Basic Facts

• A total of 60 questions to be answered in 60 minutes.
• Question in the math section contain five answer choices.
• To score a 21 on the test you need to get 30 - 31 questions correct - ~50% of the questions.
• There are not negative points for incorrect or blank answers, so answer every question on the test.

Breakdown of the Math section by concepts

Preparing for Higher Math (57 - 60%)

This category captures the more recent mathematics that students are learning, starting when students begin using algebra as a general way of expressing and solving equations. This category is divided into the following five subcategories.

Number & Quantity (7 - 10%): Demonstrate knowledge of real and complex number systems. You will understand and reason with numerical quantities in many forms, including integer and rational exponents, and vectors and matrices.

Algebra (12 - 15%): Solve, graph, and model multiple types of expressions. You will employ many different kinds of equations, including but not limited to linear, polynomial, radical, and exponential relationships. You will find solutions to systems of equations, even when represented by simple matrices, and apply your knowledge to applications.

Functions (12 - 15%): The questions in this category test knowledge of function definition, notation, representation, and application. Questions may include but are not limited to linear, radical, piecewise, polynomial, and logarithmic functions. You will manipulate and translate functions, as well as find and apply important features of graphs.

Statistics & Probability (8 - 12%): Describe center and spread of distributions, apply and analyze data collection methods, understand and model relationships in bivariate data, and calculate probabilities, including the related sample spaces.

Integrating Essential Skills (40 - 43%)

These questions address concepts typically learned before 8th grade, such as rates and percentages; proportional relationships; area, surface area, and volume; average and median; and expressing numbers in different ways. You will solve problems of increasing complexity, combine skills in longer chains of steps, apply skills in more varied contexts, understand more connections, and become more fluent.

Test Directions

Do not waste any time reading the directions at the beginning of the test. The directions are always the same as follows:

Working Math problems and bubbling answer choices uses different part of your test booklet. Complete the problems in your book, and then bubble before you turn the page.

Make 2 passes through the questions ......

1. On the first pass, answer all the problems that you know for sure and guess on all the ones you have doubts how to solve.
2. If you guess (and you cannot narrow down the answer choices), choose a letter and guess it every time.
3. Save the questions that you don't know for the end.
4. Pay attention to which questions go with a figure, chart or common information.

The December 2015 ACT, contained the following directions at the top of page 24:
       Use the following information to answer questions 41-44.
However, #44 was on page 25. It is a common mistake for students to overlook this and miss a question or two.

Numbers & Quantity and Algebra Tactics: Use them when you don't know how to solve a problem

Plug in your own numbers.

Plug in the answer choices.

Plugging in your own numbers:

Step 1: Plug in your own numbers for each variable. Make sure to write them down. Avoid using 0 and 1.

Step 2: Solve the problem using your numbers.

Step 3: Write down your answer and circle it. This is your TARGET.

Step 4: Plug in your chosen numbers into the answer choices. Make sure to check them all. The choice that matches your target is the correct answer.

Sample problems: Try to plug in your own numbers.

1. If a store sells a shirt for h dollars, how much would that shirt cost if it marked down by q%?

(A).	\(\text{hq}\)
(B).	\(\frac{1}{4}\text{hq}\)
(C).	\(\text{h(1 - }\frac{q}{100}\text{)} \)
(D).	\(\text{q(1 - }\frac{h}{100}\text{)} \)
(E).	\(\text{2hq}\)

2. If the sum of three consecutive odd integers is p, then in term of p, what is the greatest of the three integers?

(F).	\(\frac{(p - 6)}{3}\)
(G).	\(\frac{(p - 3)}{3}\)
(H).	\(\frac{p}{3}\)
(J).	\(\frac{(p + 3)}{3}\)
(K).	\(\frac{(p + 6)}{3}\)

3. If w hats costs z dollars, then how many hats could you buy with $100?

(A).	\(\frac{100}{w}\)
(B).	\(\text{100wz}\)
(C).	\(\frac{100w}{z}\)
(D).	\(\frac{100z}{w}\)
(E).	\(\text{wz}\)

---

Plug in answer choices. Be sure to check all answers!!!

The choices are usually in numerical order, so start with (C) or (H). Then determine if you should go up or down.

Sample problems:

4. What is the set of real solution for |x|² - |x| - 2 = 0 ?

(F).	{2}
(G).	{-2, 2}
(H).	{-1, 2}
(J).	{1, 2}
(K).	{-2, -1, 1, 2}

5. For what real value of x, if any, is \(log^{(x^2 + 3)}_{(x + 3)} = 2\)

(A).	-2
(B).	-1
(C).	0
(D).	1
(E).	There is no such value.

More Algebra Tips:

1. Use your common sense to eliminate illogical answers.
2. Avoid falling into traps (partial answers, simple math on difficult questions).
3. Understand, that on certain problems, you will be given extra information that is not central to solving the problem.

Sample problems:

6. The equation x² + mx + n = 0, m and n are integers. The only possible value for x is -3. What is the value of m ?

(F).	-6
(G).	-3
(H).	3
(J).	6
(K).	9

Geometry Tactics:

1. Most shapes will be drawn to scale. Use your eyes to impossible answer choices.
2. When a diagram is not given or not drawn to scale, draw one so you can answer the question.
3. Find any missing information in the figure to aid in solving the problem.

Sample problems:

7. Figure DEFG is a square. If \(\overline{EG} = 4\), what is the area of the square ?

(A).	4
(B).	\(4\sqrt{2}\)
(C).	8
(D).	16
(E).	32

8. In the picture below, ABCD is a rectangle. If the area of \(\triangle{ABE}\) is 40, what is the area of the rectangle?

(F).	10
(G).	40
(H).	48
(J).	80
(K).	112

9. In triangle ABC shown below, \( \text{sin C = } \frac{4}{5} \) and the length of \(\overline{AB}\) is 10 inches. What is the length, in inches, of \(\overline{AC} \) ?

(A).	3
(B).	\(\sqrt{41}\)
(C).	8
(D).	9
(E).	\(\frac{25}{2}\)

10. As shown below, \(\overline{BE}\) divides rectangle ACDF into 2 congruent trapezoids. he measure of \(\angle{BED}\) is 45^o. The lengths of \(\overline{BC}\), \(\overline{CD}\), and \(\overline{EF}\) are given in feet. What is the area, in square feet, of rectangle ACDF ?

(F).	10
(G).	14
(H).	60
(J).	72
(K).	84

Trigonometry Tactics: Most of the trigonometry problems can be solved with SOH-CAH-TOA or \( S\frac{O}{H} C\frac{A}{H} T\frac{O}{A} \)

\( sin \theta = \dfrac{Opposite}{Hypotenuse} \)

\( cos \theta = \dfrac{Adjacent}{Hypotenuse} \)

\( tan \theta = \dfrac{Opposite}{Adjancent} \)

Sample problems:

11. For an angle with measure \(\alpha\) in a right triangle, \(\text{sin} \alpha = \frac{112}{113} \) and \(\text{tan} \alpha = \frac{112}{15} \). What is the value of \(\text{cos} \alpha \) ?

(A).	\(\frac{15}{113}\)
(B).	\(\frac{15}{112}\)
(C).	\(\frac{15}{\sqrt{25,313}}\)
(D).	\(\frac{15}{\sqrt{12,319}}\)
(E).	\(\frac{113}{15}\)

12. Which of the following expressions gives the measure of \(\angle{STR}\) ?

(F).	\(cos^{-1} (\frac{2}{3} )\)
(G).	\(sin^{-1} (\frac{2}{3} )\)
(H).	\(cos^{-1} (\frac{3}{2} )\)
(J).	\(tan^{-1} (\frac{2}{3} )\)
(K).	\(tan^{-1} (\frac{3}{2} )\)

13. The figure below shows a lighthouse keeper looking down at a sailboat on the sea through a navigational instrument. The instrument is 60 feet above sea level and indicates an angle of depression of 53^o to the rowboat. Which of the following is closest to the horizontal distance, in feet, between the navigational instrument and the rowboat ?

(A).	36
(B).	45
(C).	48
(D).	53
(E).	80

14. Given that \(\text{2 sin a = 2}\) and \(2 cos (\frac{π}{2} - b) = 2\), which of the following could be a value, in radians, of a + b ?

(F).	0
(G).	\(\frac{π}{2} \)
(H).	2
(J).	π
(K).	2π

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