order and the result won’t change. For example, 2 + 3 = 5 and 3 + 2 = 5.
- You can also group them in any manner without changing the result. For example, (3 + 2) + 4 = 9 and 3 + (2 + 4) = 9.
- When adding integers:
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd

- When adding negative and positive numbers, you may find it advantageous to rewrite the problem as a subtraction problem. For example, 2 + (-3) = 2 – 3 = -1.
- When adding decimal numbers, always align the columns on the decimal point.
- When adding fractions, always find a common denominator. Whatever you do to the denominator, you also must do to the numerator.
- When adding fractions written as mixed numbers, you can rewrite them as improper fractions, or add the integer components and then add the fractions.

#### Subtraction

When you subtract one number from another, the result is called the difference. Just like with addition, when subtracting integers always line everything up with the units column.

Here are some more things you need to remember with subtraction.

- When subtracting integers, remember:
- Even – Even = Even
- Odd – Odd = Even
- Even – Odd = Odd
- Odd – Even = Odd

- When subtracting a negative number from a positive number (or vice versa), you may find it easier to write the problem as an addition problem. For example, 234 – (-137) = 234 + 137 = 371. In effect, the two negatives cancel each other out!
- When subtracting decimal numbers, always align the columns on the decimal point. Also remember that you may borrow from the units to the left if needed.
- When subtracting fractions, always find a common denominator first.
- When subtracting fractions written as mixed numbers, subtract the integer components and then the fraction components, or rewrite as improper fractions first. Remember to find a common denominator. You may need to simplify the fraction at the end.

#### Multiplication

You will probably do much of the multiplication with your calculator; however, it’s still important to have a firm understanding of the process. The answer to a multiplication problem is called the product.

Here are some things to remember with multiplication:

- When multiplying integers, remember:
- Even x Even = Even
- Odd x Odd = Odd
- Even x Odd = Even

- When multiplying integers with more than one digit, you must keep up with the place value.
- When multiplying positive and negative numbers, remember:
- Positive x Positive = Positive
- Negative x Negative = Positive
- Positive x Negative = Negative

- When multiplying decimals, proceed just as if you are multiplying integers. The number of places to the right of the decimal point in the product is simply the sum of the number of places to the right of the decimal point in the numbers being multiplied.
- Multiplying fractions does not require a common denominator. Just multiply the numerators and then multiply the denominators. You may have to simplify afterwards. It may help to recognize factors that you can cancel prior to multiplying.
- When multiplying mixed fractions, you must rewrite them as improper fractions first.

#### Division

Division is closely related to working with fractions. The number being divided (numerator) is called the dividend, and the number we are dividing by (the denominator) is called the divisor. The answer is the quotient. The divisor won’t always divide evenly into the dividend, leaving you with a quotient that is not a whole number. You can express this as a remainder (what’s left), a decimal or a fraction. For example, 6 divided by 4 can be expressed as: 1 with a remainder of 2 or 1.5 or 1 ½.

Here are some things to keep in mind when dividing:

- When dividing decimals, move the decimal point in the divisor all the way to the right and then in the dividend the same number of places.
- When dividing positive and negative numbers, remember:
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative

- Division with fractions is much simpler than addition and subtraction. You can convert a division problem into a multiplication problem by flipping the divisor. Finding a common denominator is not required.
- When dividing fractions written as mixed numbers, rewrite them as improper fractions and then follow the instructions above.

#### Examples

- What is -5-(-16)?
A. 1

B. 11

C. 21

D. -21

E. -11 - What is
^{9}⁄_{6}÷^{5}⁄_{10}?F.

^{3}⁄_{4}

G.^{4}⁄_{3 }H. 3

J. 4

K. 8 - What is (-.00418)⁄(-.16)?
A. .026125

B. .0026125

C. -.026125

D. -.0026125

E. -.0026125

#### Answers and Explanations

**The correct answer is B.**The problem can be rewritten as 16 – 5 = 11.

**The correct answer is H.**While you are dividing fractions, the answer won’t necessarily be a fraction. Flip the second fraction, multiply across and simplify. If you notice, choice F is the result if you forget to flip the second fraction before multiplying. Choice G is the result if you multiply and then flip the result. Both would be careless errors.

**The correct answer is A.**Before even working the problem, you can eliminate choices C, D, and E. Why? Because, the correct answer must be positive (a negative number divided by a negative number will be a positive number). It’s just a matter of making sure you’re decimal point is in the right spot. This is a good problem for a calculator.

- Time Management
- Number Properties
- Fractions
- Arithmetic
- Order of Operations
- Algebraic Manipulation
- Word Problems
- Solving Equations
- Inequalities
- Linear Equations
- Miscellaneous Algebra
- Functions
- Ratios, Proportions and Variations
- Lines & Angles
- Triangles
- Quadrilaterals
- Circles
- Graphs and Coordinate Geometry
- Probability and Statistics
- Trigonometry

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