Some students approach this section of the ACT with a lot of anxiety. Not to worry. Your confidence will rise as you go through this self-paced ACT preparation and build your skills and knowledge. Learning math is like learning to ride a bicycle. Practice leads to mastery.

This part of the ACT tests your problem-solving abilities in topics that are generally covered in high school math classes. These include algebra, geometry, probability, trigonometry and a few additional subjects. Below you will find a brief explanation about how to answer questions on each topic. Then you will answer practice questions, beginning with ones that are relatively easy. Having sharpened your skills by answering these questions, you'll be ready to move on to more challenging ACT math questions.

BASIC ALGEBRA. Let's jump right in and tackle algebra! When we see an equation such as

    7x + 14 = 84

the way to find the value of x is to isolate x on the left side of the equals sign:

Subtract 14 from both sides of the equation, which yields

    7x + 14 -14 = 84 – 14

    7x = 70

Divide both sides of the equation by 7:

    7x/7 = 70/7

     x = 10

We can check this answer by plugging the value of x back into the equation:

(7 · 10) + 14 = 84

84 = 84 confirming that indeed x = 10 is the correct answer.

Now let's complicate our example a bit. Solve for x:

    7x -20 = -3(x – 5) + 5

7x -20 = -3x + 15 + 5 Note that -3(x – 5) = -3x + 15 since –3 times –5 = 15

    10x = 40

     x = 10

FRACTIONS, ETC. When you add or subtract fractions, they must have the same denominator:

But the fractions you want to add may have different denominators, which makes everything more complicated; you've got to find a "common denominator." Suppose for example that you are adding:

A quick way to find a common denominator is to multiple all the denominators by one another: 8 x 3 x 4 = 96. But you can do better than that, by looking for the least common denominator. Try taking the largest denominator (8) and looking for a multiple of it that will be evenly divisible by the other denominators: 16 – that's divisible by 4 but not by 3. 24 – that's divisible by both 3 and 4! Then convert each of the fractions into fractions with denominator = 24:

= = =

You can convert into a mixed number if you wish: 2 .

You can also reverse direction and convert a mixed number into a fraction, by multiplying the denominator of the fraction by the whole number and then adding that to the numerator:

Fractions are often found in equations, as in:

To isolate x on the left side of the equation, we need to get rid of the fraction: 3/5. We can do that by multiplying both sides of the equation by 5/3:

= because a · b/c = a/c · b

x = 9·5 = 45

ORDER OF OPERATIONS is also important in mathematical expressions involving addition, multiplication, exponents, etc. This is the correct sequence of operations:

Parentheses

Exponents

Multiply or divide

Add or subtract

Maybe you can remember this order by remembering the word "PEMA." Or just do a lot of exercises to get familiar with the order. You begin by working inside a parenthesis; then you drop the parenthesis. Treat brackets [ ] in the same way as parentheses (). If you're solving a problem that doesn't have one of these operations, then just skip that step and go on to the next one. Here are a few examples:

5 – (10 –         5 – 10 – 8        

5 – (2)            -5 – 18                

5 -2            -13            45 + 4

3                        49

SUBSTITUTION OF A NUMBER FOR A VARIABLE

A common ACT math question will ask that you replace one or more variables in a mathematical expression with a number, and then calculate the result.

For example, if x is 3 and y is 4 evaluate the following expression:

EXPONENTS AND ALGEBRAIC EXPRESSIONS

Exponents indicate that a number is being multiplied by itself a certain given

number of times.  Examples:

5³ = 5 · 5 · 5 = 125                                      2⁵ = 2 · 2 · 2 · 2 · 2 = 32

x⁴ = x · 5 · x = 125 = · · · =

A number raised to the first power (x1) is equal to itself (x¹ = x).

And a number raised to the zero power is equal to 1 (x⁰ = 1).

If the exponent of a number is negative, then that "flips" the number as follows:

These are general rules governing exponents:

a^m x aⁿ = a^(m+n) For example a³ x a⁵ = a⁸

a^m/aⁿ = a^(m-n)             a⁷/a⁴ = a³

a^-n = 1/aⁿ = (1/a)ⁿ             a^-3 = 1/a³ = (1/a)³ 

a⁰ = 1

(ab)^m = a^mb^m                (ab)⁵ = a⁵b⁵

(a^m)ⁿ = a^mn                (a²)⁵ = a¹⁰

With a knowledge of exponents, you can simplify some algebraic expressions:

3x^-3y²z⁵ – 9xyz = 3xyz (xy – 3)            6a² -15ab – (10ab -25b²)

3x²y²z – 9xyz = 3xyz (xy – 3)            3a(2a – 5b) – 5b(2a – 5b)

3xyz (xy -3)                        (3a – 5b) (2a – 5b)

Exercises: Exponents and Algebraic Expressions

ROOTS

The inverse of squaring a number is getting the square root of a number. 8 squared is 64. The square root of 64 is 8. That is:

8² = 64 Sometimes you will encounter a cube or cube root, which works in the same way:

5³ = 125

Even if the number inside a square root sign is not a perfect square, it can sometimes be reduced:

Exercises: Roots

Decimals, Percentages, Powers of 10, and Rounding

Multiplying a number 10 by itself generates powers of 10. Ten to the 2nd power, which is written as 102, is equal to one hundred. Ten to the 3rd power, written as 103, is equal to 1000, and so on.

Notice that the number of zeros after the 1 is the same as the power of ten, as in:

10¹ = 10     10² = 100 10³ = 1000 10⁴ = 10,000 10⁵ = 100,000

Decimals are a way of writing fractions:

1/10 = .1 2/10 = .2 etc.

1/100 = .01 2/100 = .02

1/1000 = .001 2/1000 = .002 etc.

The period is called the "decimal point, and the digits that follow it are called "places." The first decimal place is the first digit after the decimal point; the 2^nd decimal place is the 2^nd digit after the decimal point, and so on.

The number 10 can also be raised to a negative power:

10^-1 = 1/10 = .01 10^-2 = 1/100 = .001 etc.