ACT Sample Test #1 Problems 31 to 60

31. What is the 100th term in the arithmetic sequence:

3, 7, 11, 15 … ?

A. 107

B. 303

C. 307

D. 399

E. 403

31. Answer: D

The nth term in an arithmetic sequence is a+(n-1)d where a is the first term and d is the difference. In this sequence 3 is the first term and the difference between each two consecutive terms is 4. Then the 100th term = 3+(100-1)4 = 399.

32. If there are 155 pages in a magazine and 31 pages are devoted to advertising, what percentage of the magazine is advertisement?

F. 18%

G. 20%

H. 31%

I. 80%

J. 82%

32. Answer: G

Part of the magazine is devoted to advertising (31 pages) and there are 155 pages total. Therefore, we can use the formula

^p/_b = ^r/₁₀₀

³¹/₁₅₅= ^r/₁₀₀

Cross multiplying we have

155 r = 31 X 100

so that

r = 20.

We can also just recognize that 31 is part of 155 and write

^31(is)/_155(of) = .2 = 20%.
33. Which graph below represents the equation x² - 4x + 4 < 0?

33. Answer: E

x² - 4x + 4 < 0

(x-2)² < 0

Since the square of any real number is positive, there is no solution.

To determine that there is no solution, however, it is not necessary to solve algebraically. Plug in numbers. If you use 5 and -5 you will eliminate answers A, B, C, and D, leaving E as the only possible answer.

34. Given the figure below, cosq =x/10.

What is sinq ?

F. (Ö10-x²)/20

G. (Ö10-x²)/10

H. (Ö100-x²)/10

I. (10-x)/10

J. (10-x)/100

34. Answer: H

If cosq = x/10 we can label the triangle as below, letting y represent the other leg of the right triangle. Using the Pythagorean Theorem we have

x²+y² = 10²

Þ y² = 10² - x²

Þ y = Ö(100 - x²)

Since sinq = y/10

sinq = (Ö(100-x²))/10

35. is equal to which

of the following?

A. cot²q

B. tan²q

C. sin²q

D. -1

E. 1

35. Answer: B

= = tan²q

36. Find the sum of (16x - 3)/9 and (-3x + 6)/27.

F. (17x - 1)/9

G. (17x + 1)/9

H. (15x - 1)/9

I. (15x+1)/9

J. 15x - 1)/3

36. Answer: H

Usually when two fractions are added the first step is to find a common denominator:

(16x-3)/9+(-3x+6)/27

=(48x-9)/27+(-3x+6)/27

=(48x-3x-9+6)/27

=(45x-3)/27

=(15x-1)/9

However, if (-3x+6)/27 had been reduced to (-x+2)/9 at the beginning, the problem would have been greatly simplified.

37. Which point lies on the line of the following equation?

- 8x + 2y - 4 = 0

A. (-2, -1)

B. (2, 1)

C. (1, 2)

D. (1, -2)

E. (-1, -2)

37. Answer: E

Solution: Plug in.

A. - 8(-2) + 2(-1) - 4 = 16 -2 -4 ¹ 0

Note: do not completely solve. Just do enough arithmetic to determine whether the answer is incorrect. For example, subtracting 2 and 4 from 16 will still be much greater than 0, so (a) cannot be the answer. In school you have to determine that 16 - 2 - 4 = 10 but even this simple calculation takes time.

B. - 8(2) + 2(1) - 4 = -16 + 2 - 4 ¹ 0

C. - 8(1) + 2(2) - 4 = -8 + 4 - 4 ¹ 0

D. - 8(1) + 2(-2) - 4 = -8 -4 -4 ¹ 0

E. - 8(-1) + 2(-2) - 4 = 8 - 4 - 4 = 0

Looking back: If you are comfortable with algebra it might be easy to simplify before doing some computation.

- 8x + 2y - 4 = 0

-4x + y - 2 = 0

Looking back: If you are good with drawing graphs quickly and don't like nit-picking computation, sketch a graph and visually eliminate answers that don't make sense.

Solve for y:

y = 4x + 2

38. A line segment has its endpoints at (1, -1) and at (10,11). What is the length of the segment?

F. 10Ö2

G. 5

H. Ö110

I. 15

J. 2Ö26

38. Answer: I

Using the distance formula we have

d = Ö(y₂ - y₁)² +(x₂ - x₁)²

= Ö(11 - (-1))²+(10 - 1)²

= Ö(12)² + (9)²

= Ö 144 + 81

= Ö 225

= 15

Looking Back:

Draw a picture. You are looking for d, the distance from (10, 11) to (1, -1). If you create a triangle as shown you can use the Pythagorean Theorem. Note that the length of the vertical leg above the x-axis is 11 units and the length of the vertical leg below the x-axis is 1 unit (from y = 0 to y = -1) and that the length of the horizontal axis is 9 (10-1).

Since the triangle is a right triangle (the Cartesian coordinate system is a 90° grid) and since the legs are in the ratio of 3:4 this must be a 3-4-5 right triangle and d = 15.

39. Find the solution set for the equation |x - 14| = 7.

A. {-21, -7}

B. {21, 7}

C. {21, -21}

D. {21, -7}

E. {21}

39. Answer: B

Note that |+a| = |-a|. Therefore

|x-14| = |-x+14|. To find the values of x that will satisfy |x-14|=7 you need to solve two equations.

x-14 = 7 Þ x = 21

and -x+14 = 7 Þ -x = -7 Þ x = 7.

40. The area of a square is (81x²)/4. What is the perimeter in terms of x?

F. 9x

G. 9x²

H. (10 ¹/₈) x

I. 18x

J. 18x²

40. Answer: I

Since the area of a square is given by A = s²,

s = ÖA

= Ö(81x²)/4

= 9x/2.

To find the perimeter, multiply the length of the side by 4.

P = 4s

= 4(9x/2)

= 18x

41. A chord of a circle is a line segment with both endpoints on the circle. The radius of a circle is 12 and a chord in the circle is 18. What is the shortest distance from the center of the circle to the center of the chord?

A. 3Ö7

B. 6Ö5

C. 5

D. 25

E. 8

41. Answer: A

Solution: Draw a picture.

Let x represent the shortest distance from the center to the chord.

Note that this segment meets the chord in a right angle, producing a right triangle with the chord and the radius.

We have x² + 9² = 12².

x² + 81 = 144

x² = 144 - 81

x² = 63

x = Ö63

x = 3Ö7

B. Did you use 18 rather than 9? The leg can't be longer than the hypotenuse, and Ö-180 is not a real number. However, Ö180 = 6Ö5.

C. Did this look like a 3-4-5 (9-12-15) triangle?

E. Good visual estimate

42. If 15/3 = x/4 then x = ?

F. 10/3

G. 20

H. 1/20

I. 45/3

J. 22

42. Answer: G

Multiply both sides of the equation by 4.

15/3 = x/4

60/3 = x

20 = x

Extra: Simplify 15/3 before doing the problem.

43. If x = -9 then (5x - 3x)(x) =

A. -648

B. -675

C. 162

D. 648

E. 675

43. Answer: C

[5(-9) - 3(-9)](-9)

= [-45 - (-27)](-9)

= (-45 + 27)(-9)

= (-18)(-9)

= 162

Extra: Simplify first. 5x - 3x = 2x.

44. √56 + √81 =

F. √137

G. √65

H. 4√7 +9

I. 2√14 + 3

J. 2√14 + 9

44. Answer: J

Solution:

√56 + √81

= √(4x14) + √81

= (√4)x(√14) + 9

=2√14 + 9

Do not add values under the radical. Note that √56+√81≠√137
45. Which of the following graphs describe the equation

y = 1/|x| ?

45. Answer: A

Solution: Plug in a few points.

x	-1	1	5
y	1	1	1/5

46. For which value of x will

x³ + 4x² - 3x - 14 = 0?

F. 1/2

G. - 2

H. 0

I. 2

J. 14

46. Answer: G.

This is plug-in. Start with easier numbers, like 0, 2, -2.

(-2)³ + 4(-2)² - 3(-2) - 14

= -8 + 4(4) + 6 - 14

= -8 + 16 + 6 - 14

= 0

47. Last week Ethan made $102 on Monday, $96 on Tuesday and Wednesday, $99 on Thursday, and $48 on Friday. What was his average daily pay for the week?

A. $86.25

B. $69

C. $88.20

D. $345

E. $444

47. Answer: C

Add the pay for five days and divide by 5. Don't forget that Tuesday and Wednesday he was paid the same amount.

($102 + $96 + $96 + $99 + 48) ÷ 5 = $88.20

A. Since the same pay was made on both Tuesday and Wednesday, $96 must be added twice, and the result divided by 5, not 4.

B. The pay for Wednesday was not included.

C. Correct.

D. Adding the numbers in the problem gives you $345.

E. Partial answer. Don't forget to divide by the number of days.

48. Which of the following demonstrates the associative property?

F. 7 + 8 = 8 + 7

G. 5(8+4) = 5x8 + 5x4

H. 9x6 = 6x9

I. (9-6)x4 = 9x4 - 9x6

J. 5 - (6+3) = (5 - 6) + 3

48. Answer: J

In general, the associative property is (a+b)+c=a+(b+c) or (axb)xc=ax(bxc).

F. commutative property

G. distributive property

H. commutative property

I. distributive property

J. associative property Correct answer.

49. Which of the following graphs describe the equation

(x-4)² + (y+4)² = 16?

A. B.

C. D.

49. Answer: A

The equation of a circle is (x-h)² + (y-k)² = r² where the center is (h,k) and r is the radius.

In this problem you are given (x-4)² + (y+4)² = 16. The center is (-4, 4) and the radius is 4.

50. log₁₀ .001 =

F. -3

G. -2

H. 1

I. 2

J. 3

50. Answer: F

log₁₀ .001 = x

Þ 10^x =.001

Þ x = -3

51.

51. Answer: A

When adding or subtracting matrices add or subtract each term.

Hint: Start in the upper left of the matrix: 4-(-6)=10. This eliminates answers C, D, E. Then eamine the upper right value to determine whether the answer is A or B.

52. If a pool is 3/4 full to begin with and 1/3 of the water is drained out on Monday, and 1/3 or the remaining water is drained out on Tuesday, then there are 2,000 gallons of water remaining in the pool. How many gallons of water are needed to fill the pool to the top?

F. 2000

G. 3000

H. 4000

I. 5000

J. 6000

52. Answer: H

Let x be the capacity of the pool, so that there is (3/4)x gallons of water in the pool. If 1/3 of the water is drained out on Monday there will be (2/3)(3/4)x gallons of water in the pool. If another 1/3 is drained out on Tuesday there will be 2/3 of the amount in the pool on Monday, or (2/3)(2/3)(3/4)x = (1/3)x gallons of water in the pool. If (1/3)x = 2000 gallons, then x = 6000 gallons and 4000 gallons are needed to fill the pool.

53. Find the last digit of 3¹²⁰.

A. 1

B. 3

C. 5

D. 7

E. 9

53. Answer: A

This number is too big to calculate so there must be a trick. Look for a pattern.

3^x	Last digit
3^{^1}	3
3²	9
3³	7
3⁴	1
3⁵	3
3⁶	9
3⁷	7
3⁸	1
3⁹	3

Note that the pattern repeats after four numbers so that 3⁴, 3⁸, 3¹² etc. all have the same last digit (which is 1); 3⁵, 3⁹, 3¹³ etc. all have the same last digit, and so on. To generalize, if x represents a positive integer, 3^4x all have the same last digit, 3^4x+1 all have the same last digit, etc. Since 120 is divisible by 4, the last digit will be 1. (Note that we can write 3 ¹²⁰ as 3 ^4x30.)

54. Let f(x)=x², g(x)=4x, and h(x)=|x-3|.

For x = (-3), which one of the following functions has the greatest value?

F. f(g(x))

G. g(f(x))

H. f(h(x))

I. h(f(x))

J. g(h(x))

54. Answer: F

Plug in.

F. f(g(-3)) = f(4(-3)) = f(-12) = (-12)² = 144

G. g(f(-3)) = g ((-3)²) = g(9) = 4(9) = 36

H. f(h(-3)) = f (|-3-3|) = f(6) = 36

I. h(f(-3)) = h ((-3)²) = h(9) = |9-3)| = |6| = 6

J. g(h(-3)) = g (|-3-3|) = g(6) = 4(6) = 24

55. Solve for x:

A. (3, -5)

B. 4

C. (-3, 5)

D. 5

E. no real value of x satisfies the equation

55. Answer: B

This is a good one to try plugging in. However, the algebraic solution follows.

Given

56. In the figure below, AB = AC, BC = CD, and angle BAC = 90°. Find the measure of angle ADB.

F. 18°

G. 22.5°

H. 25°

I. 30°

J. 45°

56. Answer: G

Since AB = AC, ABC is an isosceles triangle Ð ACB = Ð ABC.

Using the fact that there are 180° in a triangle,

ÐACB+ÐABC+ÐBAC=180°, and since Ð ACB=Ð ABC and ÐBAC=90°,

ÐACB+ÐACB+90°=180°

Þ2(ÐACB)= 180°-90°

ÞÐACB=45°

Since ÐACB and ÐBCD are supplementary angles, ÐBCD=180°-45°=135°.

Since BC=CD, CBD is an isosceles triangle andÐCDB=ÐCBD.

Now ÐBCD+ÐCDB+ÐCBD=180°

135°+ÐCDB+ÐCDB=180°

2ÐCDB=180°-135°

ÐCDB=22.5°

57. What is the largest positive value of x for which the curve

y = cos(4x) is at a minimum?

A. π/2

B. 3π/4

C. π

D. 5π/4

E. 3π/2

57. Answer: D

Answers A, C, and E are all maximum values of the curve:

A. cos(4π/2) = cos(2π) = 1

C. cos(π) = cos(4π) = 1

E. cos(4·3π/2) = cos(6π) = 1

Answers B and C are minimum values:

B. cos(4·3π/4) = cos(3π) = -1

D. cos(4·5π/4) = cos(5π) = -1

Since you are looking for the largest value of x, the answer is D since 5π/4 is larger than 3π/4.

58. If x = -1 and y = -2 then 5xy + x³y³ = ?

F. -2

G. 2

H. 10

I. 18

J. 28

58. Answer: I

Solution: Plug in.

5xy + x³y³ = 5(-1)(-2) + (-1)³(-2)³

=5(2) + (-1)(-8)

= 10 +8

= 18

59. Given a right triangle with legs of length 5y and 8x, which expression represents the length of the hypotenuse?

59. Answer: B

Let h be the hypotenuse. Then

h² = (5y)² + (8x)²

h² = 25y² + 64x²

60. What is the factored form of

9a² - 144b²?

F. (3a-12b)(3a-12b)

G. (3a+12b)(3a+12b)

H. (3a+12b)(3a-12b)

I. 9(a² - 144b²)

J. (9-144)( a² - b²)

60. Answer: H

Solution: 9a² - 144b² is the difference of two squares. The basic form of this problem is a²-b²=(a+b)(a-b).