Pre-Algebra

The Pre-Algebra (seventh or eighth-grade level) questions test basic algebraic concepts such as operations Using Whole Numbers, Fractions, and Decimals; Square Roots; Exponents; Scientific Notation; Ratios, Proportions, and Percent; Linear Equations with One Variable; Absolute Value; and Simple Probability.

1. Operations Using Whole Numbers, Decimals, and Fractions

The ACT Mathematics Test will require you to add, subtract, multiply, and divide whole numbers, fractions, and decimals. When performing these operations, be sure to keep track of negative signs and line up decimal points in order to eliminate careless mistakes. 

The following are some simple rules to keep in mind regarding whole numbers, fractions, and decimals:

1. Ordering is the process of arranging numbers from smallest to greatest or from greatest to smallest. The symbol > is used to represent “greater than,” and the symbol < is used to represent “less than.” To represent “greater than or equal to,” use the symbol ≥; to represent “less than or equal to,” use the symbol ≤.

2. The Commutative Property of Multiplication is expressed as a × b = b × a, or ab = ba.

3. The Distributive Property of Multiplication is expressed as a(b + c) = ab + ac.

4. The order of operations for whole numbers can be remembered by using the acronym PEMDAS: 

  • P - First, do the operations within the parentheses, if any.
  • E - Next, do the exponents.
  • MD - Next, do the multiplication and division, in order from left to right.
  • AS - Finally, do the addition and subtraction, in order from left to right.

5. When a number is expressed as the product of two or more numbers, it is in factored form. Factors are all of the numbers that will divide evenly into one number.

6. A number is called a multiple of another number if it can be expressed as the product of that number and a second number. For example, the multiples of 4 are 4, 8, 12, 16, etc., because 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, 4 × 4 = 16, etc.

7. The Greatest Common Factor (GCF) is the largest integer that will divide evenly into any two or more integers. The Least Common Multiple (LCM) is the smallest integer into which any two or more integers will divide evenly. For example, the Greatest Common Factor of 24 and 36 is 12, because 12 is the largest integer that will divide evenly into both 24 and 36. The Least Common Multiple of 24 and 36 is 72, because 72 is the smallest integer into which both 24 and 36 will divide evenly.

8. Multiplying and dividing both the numerator and the denominator of a fraction by the same nonzero number will result in an equivalent fraction.

9. When multiplying fractions, multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product.

10. To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

11. When adding and subtracting like fractions, add or subtract the numerators and write the sum or difference over the denominator.

12. When adding or subtracting unlike fractions, first find the Lowest Common Denominator. The Lowest Common Denominator is the smallest integer into which all of the denominators will divide evenly.

13. Place value refers to the value of a digit in a number relative to its position. Moving left from the decimal point, the values of the digits are 1’s, 10’s, 100’s, etc. Moving right from the decimal point, the values of the digits are 10ths, 100ths, 1000ths, etc.

14. When converting a fraction to a decimal, divide the numerator by the denominator.

2. Square Roots

A square root is written as √n, and is the non-negative value a that fulfils the expression a2 = n.

For example, the square root of 25 would be written as √25, which is equivalent to 52, or 5 × 5.

A number is considered a perfect square when the square root of that number is a whole number. So, 25 is a perfect square because the square root of 25 is 5.

3. Exponents

When a whole number is multiplied by itself, the number of times it is multiplied is referred to as the exponent.

The exponent of 52 is 2 and it signifies 5 × 5. Any number can be raised to any exponential value. For example, 76 = 7 × 7 × 7 × 7 × 7 × 7 = 117,649.

4. Scientific Notation

When numbers are very large or very small, scientific notation is used to shorten them. To form the scientific notation of a number, the decimal point is moved until it is placed after the first nonzero digit from the left in the number.

For example, 568,000,000 written in scientific notation would be 5.68 × 108, because the decimal point was moved 8 places to the left. Likewise, 0.0000000354 written in scientific notation would be 3.54 × 10−8, because the decimal point was moved 8 places to the right.

5. Ratio, Proportion, and Percent

A ratio is the relation between two quantities expressed as one divided by the other. For example, if there are 3 blue cars and 5 red cars, the ratio of blue cars to red cars is 3/5, or 3:5.

A proportion indicates that one ratio is equal to another ratio. For example, if the ratio of blue cars to red cars is 3/5, and there are 8 total cars, you could set up a proportion to calculate the percent of blue cars, as follows:

3 cars is to 8 cars as x percent is to 100 percent.

3/8 = x/100

Solve for x

8x = 300

x = 37.5%

A percent is a fraction whose denominator is 100. The fraction 55/100 is equal to 55%.

6. Linear Equations with One Variable

In a linear equation with one variable, the variable cannot have an exponent or be in the denominator of a fraction. An example of a linear equation is 2x + 13 = 43. The ACT Mathematics Test will most likely require you to solve for x in that equation. Do this by isolating x on the left side of the equation, as follows:

2x + 13 = 43

2x = 43 - 13

2x = 30

x = 15

One common ACT example of a linear equation with one variable is in questions involving speed of travel. The basic formula to remember is:

Rate × Time = Distance

The question will give you two of these values and you will have to solve for the remaining value.

7. Absolute Value

The absolute value of a number is notated by placing that number inside two vertical lines. For example, the absolute value of 10 is written as |10|. Absolute value can be defined as the numerical value of a real number without regard to its sign.

This means that the absolute value of 10, |10|, is the same as the absolute value of –10, |–10|, in that they both equal 10. Think of it as the distance from –10 to 0 on the number line and the distance from 0 to 10 on the number line: both distances equal 10 units.

8. Simple Probability

Probability is used to measure how likely an event is to occur. It is always between 0 and 1; an event that will definitely not occur has a probability of 0, whereas an event that will certainly occur has a probability of 1.

To determine probability, divide the number of outcomes that fit the conditions of an event by the total number of outcomes. For example, the chance of getting heads when flipping a coin is 1 out of 2, or 1/2. There are two possible outcomes (heads or tails) but only one outcome (heads) that fits the conditions of the event. Therefore, the probability of the coin toss resulting in heads is 0.5, or 50%.

When two events are independent, meaning the outcome of one event does not affect the other, you can calculate the probability of both occurring by multiplying the probabilities of each of the events together. For example, the probability of flipping three heads in a row would be 1/2 × 1/2 × 1/2 or 1/8. The ACT Mathematics Test will assess your ability to calculate simple probabilities in everyday situations.