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What You See Is What You Get: The Format and Breakdown of the Math Test

No, the “breakdown” in the preceding heading doesn’t refer to your (nervous) breakdown, but rather to the breakdown of the number and types of problems in the Mathematics Test. This 60-minute test features 60 questions (which makes figuring out your time per problem convenient, no?). The questions fall into pretty standard categories.

In the ACT bulletin and in many ACT study books, you may have to slog through incredibly detailed analyses of the exact number of each question type on the test: 14 plane geometry questions, 4 trigonometry questions, blah, blah, blah. We refuse to put you to sleep with that sort of detail. The truth is that the number of questions of each concept varies from test to test. And it’s not as if you have any control over the distribution of questions, right? (We can just see the letter: “Dear ACT: Please be sure that I have more geometry and fewer algebra problems — thanks . . . .”)

The following is the short ‘n’ sweet version of the kinds of math questions you encounter in the dark alleyways of the ACT:

Pre-algebra: (Normal people refer to this as arithmetic.) About one-fourth of the questions cover basic arithmetic, including such concepts as fractions, decimals, and subtracting negative numbers.
Elementary algebra: You learn this type of material in your first semester or two of algebra. These questions test your ability to work with variables, set up algebraic formulas, solve linear equations, and do the occasional FOIL problem. About one-eighth of the questions cover elementary algebra.
Intermediate algebra/coordinate geometry: A little less than half of the questions cover more difficult quadratic problems, as well as inequalities, bases, exponents, radicals, and basic graphing (finding points on an x, y–coordinate graph).
Plane geometry and trigonometry: A little over one fourth of the questions cover plane figures (what you think of as “just plain figures,” like triangles, circles, quadrilaterals, and so on) and trigonometry. The trig questions make up no more than 10 percent of the test, so if you haven’t had trig yet, don’t despair. At least half of the trig questions are very basic, covering trig ratios and basic trigonometric identities.

Confused? Don’t worry about the exact number of questions. Just remember two important points:

You have 60 minutes to do 60 questions.
About one-third of the questions are arithmetic, one-third are algebra, and one-third are geometry.

Getting into the Grind: The Approach

You’ve done multiple-choice math problems all your life. In fact, you probably don’t have much more to learn about doing multiple-choice math questions. However, the following common-sense steps can help you stay focused as you move quickly through the Math Test:

Identify the point of the question.

Yes, even the stupid word problems have a point. Each question is trying to get you to supply a specific piece of information. It helps to read the end question first to determine what you're solving for. Does the question ask you to find the circumference or an area? Do you have to state the value of x or of 2x? Circle precisely what the question asks for. After you finish the problem, go back and double-check that your answer provides the circled information.

We just said that the ACT is not out to trap you — but that doesn’t mean you can’t trap yourself. Among the answer choices are answers that you get by making careless errors. Suppose, for example, that the problem asks for the product of numbers, and you find the sum. Your answer will undoubtedly be there with the other wrong answers (and the one right one, of course). If the question asks for one-half of a quantity and you solve for twice the quantity, that answer will also likely be there. Because these types of answer choices are available to you, it’s especially important that you identify exactly what the problem asks for and supply only that information.

To help you focus, especially on word problems, read the final lines of the question first. Knowing what you're solving for as you read the rest of the question saves time.
Budget your time and brain strain: Decide whether the problem is worth your time and effort.

You don’t have to do every math problem in order, you know. Read the question and then predict how time-consuming it will be to solve. If you know you have to take several steps to answer the question, you may want to skip the problem, mark it, and go back to it later. If you’re not even sure where to start the problem, don’t sit there gnawing at your pencil as if it were an ear of corn (unless you’re Pinocchio, wood really isn’t brain food). Guess and go.

Guess, guess, guess! The ACT has no penalty for wrong answers. You’re going to (or already have) read that statement hundreds of times throughout this book. We say it every chance we get to remind you that you can guess without fear of reprisal. Whenever you skip a problem, choose an answer, any answer, mark it on your answer sheet, and hope that you get lucky. Put a big arrow in the margin of the test booklet next to the question (not on the answer sheet, because it may mess up the computer grading) to remind yourself that you made a wild guess. But if you run out of time and don’t get back to the question, at least you have a chance of guessing the answer right.
Look before you leap: Preview the answer choices.

Look at the answer choices before you begin doing any pencil-pushing. Often, the choices are variations on a theme, like 0.5, 5, 50, 500, and 5,000. If you see those answers, you know you don’t have to worry about the digit, only the decimal. Maybe the answers are very far apart, like 1, 38, 99, 275, and 495. You probably can make a wild estimate and get that answer correct. But if you see that the answers are close together (like 8, 9, 10, 11, and 12), you know you have to invest a little more time and effort into being extra careful when solving the problem.

We’d be wealthy if we had a nickel for every student who groaned and complained as he or she looked at the answer choices, “Man, I didn’t really have to work that whole problem out. I could’ve just estimated from the answer choices.” Absolutely true.
Give yourself a second chance: Use your answer to check the question.

Think of this step as working forward and backward. First, work forward to come up with the answer to the question. Then plug the answer into the question and work backward to check it. For example, if the question asks you to solve for x, work through the equation until you get the answer. Then plug that answer back into the equation, and make sure it works out. This last step takes less time than you may think and can save you a lot of points.

Not all the math questions on the ACT contain only one sentence or request that you simply “solve for x.” In fact, many math questions require you to sift through a bunch of information to figure out what the real question is. These word problems, as they’re called, require you to translate words into numbers and then arrange them in a way that makes mathematical sense. You know what we’re talking about — those problems that tell you how fast Train A travels and what speed Train B moves at and then expect you to figure out exactly what hour the two trains will collide. Watch out!

Don’t worry; we’re here to help you make sense of all these words. Several words translate nicely into mathematical expressions, and many types of word problems lend themselves perfectly to specific formulas or strategies.

Translating English into Math

When you see a word problem on the Math Test, you may feel a little lost at first. Straightforward math equations seem so much more, well, straightforward. Even though word problems are written in English, they may seem like they’re written in a foreign language. To help you with the translation, Table 11-1 provides you with some of the more common words you encounter in word problems and tells you what they mean (and look like!) in math terms.

images — Table 11-1 Common Words and Their Math Counterparts

Plain English	Math Equivalent
More than, increased by, added to, combined with, total of, sum of	Add (+)
Decreased by, diminished by, reduced by, difference between, taken away from, subtracted from, less than, fewer than	Subtract (–)
Of, times, product of	Multiply ()
Ratio of, per, out of, quotient	Divide ( or /)
Percent	100
Is, are, was, were, becomes, results in	Equals (=)
How much, how many, what, what number	Variable (x, y)

Subtraction phrases such as “taken away from,” “subtracted from,” “less than,” and “fewer than” require you to switch the order of the quantities you’re subtracting. For example, “Ten decreased by six” means 10 – 6 (which equals 4), but “Ten subtracted from six” means 6 – 10, or –4.

As you read through a word problem, analyze its language to determine what math operations it involves. Keep this general process in mind:

Determine what you’re supposed to solve for, specifically what the x is in the equation.
Analyze the rest of the information to figure out how to arrange the equation to solve for x.

Many of the word problems on the ACT like the following example, concern percentages.

The problem asks for the total amount (that’s the x) of Ms. Bond’s loan with (which means +) interest, so you have to add what she owes in interest to the original amount of the loan. Before you add the interest amount, you must find out what the amount of interest is. The language of the problem tells you that Ms. Bond has to pay an interest rate of (meaning ´) 8% (which means you divide 8 by 100). Written with numbers rather than words, the problem looks something like this:

Perform the operation in parentheses first (as we explain in Chapter 8) to get 0.08. Next, multiply $650 by 0.08 to get $52. Ms. Bond pays $52 in interest. Add the interest amount to the loan amount to get your final answer: images . The correct answer is Choice (C).

If you picked Choice (A), you added 8 to $650, which isn’t the proper way to determine interest. Choice (B) is the correct interest amount but not the total amount of the loan plus the interest. If you opted for Choice (D), you incorrectly divided 8 by 10 rather than by 100 to come up with the interest amount. And Choice (E) is just $650 doubled.

Time Flies When You’re Having Fun: Timing Tips

The most common complaint we hear from students about the Mathematics Test is, “There’s just not enough time. If I had more time, I could probably ace every single question, but I always run out of time.” True enough. Although having one minute per question (60 math questions, 60-minute section) sounds good, you’ll be surprised how fast time goes by. We have a few suggestions to help you make the best use of your time.

Skim for your favorite questions

We think of this technique as eating dessert first (something we always do). Go for the chocolate cake first (the easy questions) to make sure time doesn’t run out before you get to the good stuff. Leave the green beans (the harder problems) for the end. If you run out of time (which happens to most test takers), at least you’ll have finished the questions that you had the best chance of answering correctly.

In fact, consider approaching the last 20 questions in the Math Test (traditionally the hardest) in two passes. When you get to about question 35 or 40, follow these steps:

Skim the problem to discover the answer to these two questions:
- Is the concept tested one I feel comfortable with?
- Will it take me less than one minute to solve it?
If the answer is yes to both questions, go for it! Answer the question, mark your answer on your answer sheet and move on.
If the answer to either question is no, skip it! Draw a big X next to the question in your test booklet and check out the next question.
Answer the remaining questions in the same way, never spending more than about 30 seconds on any one question and marking questions you skip. If a question you first think will be easy turns out to be time-consuming, skip it, mark it, and move on.
When you've seen all 60 questions, go back to the ones you've marked. The point is to make it all the way through the section because some of the later questions in the section can be really easy. You would hate to miss them because you've spent too much time on one or two dumb questions.
Make sure you've marked answers for every question on your answer sheet.

If you decide to skip a question in any section on the ACT, make sure you mark an answer for it on your answer sheet before you move on. You won’t receive a penalty for wrong answers, so make sure you’ve bubbled in an answer for every question in the section before the proctor calls time. You can always erase the guess later and replace it with the answer you've selected.

Backsolve when the answers are actual values.

On many problems, you can simply plug in the answer choices to see which one fits. This technique is call backsolving. If you find yourself thinking that there must be some sort of equation you can come up with to solve a complex problem, but you can’t actually come up with that formula or you know what to do but the computations will be super time-consuming, try plugging in the answers. This will only work if the answers contain no variables, of course, and will work best if the answers are all integers.

The ACT arranges answer choices from greatest to least or least to greatest, so unless there's some other compelling reason to start with another answer choice, begin with Choice (C), the middle value.

For example, suppose that the question is something like this:

Example The entrance fee for Great Mountains National Park is $10 for vans and $5 for cars. In one summer the park collected x van fees and y car fees. Which of the following is an expression for the total amount in dollars that the park collected for the entire summer?

(A) images

(B) images

(D) images

(E) images

You may recognize the correct answer immediately, but bear with us and learn the approach so you'll be prepared for trickier questions later.

Give your variables values. Say and . Don't worry that 2 van fees and 4 car fees in a whole summer doesn't make a lot of sense in real life. The goal is to keep your calculations quick and easy breezy.
Substitute the values for the variables to figure out the total dollar amount that would be collected for 2 vans and 4 cars: 2 van fees at $10 is $20 and 4 car fees at $5 is $20. So the total amount collect is $40.
Plug 10 for v and 5 for y in each answer choice to see which one results in $40.

Check Choice (A). images . That's your answer! You could try the others just to be sure, but you'll see that no other answer results in $40.

Kindly refrain from showing off everything you know

Some of the ACT problems have extraneous information. For example, a geometry problem may list all sorts of numbers, including lengths of sides and measures of interior angles. If you read the question first and it asks you to find the area of a trapezoid, you know you need just the numbers for base and height. (Remember the formula? The area of a trapezoid is images .) Extra red-herring info can make you waste a lot of time. We already know that you’re brilliant (you bought this book, didn’t you?); you don’t need to prove it by doing more than you’re asked during the test. If you convert every problem into two or three new problems, you’ll never finish the Math Test on time.

Put aside two minutes to fill in the remaining ovals

The ACT assesses no penalty for guessing. We like to say that over and over and over again until you’re so exasperated that you want to cut off our air supply. It’s critical to remember that you don’t lose points for wrong answers; always keep in mind that wild guesses are worth making. Nothing is worse than that sinking feeling you get when the proctor calls time and you still have five problems you haven’t even looked at. If you save a few minutes at the end, you can wildly fill in some answers for those last ten problems. You have a good chance of getting at least a two of them correct.

The proctor is supposed to tell you when you have only five minutes left in the test. Before the test actually begins, you may want to remind your proctor to do so.

Do’s, Don’ts, and Darns: What to Do and Not Do on the Math Test

Although the math questions are pretty straightforward, a few basic do’s and don’ts are worth noting here.

Do get the lead out

Give your pencil a workout. If you have to solve a geometry problem, jot down the formula first and then just fill in the numbers. If you have the formula staring at you, you’re not as likely to make a careless mistake as you would be if you tried to keep everything in your head. If the geometry problem has words, words, words, but no picture, draw the picture yourself. When you plug in the answer choices or make up your own numbers to substitute for variables, write down what you plugged in and tried. We see students redoing the same things over and over because they forgot what they’d already plugged in. Doodle away. You get no scratch paper for the ACT, but the test booklet has plenty of white space.

Don’t start working until you’ve read the entire problem

So you read the first part of a problem and start trying to solve for the area of the triangle or the circumference of the circle. But if you read further, you may find that the question asks only for a ratio of the areas of two figures, which you can figure out without actually finding the precise areas. Or you may solve a whole algebraic equation, only to realize that the question didn’t ask for the variable you found, but for something else entirely. Reading the question first can prevent this messiness. As we say in the “Getting into the Grind: The Approach” section, earlier in this chapter, read the last line or two of the problem first and circle the part of the problem that specifies exactly what you're solving for.

Do reread the problem with your answer inserted

Very few students take this last critical step. Most test takers are so concerned with finishing on time that they solve the problem and zoom on to the next question. Big tactical error. Rereading the question in light of your answer can show you some pretty dumb mistakes. For example, if the question asks you for the average of 5, 9, 12, 17, and 32 and your answer is 75, you can immediately realize that you found the sum but forgot to divide by the number of terms. (And of course, 75 is one of the answer choices.) Maybe the question asks you to find one interior angle of a figure, and your answer is 190. If you look at the angle and see that it is acute (less than 90 degrees), you’ve made a mistake somewhere.

Don’t strike out over a difficult question early on

Most standardized exams put their questions in order of difficulty, presenting the easy ones first, then the medium ones, and finally the hard ones. Things aren’t as cut and dried on the ACT Mathematics Test. You may find a question that you consider really tough very early in the exam. Although easy and hard are subjective, many of our students over the years have been furious with themselves because they never looked at the last several questions — reasoning that if they couldn’t get the earlier ones right, they obviously couldn’t get the later ones at all. Wrong. We’ve seen some relatively simple questions, especially basic geometry questions, close to the end of the exam.

Numb and Number: Acing the Mathematics Test