Pre ACT Math

The Preparing for Higher Math category includes single-skill questions on the following topics:

• Number and Quantity

• Algebra

• Functions

• Geometry

• Statistics and Probability

INTEGRATING ESSENTIAL SKILLS (9–16%)

The Integrating Essential Skills category is made up of compound questions that require the

integration of two or more skills to find the correct answer. These questions are likely to build on

any of the skills listed in the Preparing for Higher Math section.

Topics likely to appear in the Integrating Essential Skills questions include the following:

• Rates and percentages

• Proportional relationships

• Area, surface area, and volume

• Average and median

• Expressing numbers in different ways

MODELING (25%)

Modeling questions require the test taker to represent real situations using a mathematical model.

Modeling questions also count toward the other reporting categories listed above.

CALCULATOR POLICY

All of the questions on the PreACT 8/9 can be answered without a calculator, but a calculator is

allowed. That means you’ll have to use your best judgement on when and when not to use one to

keep your pace for the test. The questions also vary by complexity.

Number and Quantity

NUMBER BASICS

CLASSIFICATIONS OF NUMBERS

Numbers are the basic building blocks of mathematics. Specific features of numbers are identified

by the following terms:

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Integer – any positive or negative whole number, including zero. Integers do not include

fractions (1

3), decimals (0.56), or mixed numbers (7 3

4).

Prime number – any whole number greater than 1 that has only two factors, itself and 1; that is, a

number that can be divided evenly only by 1 and itself.

Composite number – any whole number greater than 1 that has more than two different factors; in

other words, any whole number that is not a prime number. For example: The composite number 8

has the factors of 1, 2, 4, and 8.

Even number – any integer that can be divided by 2 without leaving a remainder. For example: 2,

4, 6, 8, and so on.

Odd number – any integer that cannot be divided evenly by 2. For example: 3, 5, 7, 9, and so on.

Decimal number – any number that uses a decimal point to show the part of the number that is

less than one. Example: 1.234.

Decimal point – a symbol used to separate the ones place from the tenths place in decimals or

dollars from cents in currency.

Decimal place – the position of a number to the right of the decimal point. In the decimal 0.123, the

1 is in the first place to the right of the decimal point, indicating tenths; the 2 is in the second place,

indicating hundredths; and the 3 is in the third place, indicating thousandths.

The decimal, or base 10, system is a number system that uses ten different digits (0, 1, 2, 3, 4, 5, 6,

7, 8, 9). An example of a number system that uses something other than ten digits is the binary, or

base 2, number system, used by computers, which uses only the numbers 0 and 1. It is thought that

the decimal system originated because people had only their 10 fingers for counting.

Rational numbers include all integers, decimals, and fractions. Any terminating or repeating

decimal number is a rational number.

Irrational numbers cannot be written as fractions or decimals because the number of decimal

places is infinite and there is no recurring pattern of digits within the number. For example, pi (π)

begins with 3.141592 and continues without terminating or repeating, so pi is an irrational

number.

Real numbers are the set of all rational and irrational numbers.

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NUMBERS IN WORD FORM AND PLACE VALUE

When writing numbers out in word form or translating word form to numbers, it is essential to

understand how a place value system works. In the decimal or base-10 system, each digit of a

number represents how many of the corresponding place value—a specific factor of 10—are

contained in the number being represented. To make reading numbers easier, every three digits to

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the left of the decimal place is preceded by a comma. The following table demonstrates some of the

place values:

Power of 10 103 102 101 100 10−1 10−2 10−3

Value 1,000 100 10 1 0.1 0.01 0.001

Place thousands hundreds tens ones tenths hundredths thousandths

For example, consider the number 4,546.09, which can be separated into each place value like this:

4: thousands

5: hundreds

4: tens

6: ones

0: tenths

9: hundredths

This number in word form would be four thousand five hundred forty-six and nine hundredths.

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RATIONAL NUMBERS

The term rational means that the number can be expressed as a ratio or fraction. That is, a number,

𝑟, is rational if and only if it can be represented by a fraction 𝑎

𝑏 where 𝑎 and 𝑏 are integers and 𝑏

does not equal 0. The set of rational numbers includes integers and decimals. If there is no finite

way to represent a value with a fraction of integers, then the number is irrational. Common

examples of irrational numbers include: √5, (1 + √2), and 𝜋.

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NUMBER LINES

A number line is a graph to see the distance between numbers. Basically, this graph shows the

relationship between numbers. So a number line may have a point for zero and may show negative

numbers on the left side of the line. Any positive numbers are placed on the right side of the line.

For example, consider the points labeled on the following number line:

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We can use the dashed lines on the number line to identify each point. Each dashed line between

two whole numbers is 1

4. The line halfway between two numbers is 1

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ABSOLUTE VALUE

A precursor to working with negative numbers is understanding what absolute values are. A

number’s absolute value is simply the distance away from zero a number is on the number line. The

absolute value of a number is always positive and is written |𝑥|. For example, the absolute value of

3, written as |3|, is 3 because the distance between 0 and 3 on a number line is three units.

Likewise, the absolute value of –3, written as |−3|, is 3 because the distance between 0 and –3 on a

number line is three units. So |3| = |−3|.

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OPERATIONS

An operation is simply a mathematical process that takes some value(s) as input(s) and produces

an output. Elementary operations are often written in the following form: value operation value. For

instance, in the expression 1 + 2 the values are 1 and 2 and the operation is addition. Performing

the operation gives the output of 3. In this way we can say that 1 + 2 and 3 are equal, or 1 + 2 = 3.

ADDITION

Addition increases the value of one quantity by the value of another quantity (both called

addends). Example: 2 + 4 = 6 or 8 + 9 = 17. The result is called the sum. With addition, the order

does not matter, 4 + 2 = 2 + 4.

When adding signed numbers, if the signs are the same simply add the absolute values of the

addends and apply the original sign to the sum. For example, (+4) + (+8) = +12 and (−4) +

(−8) = −12. When the original signs are different, take the absolute values of the addends and

subtract the smaller value from the larger value, then apply the original sign of the larger value to

the difference. Example: (+4) + (−8) = −4 and (−4) + (+8) = +4.

SUBTRACTION

Subtraction is the opposite operation to addition; it decreases the value of one quantity (the

minuend) by the value of another quantity (the subtrahend). For example, 6 − 4 = 2 or 17 − 8 =

9. The result is called the difference. Note that with subtraction, the order does matter, 6 − 4 ≠ 4 −

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For subtracting signed numbers, change the sign of the subtrahend and then follow the same rules

used for addition. Example: (+4) − (+8) = (+4) + (−8) = −4

MULTIPLICATION

Multiplication can be thought of as repeated addition. One number (the multiplier) indicates how

many times to add the other number (the multiplicand) to itself. Example: 3 × 2 = 2 + 2 + 2 = 6.

With multiplication, the order does not matter, 2 × 3 = 3 × 2 or 3 + 3 = 2 + 2 + 2, either way the

result (the product) is the same.

If the signs are the same, the product is positive when multiplying signed numbers. Example:

(+4) × (+8) = +32 and (−4) × (−8) = +32. If the signs are opposite, the product is negative.

Example: (+4) × (−8) = −32 and (−4) × (+8) = −32. When more than two factors are multiplied

together, the sign of the product is determined by how many negative factors are present. If there

are an odd number of negative factors then the product is negative, whereas an even number of

negative factors indicates a positive product. Example: (+4) × (−8) × (−2) = +64 and

(−4) × (−8) × (−2) = −64.

DIVISION

Division is the opposite operation to multiplication; one number (the divisor) tells us how many

parts to divide the other number (the dividend) into. The result of division is called the quotient.

Example: 20 ÷ 4 = 5. If 20 is split into 4 equal parts, each part is 5. With division, the order of the

numbers does matter, 20 ÷ 4 ≠ 4 ÷ 20.

The rules for dividing signed numbers are similar to multiplying signed numbers. If the dividend

and divisor have the same sign, the quotient is positive. If the dividend and divisor have opposite

signs, the quotient is negative. Example: (−4) ÷ (+8) = −0.5.

Review Video: Mathematical Operations

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PARENTHESES

Parentheses are used to designate which operations should be done first when there are multiple

operations. Example: 4 − (2 + 1) = 1; the parentheses tell us that we must add 2 and 1, and then

subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an

answer of 3).

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EXPONENTS

An exponent is a superscript number placed next to another number at the top right. It indicates

how many times the base number is to be multiplied by itself. Exponents provide a shorthand way

to write what would be a longer mathematical expression, Example: 24 = 2 × 2 × 2 × 2. A number

with an exponent of 2 is said to be “squared,” while a number with an exponent of 3 is said to be

“cubed.” The value of a number raised to an exponent is called its power. So 84 is read as “8 to the

4th power,” or “8 raised to the power of 4.”

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ROOTS

A root, such as a square root, is another way of writing a fractional exponent. Instead of using a

superscript, roots use the radical symbol (√ ) to indicate the operation. A radical will have a

number underneath the bar, and may sometimes have a number in the upper left: √𝑎

𝑛 , read as “the

𝑛th root of 𝑎.” The relationship between radical notation and exponent notation can be described by

this equation:

√𝑎

𝑛 = 𝑎1

𝑛

The two special cases of 𝑛 = 2 and 𝑛 = 3 are called square roots and cube roots. If there is no

number to the upper left, the radical is understood to be a square root (𝑛 = 2). Nearly all of the

roots you encounter will be square roots. A square root is the same as a number raised to the one-

half power. When we say that 𝑎 is the square root of 𝑏 (𝑎 = √𝑏), we mean that 𝑎 multiplied by itself

equals 𝑏: (𝑎 × 𝑎 = 𝑏).

A perfect square is a number that has an integer for its square root. There are 10 perfect squares

from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1 through 10).

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WORD PROBLEMS AND MATHEMATICAL SYMBOLS

When working on word problems, you must be able to translate verbal expressions or “math

words” into math symbols. This chart contains several “math words” and their appropriate

symbols:

Phrase Symbol

equal, is, was, will be, has, costs, gets to, is the same as, becomes =

times, of, multiplied by, product of, twice, doubles, halves, triples ×

divided by, per, ratio of/to, out of ÷

plus, added to, sum, combined, and, more than, totals of +

subtracted from, less than, decreased by, minus, difference between −

what, how much, original value, how many, a number, a variable 𝑥, 𝑛, etc.

EXAMPLES OF TRANSLATED MATHEMATICAL PHRASES

• The phrase four more than twice a number can be written algebraically as 2𝑥 + 4.

• The phrase half a number decreased by six can be written algebraically as 1

2 𝑥 – 6.

• The phrase the sum of a number and the product of five and that number can be written

algebraically as 𝑥 + 5𝑥.

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• You may see a test question that says, “Olivia is constructing a bookcase from seven boards.

Two of them are for vertical supports and five are for shelves. The height of the bookcase is

twice the width of the bookcase. If the seven boards total 36 feet in length, what will be the

height of Olivia’s bookcase?” You would need to make a sketch and then create the equation

to determine the width of the shelves. The height can be represented as double the width.

(If 𝑥 represents the width of the shelves in feet, then the height of the bookcase is 2𝑥. Since

the seven boards total 36 feet, 2𝑥 + 2𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 = 36 or 9𝑥 = 36; 𝑥 = 4. The

height is twice the width, or 8 feet.)

SUBTRACTION WITH REGROUPING

A great way to make use of some of the features built into the decimal system would be regrouping

when attempting longform subtraction operations. When subtracting within a place value,

sometimes the minuend is smaller than the subtrahend, regrouping enables you to ‘borrow’ a unit

from a place value to the left in order to get a positive difference. For example, consider subtracting

189 from 525 with regrouping.

First, set up the subtraction problem in vertical form:

525

– 189

Notice that the numbers in the ones and tens columns of 525 are smaller than the numbers in the

ones and tens columns of 189. This means you will need to use regrouping to perform subtraction:

5 2 5

– 1 8 9

To subtract 9 from 5 in the ones column you will need to borrow from the 2 in the tens columns:

5 1 15

– 1 8 9

Next, to subtract 8 from 1 in the tens column you will need to borrow from the 5 in the hundreds

column:

4 11 15

– 1 8 9

3 6

Last, subtract the 1 from the 4 in the hundreds column:

4 11 15

– 1 8 9

3 3 6

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ORDER OF OPERATIONS

The order of operations is a set of rules that dictates the order in which we must perform each

operation in an expression so that we will evaluate it accurately. If we have an expression that

includes multiple different operations, the order of operations tells us which operations to do first.

The most common mnemonic for the order of operations is PEMDAS, or "Please Excuse My Dear

Aunt Sally." PEMDAS stands for parentheses, exponents, multiplication, division, addition, and

subtraction. It is important to understand that multiplication and division have equal precedence,

as do addition and subtraction, so those pairs of operations are simply worked from left to right in

order.

For example, evaluating the expression 5 + 20 ÷ 4 × (2 + 3)2 − 6 using the correct order of

operations would be done like this:

• P: Perform the operations inside the parentheses: (2 + 3) = 5

• E: Simplify the exponents: (5)2 = 5 × 5 = 25

o The expression now looks like this: 5 + 20 ÷ 4 × 25 − 6

• MD: Perform multiplication and division from left to right: 20 ÷ 4 = 5; then 5 × 25 = 125

o The expression now looks like this: 5 + 125 − 6

• AS: Perform addition and subtraction from left to right: 5 + 125 = 130; then 130 − 6 = 124

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FRACTIONS, DECIMALS, AND PERCENTAGES

FRACTIONS

A fraction is a number that is expressed as one integer written above another integer, with a

dividing line between them (𝑥

𝑦). It represents the quotient of the two numbers “𝑥 divided by 𝑦.” It

can also be thought of as 𝑥 out of 𝑦 equal parts.

The top number of a fraction is called the numerator, and it represents the number of parts under

consideration. The 1 in 1

4 means that 1 part out of the whole is being considered in the calculation.

The bottom number of a fraction is called the denominator, and it represents the total number of

equal parts. The 4 in 1

4 means that the whole consists of 4 equal parts. A fraction cannot have a

denominator of zero; this is referred to as “undefined.”

Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing

(but not adding or subtracting) both the numerator and denominator by the same number. If you

divide both numbers by a common factor, you are reducing or simplifying the fraction. Two

fractions that have the same value but are expressed differently are known as equivalent

fractions. For example, 2

10, 3

15, 4

20, and 5

25 are all equivalent fractions. They can also all be reduced or

simplified to 1