ACT Math Questions

What is the smallest possible value for the product of 2 real numbers that differ by 6? A. −9 B. −8 C. −5 D. 0 E. 7

A. If 2 numbers, x and y, differ by 6, that means x−y= 6. Multiplying the two numbers, (x)(y), will yield the product. Solve the first equation for x. x−y = 6x = y+6 Substitute the result for x in the second equation. (y+6)y Since one of the answer choices must be the solution to that equation, plug in the answer choices, starting with the smallest value (note that the answer choices are in ascending order): (y+6)y=−9y^2+6y+9=0(y+3)2=0y=−3 Now, substitute −3 for y in the first equation and solve for x: x−(−3)=6x=3

If a, b, and c are nonzero real numbers, what is the least number of times a function f(x) = ax^2 + bx + c may cross the x-axis? A. 0 B. 1 C. 2 D. 3 E. Cannot be determined without the values of a, b, c

A. The resulting graph would be a parabola, which could be completely within the first quadrant.

For which values of x will 3(x + 4) ≥ 9(4 + x)? A. x ≤ −4 B. x ≥ −4 C. x ≥ −16 D. x ≤ 4 E. x ≤ −16

A. To answer this question, solve the inequality for x, as follows: 3(x−4)≥9(4+x) 3x+12≥36+9x −6x≥24 x≤−4

If f(x)=x^3-4, then f(x+h)=? A. x^3+3x^2h+3xh^2+h^3-4 B. x^3-4+h C. x^3+3xh+h^3+4 D. x^3+x^3h^3+h^3+4 E. x^3+x^3h^3+h^3+4

A. f(x+h)=(x+h)^3-4 = (x^3+3x^2h+3xh^2+h^3)-4

If X, Y, and Z are real numbers, and XYZ = 1, then which of the following conditions must be true? A. XZ=1Y B. X, Y, and Z > 0 C. Either X = 1, Y = 1, or Z = 1 D. Either X = 0, Y = 0, or Z = 0 E. Either X < 1, Y < 1, or Z < 1

A. If XYZ = 1, then Z cannot equal 0. If Z (or X or Y, for that matter) were 0, then XYZ would equal 0. Both sides of the equation can be divided by Y, which gives you XZ=1Y, answer choice A.

What is the 400th term in the sequence 18, 21, 9, 24, 5, 18, 21, 9, 24, 5, ...? A. 5 B. 9 C. 18 D. 21 E. 24

A. The sequence repeats every five terms, so for example, the 5th, 10th, 15th, 20th, and 25th terms will all be 5. Further, since 400 is a multiple of 5, the sequence will "restart" at 18 on the 401st term, and the 400th term will be 5.

If xy^2<0, then which of the following statements MUST be true? A. The value of x is negative, while the value of y is positive or negative. B. The value of y is negative, while the value of x is positive or negative. C. The sign of x must be the same as the sign of y. D. The sign of x must be different from the sign of y. E. The values of x and y must be the same.

A. The square of any number is always positive. Therefore, if x y^2 is negative, the value of x must be negative.

Casey has buckets of 3 different sizes. The total capacity of 12 of the buckets is g gallons, the total capacity of 8 buckets of another size is g gallons, and the total capacity of 4 buckets of the third size is also g gallons. In terms of g when g > 0, what is the capacity, in gallons, of each of the smallest-sized buckets? A. g/12 B. g/8 C. g/4 D. 12g E. 8g

A. To solve this problem, first determine the size of each bucket. Because the total capacity of 12 buckets is g gallons, each bucket can hold (1/12)g, or g/12 gallons. Because the total capacity of 8 buckets is g gallons, then each of those buckets can hold (1/8)g, or g/8 gallons. If the total capacity of 4 buckets is g gallons, then each bucket is (1/4)g, or g/4 gallons. Therefore, the capacity of the smallest buckets is g/12.

When x is divided by 7, the remainder is 4. What is the remainder when 2x is divided by 7? A.1 B. 4 C. 5 D. 7 E. 8

A. You are given that x divided by 7 leaves a remainder of 4. The easiest approach to this problem is to assume that 7 goes into x one time, with a remainder of 4. Therefore, x is equal to 11. If x = 11, then 2x = 22. When 22 is divided by 7, the remainder is 1.

Given that y−5=12x+1 is the equation of a line, at what point does the line cross the x-axis? A. −15 B. −12 C. 1 D. 4 E. 6

B. A graph crosses the x-axis at the point when y = 0. Given that y−5=x^2+1, let y = 0 such that −5=x^2+1. Subtracting 1 from both sides yields −6=x^2. Multiplying by 2 yields −12 = x.

It is estimated that, from the beginning of 1993 to the end of 1997, the average number of CDs bought by teenagers increased from 7 per year to 15 per year. During the same time period, the average number of video games purchased by teenagers increased from 6 per year to 18 per year. Assuming that in each case the rates or purchase are the same, in what year did teenagers buy the same average number of CDs and video games? A. 1993 B. 1994 C. 1995 D. 1996 E. 1997

B. First, find the rate of sale change per year for CDs, and for video games. Rate of Sales Change per Year = Total Change/Total Years You can create an equation to represent the number of CDs sold, depending on the amount of time since the beginning of 1993: y=7+85xy=7+85x Where 7 is the starting CD sales, and 8585 is the rate of sales change per year. You can create an equation to represent the number of video games sold, depending on the amount of time since the beginning of 1993: y=6+125xy=6+125x 7+8 5x=6+125x7+8 5x=6+125x x = 1.25 yrs after the beginning of 1993, which would be the year 1994.

If one of the solutions to the equation (x − 5)(x + 2) (ax − b) = 0 is −7, then which of the following pairs of values for a and b is possible? A. a = −7, b = 1 B. a = 1, b = −7 C. a = 7, b = 1 D. a = 1, b = 7 E. a = −1, b = −7

B. If ax − b = 0, then ax = b and x=b/a. Therefore, a and b can have any values where b/a=−7. Of the pairs of values given, you need a pair where b is 7 or −7 and a has the opposite sign; only choice B meets these conditions.

Which of the following points is on the graph of the function f(x) = (x − 5)(x + 3)(x − 1)(x + 10)? A. (0, 0) B. (5, 0) C. (0, −3) D. (−1, 0) E. (0, 10)

B. Since x − 5 is a factor, 5 must be a zero of the function, and (5, 0) is on its graph.

Solve the following equation for x: 54! (x − 500) = 56! A. 2,580 B. 3,580 C. 501 D. −499 E. 611

B. The "!" after a number is called a "factorial," which is a command to take the given number times every integer less than it down to 1. When 56! is divided by 54!, all the numbers will cancel except 56 and 55: (x − 500) = 55 × 56 x − 500 = 3,080 x = 3,580

If a + b = 25 and a > 4, then which of the following must be true? A. a = 22 B. b < 21 C. b > 4 D. b = 0 E. a < 25

B. The correct answer will be the statement that is always true. Because a is greater than 4, and 25 − 4 = 21, b must always be less than 21.

Given the equation 2(x + 1) = −(y + 2), when y = 0, which of the following is the value of x? A. −4 B. −2 C. 0 D. 3 E. 5

B. When y = 0, 2x + 2 = −2, which simplifies to 2x = −4. This equation has a solution of x = −2.

For the function f(x)=x^2-3x, what is the value of f(5)? A. -10 B. 10 C. 25 D. 40

B. f(x) = x^2-3x f(5) = (5)^2-(3)(5) f(5) = 25-15 f(5) = 10

Let a, b, and c represent any three real numbers where a < 0. If a quadratic function f(x) = ax^2 + bx + c has x-intercepts of (3, 0) and (9, 0), then which of the following must be true about f(4)? A. The value of f(4) is negative. B. The value of f(4) is positive. C. The value of f(4) is zero. D. The value of f(4) is between −5 and 5. E. The value of f(4) is undefinable.

B. If a < 0, the graph is facing downward, and for any x value between 3 and 9, f(x) > 0.

What is the product of the 2 solutions of the equation x^2 + 4x − 21 = 0? A. −​63 B. −21 C. −20 D. 20 E. 21

B. The easiest way to solve this problem is to remember that when two binomial expressions are multiplied, there is a predictable result. (x + a)(x − b) = x^2 − bx + ax − ab. (x + a)(x − b) = 0, then x = −a and x = b. The product of the solutions is −ab. With this expression, x^2 + 3x − 21 = 0: (x + 7)(x − 3) = 0 x = −7, x = 3 The product of the solutions (−3 × 7) is −21.

If f(x) = 1/2x^2 and g(x) = 2x, then for what value or values of x is f(g(x)) = 8? A. −2 only B. −2 and 2 only C. 2 only D. 2√22 only E. −2√−22 and 2√22 only

B. The equation f(g(x)) = 8 is equivalent to 1/2(2x)^2 = 8, or 1/2 (4x^2) = 8. Dividing both sides by 2, x^2 =4, so x = +/-√4 = +/-2

How many real number solutions does the equation 2x3 = 9 have? A. 0 B. 1 C. 2 D. 3 E. 9

B. The given equation is equivalent to x^3=9/2, which has one solution, x=3√9/2.

If 3m − 4 = 6n − 8, which of the following expressions is equivalent to 3m − 8? A. 6n B. 6n − 12 C. 6n − 8 D. 6n − 4 E. 6n + 4

B. To get 3m − 8 from the given 3m − 4, we must subtract 4 from both sides of the equation. On the right side, 6n − 8 − 4 = 6n − 12.

Every point on the line in the (x,y) coordinate plane has the form (x, x-1/4). Which of the following is the equation of this line in standard form? A. y − 4x = −1 B. 4y − x = −1 C. 4y − x = 4 D. 4y − 4x = 4 E. 4y − 4x = 1

B. We are given the point y=x-1/4 x/4 -1/4=1/4x - 1/4 To put the equation in standard form, multiply both sides by 4, and bring the x and y terms to the same side: 4y-x=-1

g(x) is a transformation that moves f(x) 2 units in the negative x-direction and 3 units in the positive y-direction in the standard coordinate plane. What is g(x)? A. g(x) = f(x − 2) − 3 B. g(x) = f(x + 2) + 3 C. g(x) = f(x + 3) − 2 D. g(x) = f(x − 3) + 2 E. g(x) = f(x − 2) + 3

B. f (x + 2) shifts the function 2 units to the left. Adding 3 to that shifts g(x) up by 3 units.

Which of the following represents the values of x that are solutions for the inequality (x − 1)(4 − x) < 0? A. −14<x<1 B. x < 1 or x > 4 C. −1<x<14 D. −4 < x < 1 E. 14<x<4

B. x = 1 and x = 4. Only answer choice B has both of these numbers, so answer choice B is correct. To make sure, choose a number that is greater than 4, like 5, and see if it is a solution to the inequality (x − 1)(4 − x) < 0:

How many imaginary roots does the function g(x) = x^2 + 1 have? A. 0 B. 1 C. 2 D. 3 E. 4

C. If x^2 + 1= 0, then x^2 = −1 and x=±√-1=±i.

If 2p+q/+p>5 and p > 2, Which of the following inequalities MUST be true? A. q > 0 B. q > 2 C. q > 6 D. q > 10 E. q > 12

C. Rewriting the first inequality shows that 2p + q > 5p. Collecting terms, q > 3p. Therefore, when p > 2, q > 6.

In the (x, y) coordinate plane, which of the following functions would have a graph that crosses or touches the x-axis at only one point? A. x^2 + 5 B. x^2 − 6 C. x^2 − 8x + 16 D. x^2 + 2x + 18 E. −x^2 + 4x −10

C. To touch the x-axis at only one point and be a quadratic function as shown in the answer choices, the function would have to be of the form (x + c)^2 = x^2 + 2xc + c^2 or (x − c)^2 = x^2 − 2xc + c^2 for some value of c. The function in answer choice C is of this form, since 8 = 2(4) and 4^2 = 16.

If f(x)=x^2, find f(x+1). A. x^2-2x+1 B. x^2-2x-1 C. x^2+2x+1 D. x^2-1

C. f(x+1)(x+1)^2 (x+1)(x+1) x^2+x+x+1 x^2+2x+1

Which of the following expressions in place of the percent would make the expression 10n^3-5^2=%(2n^2-n) true for any value of n? A. 5 B. n C. 5n D. 5n^2 E. n^3

C. On the right side of the equation, both coefficients have been reduced by a factor of 5. Additionally, the power on n was reduced by 1 on each term. Therefore, 5n must have been the term factored out.

Which of the following expressions is a polynomial factor of a^16 − 16? A. a^4 − 4 B. a^4 + 4 C. a^4 + 2 D. a + 2 E. a − 2

C. Remember that a difference of squares factors easily, such as: a^2 − b^2 = (a + b)(a − b). Using the same technique, you can factor a^16 − 16 into (a^8 + 4)(a^8 − 4). The factor (a^8 − 4) is another difference of squares, so it can be factored further into itself: (a^8 − 4) = (a^4 + 2)(a^4 − 2). Of these factors, only (a^4 + 2) is an answer choice.

The 1st term of an arithmetic sequence is m, and each subsequent term is found by adding n to the previous term. Which of the following expressions represents the value of the 99th term? A. nm^99 B. n^99m C. m + 98n D. 98m + n E. 98mn

C. The nth term of any arithmetic sequence is found with the formula an = a1 + d(n − 1), where d is the common difference. In this equation would be m, the variable n would be the difference, and it would be multiplied by 99 − 1, yielding m + 98n.

What is the 101st term of the sequence 5, 4, 3, 0, 8, 5, 4, 3, 0, 8, ...? A. 0 B. 3 C. 4 D. 5 E. 8

D. Every 5th term is 8. Therefore, the 100th term will be 8, since 100 is a multiple of 5, and the next term will be 5.

In the standard (x, y) coordinate plane, if the x-coordinate of each point on a line is 5 more than half the y-coordinate, what is the slope of the line? A. −5 B. −13 C. −12 D. 2 E. 5

D. If the x-coordinate of each point on a line is 5 more than half the y-coordinate, then x= y^2+5. To find the slope of the line, solve for y and put the equation in slope-intercept form (y = mx + b, where m is the slope). To do so, first subtract 5 from both sides, then multiply the entire equation by 2 to get y = 2x − 10. The slope is 2.

If the function f(x) = x^2 − 9x + 8 is graphed in the (x, y) coordinate plane, then for which of the following values of x will the graph of f(x) be below the x-axis? A. x < 1 B. x < 8 C. x < 1 and x > 8 D. 1 < x < 8 E. The graph is not below the x-axis for any values of x.

D. Since the coefficient on the first term is positive, the graph of the function is a parabola opening up. Additionally, since x^2 − 9x + 8 = (x − 8)(x − 1), the function crosses the x-axis at x = 8 and x = 1. Therefore, for values of x between 1 and 8, f(x) will be negative.

Which operation in place of the # would make the equation true for all nonzero values of x? x^2 # 2x = x/2 A. x B. + C. - D. / E. Cannot be determined

D. Since the term on the right side of the equation has a lower degree than the first term on the left side of the equation, the operation must have been division.

Which of the following is a factor of x^4 − 16? A. x − 16 B. x − 8 C. x − 4 D. x − 2 E. x

D. The expression x^4 − 16 is a difference of squares, so it factors into (x^2 + 4)(x^2 − 4), which factors further into (x^2 + 4)(x + 2)(x − 2).

What is the y-coordinate of the point in the standard (x, y) coordinate plane at which the 2 lines y=x2+3 and y = 3x − 2 intersect? A. 1 B. 2 C. 3 D. 4 E. 5

D. To find the y-coordinate where the 2 lines y=x^2+3 and y = 3x − 2 intersect, you could set x^2+3 equal to 3x − 2 because they are both already solved for y. Where they would intersect, their y-coordinates would be equal. To solve x^2+3=3x−2, you could add 2 and subtract x^2 to both sides to get 5x^2=5, then multiply by 25 (the reciprocal of 52 ) to get x = 2. Then simply substitute 2 for x into either of the initial equations to get y = 4.

If f(x) = x + 2 and g(x) = x^2 − x − 6, then which of the following is equivalent to g(x)/f(x) for all values of x ≠ −2? A. x − 12 B. x − 8 C. x − 4 D. x − 3 E. x + 4

D. g(x)/f(x) =x^2−x−6/x+2 =(x−3)(x+2)/x+2 =x−3

If mn=4, then 6(m^2n^2-1)= A. 15 B. 23 C. 42 D. 90 E. 95

D. 6(m^2n^2-1) = 6(4^2-1) = 6(16-1) = 6(15) = 90

What is the least common denominator of 3/8a and 1/6a? A. 6a B. 8a C. 14a D. 24a E. 48a

D. Multiples of 8a include 16a, 24a, 32a, etc. Multiples of 6a include 12a, 18a, 24a, etc. The smallest multiple shared on these two lists is 24a.

For every positive 2-digit number, a, with tens digit x and units digit y, let b be the 2-digit number formed by reversing the digits of a. Which of the following expressions is equivalent to a - b? A. 0 B. 9x − y C. 9y − x D. 9(x − y) E. 9(y − x)

D. You are given that a is a number with tens digit x and units digit y. Therefore, x is equivalent to 10 times y, and a = xy = 10x + y. You are given that b is formed by reversing the digits of a. Therefore, b = yx = 10y + x. Set up an equation and solve for a − b as follows: a − b = (10x + y) − (10y + x) = 10x + y − 10y − x = 9x − 9y = 9(x − y)

Which of the following expressions represents a polynomial with exactly one real root? A. x^2 − 3 B. x^2 − 1 C. x^2 + 1 D. x^2 + 2x + 1 E. x^2 + 3x + 2

D. x^2 + 2x + 1 = (x + 1)^2, which has a single root of x = −1

The first and second terms of a geometric sequence are a and ab, in that order. What is the 643rd term of the sequence? A. (ab)^642 B. (ab)^643 C. a^642b D. a^643b E. ab^642

E.

Which of the following is a root of −5x + x^2? A. −5 B. −3 C. 1 D. 3 E. 5

E. A root of a function is a solution to the equation formed when setting the function equal to zero. In this case, it would be a solution to −5x + x^2 = 0, which is equivalent to x(−5 + x) = 0 when x is factored out. Finally, by the zero product rule, the roots are x = 0 and x = 5.

If loga x = n and loga y = p, then loga (xy)2 =? A. np B. 2np C. 4np D. n + p E. 2(n + p)

E. By definition, loga(xy)2 = 2 loga(xy) = 2(logax + loga y). Because you are given that logax = n and loga y = p, the correct answer is 2(n + p).

In the (x, y) coordinate plane, the graphs of f(x) = 2x^2 + 14 and g(x) = x^2 + 30 intersect at the point (a, b). If a < 0, what is the value of b? A. 22 B. 28 C. 30 D. 39 E. 46

E. Setting the equations equal to each other, 2x^2 + 14 = x^2 + 30. Collecting terms gives us x^2 = 16. This equation has solutions x = 4, x = −4. But since a < 0, we will use x = −4. At x = −4, b = f(−4) = 2(16) + 14 = 46.

For any real number a, the equation |x − 2a| = 5. On a number line, how far apart are the 2 solutions for x? A. 2a B. 5 + 2a C. 10a D. 5 E. 10

E. The absolute value of a number is indicated when a number is placed within two vertical lines. Absolute value can be defined as the numerical value of a real number without regard to its sign. This means that the absolute value of 10, |10|, is the same as the absolute value of −10, | − 10|, in that they both equal 10. If the absolute value of x − 2a = 5, then x − 2a must also equal −5. If x − 2a = 5, then x = 2a + 5. If x − 2a = −5, then x = 2a − 5.

How many ordered pairs (x, y) of real numbers will satisfy the equation 5x − 7y = 13? A. 0 B. 1 C. 2 D. 3 E. Infinitely many

E. The equation 5x − 7y = 13 defines a line. Since there are an infinite number of points in a line, there are an infinite number of ordered pairs (x, y) of real numbers that satisfy the equation 5x − 7y = 13.

If a is inversely proportional to b and a = 36 when b = 12, what is the value of a when b = 48? A. 0 B. 13 C. 14 D. 4 E. 9

E. By definition, if a and b are inversely proportional, then a1b1 = a2b2. Therefore, (36)(12) = (48)a. Solve for a as follows: (36)(12) = (48)a 432 = 48a 9 = a

Melissa had 3 fewer apples than Marcia. Then, she gave 2 apples to Marcia. Now how many fewer apples does Melissa have than Marcia? A. 0 B. 2 C. 3 D. 5 E. 7

E. If Marcia has 10 apples, Melissa has 10 − 3 = 7 apples. If Melissa gives 2 of her 7 apples to Marcia, Melissa is left with 7 − 2 = 5 apples. When Marcia receives 2 more apples, she has 10 + 2 = 12 apples. Since Marcia now has 12 apples and Melissa now has 5 apples, Melissa has 12 − 5 = 7 fewer apples than Marcia.

If x = 6a + 3 and y = 9 + a, which of the following expresses y in terms of x? A. y = x + 51 B. y = 7x + 12 C. y = 9 + x D. y = 57+x/6 E. y = 51+x/6

E. The first step in solving this problem is to solve x = 6a + 3 for a: x=6a+3 x−3=6a x−3/6=a Now you can substitute x−36 for a in the second equation, and solve for y: y=9+a y=9+(x−3)/6 y=546+x−3/6 y=54+x−3/6 y=51+x/6

If the point (a, b) lies on the graph of a function f(x) in the standard (x, y) coordinate plane, then which of the following points lies along the graph of f(x − 2)? A. (−2a, b) B. (a, −2b) C. (a − 2, b) D. (a, b − 2) E. (a + 2, b)

E. The graph of f(x − 2) would be the graph of f(x) shifted to the right by 2 units. Therefore, every x-coordinate would be increased by 2.

For what values of x is x^2 − 3x − 10 = 0? A. x = 3 B. x = 10 C. x = 5 D. x = 3 and x = 10 E. x = 5 and x = −2

E. The quadratic function x^2 − 3x − 10 factors into (x − 5)(x + 2). This can equal zero only if x = 5 or x = −2.

For all numbers x and y, let the operation 0 be defined as x 0 y=2xy-4x. If a and b are positive integers, which of the following can be equal to zero? I. a 0 b II. (a-b) 0 b III. b 0 (a-b) A. I only B. II only C. III only D. I and II only E. I, II, and III only

E. To solve this problem, substitute a for x and b for y and set it equal to zero.

Which of the following is equivalent to g(a^2-10) when g(x)=x^2 + x - 5? A. a^4 + 20a^2 + 85 B. a^4 + a^2 − 115 C. a^4 + a^2 + 85 D. a^4 − 19a^2 − 115 E. a^4 − 19a^2 + 85

E. g(a^2-10) =(a^2-10)^2+(a^2-10)-5 =a^4-20a^2+100+a^2-10-5 =a^4-19a^2+85