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ACT Prep: Practice & Study Guide43 chapters | 347 lessons

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Lesson Transcript

Instructor:
*Elizabeth Foster*

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Worried about what formulas you'll need to memorize for the ACT? Some good news: it's probably not as bad as you think! Here's a checklist to make sure you're on top of everything.

The list of topics covered on the ACT math test sounds pretty intimidating, so you might think you'll need to cram your head full of every formula in the book. On the ACT Math test, you won't need to memorize a huge number of special formulas. In fact, you probably already know most of what you need for the test! The ACT isn't really about cramming your head full of new equations; it's more about knowing how to apply the stuff you've already learned.

Just to make sure, though, here's a quick checklist of the formulas you'll want to brush up on for the test. If anything looks unfamiliar or you'd just like to review it, check out the math lessons for some more in-depth coverage.

Let's start with some basic formulas, covering simple averages, statistics, probability and distances.

**Distance and work:**

Distance traveled or work done equals rate times time (*d* = *r* * *t*). If you have any two parts of this equation, you can just plug them in to solve for the third. For example, if you travel for four hours at 60 miles per hour, you would travel a total distance of 240 miles.

**Averages:**

To find the average, also called the arithmetic mean, take the sum of all the numbers and divide it by the number of numbers. For example, the average of 1, 2 and 9 is 4.

**Probability:**

The probability of a desired outcome equals the number of desired outcomes over the number of total possible outcomes. For example, if you flip a coin, the probability of getting tails is 1/2, since there are two possible outcomes (heads or tails) and only one of them is desired (tails). To find the probability that two independent events will both happen, multiply the individual probabilities together. For example, if you flip a coin twice, the probability that you'll get tails both times is 1/2 * 1/2 or 1/4.

Now let's take a quick look at exponents. Exponents can be tricky if you're not comfortable with them, but having a solid grasp of the formulas should help.

**General exponent rules:**

- If you multiply two exponential expressions with the same base, add the exponents.
*x*^*a***x*^*b*=*x*^(*a*+*b*) - If you divide two expressions, subtract the exponents.
*x*^*a*/*x*^*b*=*x*^(*a*-*b*) - If you raise an exponential expression to a power, multiply the exponents. (
*x*^*a*)^*b*=*x*^(*a**b*) - If you take the root of an exponential expression, divide the exponent inside the square root by the one outside.
*b*root(*x*^*a*) =*x*^(*a*/*b*)

**FOIL and Factoring:**

**FOIL** stands for **F**irst, **O**utside, **I**nside, **L**ast. This is the method for multiplying polynomials.

For example: (*x* + *y*)(*a* + *b*) = *ax* + *bx* + *ay* + *by*. You can see how first we multiplied the first terms in each set of parentheses to get *ax*. Then we multiplied the outside terms to get *bx*, then the inside to get *ay* and finally the last terms to get *by*.

One more equation to know when you're factoring is a special factor called the **difference of squares**. This is what you get when you factor *x*^2 - *y*^2.

*x*^2 - *y*^2 = (*x* + *y*)(*x* - *y*).

**Log rules:**

You won't need these unless you're going for a really elite score on the math, but in case you are, remember that if *x* equals log base *a* of *b*, then *a* to the *x* power equals *b*.

Next up: shapes. Again, it's not really about memorizing everything in the world as it is about knowing the basics really well and being prepared to apply them whenever you need them. Here we'll break it all down by shape.

**Triangles:**

The area of a triangle equals one-half times the base times the height: *A* = 1/2(*b* * *h*).

The **Pythagorean theorem** states that in a triangle with side lengths *a*, *b* and *c* (where *c* is the longest side), *a*^2 + *b*^2 = *c*^2. Two triangles are worth memorizing so you don't have to do this every time: the 3-4-5 right triangle and the 5-12-13 right triangle.

You'll also want to know the special right triangles backwards and forwards. In a 45-45-90 triangle, the side lengths opposite each angle are *x*, *x* and *x* root 2, respectively. In a 30-60-90 triangle, the side lengths are *x*, *x* root 3 and 2*x*, respectively.

**Circles:**

The area of a circle is pi * *r*^2. The circumference is 2pi * *r*.

**Rectangles and squares:**

The area of a rectangle is *l* * *w*. The area of a square is just one side squared. To find the perimeter of either, just add all the sides together.

**3D shapes:** The volume of a cube, rectangular prism or cylinder is the area of the base times the height.

**Lines and graphing:**

To find the slope of a line, just pick two points on the line, and look at the distance between them vertically and horizontally. The vertical distance is called the rise, and the horizontal distance is called the run. Slope is just rise over run. The equation for a line in point-slope form is *y* = *mx* + *b*, where *m* is the slope and *b* is the *y*-intercept.

If you have to deal with a circle on a graph, the equation is (*x* - *h*)^2 + (*y* - *k*)^2 = *r*^2, where *h* and *k* are the coordinates of the center, and *r* is the radius.

Almost done! Now it's time for the miscellaneous box: everything that didn't quite fit anywhere else.

**Trigonometry:**

There are only a couple trig problems on the ACT, so you can ignore them completely if you're not aiming for a near-perfect score. But if you're reaching for the stars, remember **SOH-CAH-TOA**, the abbreviation that gives you the sine, cosine and tangent for any angle theta.

The sine of theta equals the opposite side over the hypotenuse. The cosine equals the adjacent side over the hypotenuse, and the tangent equals the opposite side over the adjacent side.

**Radians:**

If you're aiming for a really high score, you'll also want to know radians. **Radians** are an alternative way of measuring the angles of a circle. 2pi radians equals 360 degrees, or the total number of degrees in a single circle. To convert from radians to degrees, just multiply the number of radians by 180 over pi, or multiply the number of degrees by pi over 180.

In this lesson, you got a lightning-fast review of some formulas you'll need on the ACT Math section. Hopefully most of them should already be familiar - the goal of the test isn't to get you breaking your brain with hours of memorization. It's really more about application: you'll have to take these basic formulas and use them in all kinds of ways to show that you really understand how they work.

If there's anything here that didn't ring a bell, make sure to check out the math lessons for a review - but first start with the quiz questions in this lesson to test your knowledge.

After studying this video on ACT preparation, you should be able to:

- Recognize that the ACT Math section assesses your ability to apply what you already know
- Calculate distance and work, averages and probability
- List the rules of exponents and logs
- Apply the appropriate formulas to geometric shapes
- Identify some other formulas that may be on the ACT

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