Your FREE Praxis Middle School Mathematics (5164) Practice Test 2026 – 150+ Q&A

Praxis 5164 Practice Questions

1. What is the value of $8^{2/3}$?
   • 2
   • 4
   • 16
   • 64

   Correct answer: 4

   Correct answer: 4. Explanation: The expression represents $\sqrt[3]{8^2}$, which is $64$, and $\sqrt[3]{64}$ is $4$.

2. If $\frac{5}{x} = \frac{x}{20}$, what is the value of $x$?
   • 4
   • 10
   • 20
   • 25

   Correct answer: 10

   Correct answer: 10. Explanation: Solve the proportion by cross-multiplying and simplifying the resulting quadratic equation.

3. What is the result of $\left(\frac{2}{3}\right)^{-2}$?
   • $\frac{4}{9}$
   • $\frac{9}{4}$
   • $\frac{3}{2}$
   • 2.25

   Correct answer: $\frac{9}{4}$

   Correct answer: $\frac{9}{4}$. Explanation: The negative exponent means take the reciprocal of the base and then square it.

4. Simplify: $\sqrt[3]{-125}$.
   • -5
   • 5
   • -25
   • 25

   Correct answer: -5

   Correct answer: -5. Explanation: Find $\sqrt[3]{-125}$, which is -5.

5. What is the least common multiple of 6, 8, and 12?
   • 24
   • 48
   • 72
   • 96

   Correct answer: 24

   Correct answer: 24. Explanation: Find the smallest number that is a multiple of all three given numbers.

6. Evaluate: $\frac{6!}{4! \cdot 2!}$.
   • 15
   • 18
   • 30
   • 45

   Correct answer: 15

   Correct answer: 15. Explanation: Calculate the factorial expressions and simplify the fraction.

7. What is the greatest common divisor (GCD) of 54 and 81?
   • 9
   • 18
   • 27
   • 36

   Correct answer: 27

   Correct answer: 27. Explanation: The GCD of 54 and 81 is the largest number that divides both, which is 27.

8. What is the least common multiple (LCM) of 24, 36, and 48?
   • 144
   • 288
   • 432
   • 576

   Correct answer: 144

   Correct answer: 144. Explanation: The LCM of 24, 36, and 48 is the smallest number that is a multiple of all three.

9. What is the decimal equivalent of $\frac{1}{8} + \frac{1}{4}$?
   • 0.125
   • 0.25
   • 0.375
   • 0.5

   Correct answer: 0.375

   Correct answer: 0.375. Explanation: Add the fractions and convert the sum to a decimal.

10. A certain number is increased by 25% to become 125. What was the original number?
    • 96
    • 100
    • 104
    • 110

    Correct answer: 100

    Correct answer: 100. Explanation: Calculate the original number by reversing the percentage increase.

11. What is the value of $x$ in the equation $2x^2 - 8x = 0$?
    • 0 and 2
    • 0 and 4
    • -4 and 4
    • 2 and 4

    Correct answer: 0 and 4

    Correct answer: 0 and 4. Explanation: Factor the quadratic equation and solve for $x$.

12. What is the vertex of the parabola represented by the equation $y = 2x^2 - 4x + 1$?
    • $(1, -1)$
    • $(-1, 1)$
    • $(1, 3)$
    • $(-1, -3)$

    Correct answer: $(1, -1)$

    Correct answer: $(1, -1)$. Explanation: Use the vertex formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex and then substitute to find the y-coordinate.

13. Given $f(x) = x^2 - 4x + 3$, find the value of $f(2a)$.
    • $4a^2 - 8a + 3$
    • $4a^2 - 16a + 3$
    • $a^2 - 4a + 3$
    • $4a^2 - 8a - 1$

    Correct answer: $4a^2 - 8a + 3$

    Correct answer: $4a^2 - 8a + 3$. Explanation: Substitute $2a$ for $x$ in $f(x)$ and simplify.

14. Which of the following is the factored form of $x^2 - 5x - 6$?
    • $(x - 6)(x + 1)$
    • $(x - 2)(x - 3)$
    • $(x + 6)(x - 1)$
    • $(x + 2)(x + 3)$

    Correct answer: $(x - 6)(x + 1)$

    Correct answer: $(x - 6)(x + 1)$. Explanation: Factor the quadratic expression by finding two numbers that multiply to -6 and add to -5.

15. Find the inverse function of $f(x) = \frac{3x - 4}{2}$.
    • $f^{-1}(x) = \frac{2x + 4}{3}$
    • $f^{-1}(x) = \frac{2x - 4}{3}$
    • $f^{-1}(x) = \frac{3x + 4}{2}$
    • $f^{-1}(x) = \frac{3x - 4}{2}$

    Correct answer: $f^{-1}(x) = \frac{2x + 4}{3}$

    Correct answer: $f^{-1}(x) = \frac{2x + 4}{3}$. Explanation: Swap $x$ and $y$ in the original function and solve for $y$.

16. If h(x) = 2|x − 3|, what is the value of h(0)?
    • 6
    • -6
    • 3
    • -3

    Correct answer: 6

    Correct answer: 6. Explanation: Substitute 0 for x in h(x) and simplify, remembering that absolute values are always non-negative.

17. If h(t) = 3t² − 2t + 1, what is h(−2)?
    • 17
    • 13
    • 11
    • 7

    Correct answer: 17

    Correct answer: 17. Explanation: Substitute −2 for t in the equation and simplify.

18. Determine the zeros of the quadratic function f(x) = x² − 6x + 8.
    • 2 and 4
    • 1 and 7
    • 3 and 5
    • 4 and 6

    Correct answer: 2 and 4

    Correct answer: 2 and 4. Explanation: Factor the quadratic and solve for x.

19. The function f(x) = 4x − 7 is applied to a number, and the result is 9. What was the original number?
    • 2
    • 3
    • 4
    • 5

    Correct answer: 4

    Correct answer: 4. Explanation: Set f(x) = 9 and solve for x.

20. If $a$ and $b$ are solutions to the equation $x^2 - 5x + 6 = 0$, what is $a + b$?
    • 1
    • 5
    • 6
    • 11

    Correct answer: 5

    Correct answer: 5. Explanation: Use the sum of roots formula for a quadratic equation.

21. If $g(x) = \frac{1}{x-2}$, for which values of $x$ is $g(x)$ undefined?
    • $x = 0$
    • $x = 1$
    • $x = 2$
    • $x = -2$

    Correct answer: $x = 2$

    Correct answer: $x = 2$. Explanation: The function is undefined where the denominator equals zero, which is at $x = 2$.

22. What is the range of the function $h(x) = x^2 + 4$ for $x \geq 0$?
    • $y \geq 4$
    • $y > 0$
    • $y \geq 0$
    • $y > 4$

    Correct answer: $y \geq 4$

    Correct answer: $y \geq 4$. Explanation: The minimum value of $x^2$ is 0, so the minimum value of $h(x)$ is 4.

23. The function $f(x) = 2^x$ represents an exponential growth. What is the rate of growth?
    • 100\%
    • 200\%
    • 50\%
    • 20\%

    Correct answer: 100\%

    Correct answer: 100\%. Explanation: The function doubles for each increase in $x$, indicating a 100\% growth rate.

24. What is the inverse function of $f(x) = 3x + 2$?
    • $f^{-1}(x) = \frac{x - 2}{3}$
    • $f^{-1}(x) = \frac{x + 2}{3}$
    • $f^{-1}(x) = 3x - 2$
    • $f^{-1}(x) = \frac{3}{x + 2}$

    Correct answer: $f^{-1}(x) = \frac{x - 2}{3}$

    Correct answer: $f^{-1}(x) = \frac{x - 2}{3}$. Explanation: Solve $y = 3x + 2$ for $x$ and replace $y$ with $f^{-1}(x)$.

25. If the function k(x) = |x − 5| is graphed, what is its vertex?
    • (5, 0)
    • (0, 5)
    • (−5, 0)
    • (0, −5)

    Correct answer: (5, 0)

    Correct answer: (5, 0). Explanation: The vertex of the graph of |x − a| is at (a, 0).

26. What is the domain of the function $f(x) = \sqrt{x + 3}$?
    • $x \geq -3$
    • $x > -3$
    • $x \geq 3$
    • $x > 3$

    Correct answer: $x \geq -3$

    Correct answer: $x \geq -3$. Explanation: The square root function is defined for non-negative values.

27. If f(x) = x³ and g(x) = x², what is (f ∘ g)(x)?
    • x⁵
    • x⁶
    • x⁸
    • x⁹

    Correct answer: x⁶

    Correct answer: x⁶. Explanation: Substitute g(x) into f(x) and simplify.

28. The function $f(x) = \log_2(x)$ is defined for what range of $x$?
    • $x > 0$
    • $x \geq 1$
    • $x > 1$
    • $x \geq 0$

    Correct answer: $x > 0$

    Correct answer: $x > 0$. Explanation: Logarithmic functions are defined for positive values of $x$.

29. Which function represents a linear growth with a slope of 3 and a y-intercept of −2?
    • f(x) = 3x − 2
    • f(x) = 3x + 2
    • f(x) = 2x + 3
    • f(x) = 2x − 3

    Correct answer: f(x) = 3x − 2

    Correct answer: f(x) = 3x − 2. Explanation: The function is in the form y = mx + b, where m is the slope and b is the y-intercept.

30. Given the function $h(x) = \frac{2x}{x^2 - 1}$, for which values of $x$ is $h(x)$ undefined?
    • $x = 1$ and $x = -1$
    • $x = 0$
    • $x = 2$ and $x = -2$
    • $x = 1$ and $x = -2$

    Correct answer: $x = 1$ and $x = -1$

    Correct answer: $x = 1$ and $x = -1$. Explanation: A function becomes undefined where its denominator is zero.

31. If the graph of y = f(x) passes through the point (2, 3) and is a horizontal line, what is f(5)?
    • 2
    • 3
    • 5
    • Cannot be determined

    Correct answer: 3

    Correct answer: 3. Explanation: A horizontal line indicates a constant function.

32. Which of the following represents the inverse function of $f(x) = 2x + 3$?
    • $f^{-1}(x) = \frac{x}{2} + 3$
    • $f^{-1}(x) = \frac{x - 3}{2}$
    • $f^{-1}(x) = 2x - 3$
    • $f^{-1}(x) = \frac{x + 3}{2}$

    Correct answer: $f^{-1}(x) = \frac{x - 3}{2}$

    Correct answer: $f^{-1}(x) = \frac{x - 3}{2}$. Explanation: Solve for $x$ in the equation $y = 2x + 3$ to find the inverse.

33. What is the range of the function $f(x) = \frac{1}{x - 2}$?
    • All real numbers except $x = 2$
    • All real numbers except $x = -2$
    • All real numbers except $f(x) = 0$
    • All real numbers

    Correct answer: All real numbers except $f(x) = 0$

    Correct answer: All real numbers except $f(x) = 0$. Explanation: Identify values that the function cannot take.

34. If $f(x) = x^2 - 4x$, what are the x-intercepts of the graph of $f$?
    • $x = 0$ and $x = 4$
    • $x = -2$ and $x = 2$
    • $x = 2$ and $x = -2$
    • $x = 0$ and $x = -4$

    Correct answer: $x = 0$ and $x = 4$

    Correct answer: $x = 0$ and $x = 4$. Explanation: Set $f(x)$ equal to 0 and solve for $x$.

35. If $g(x) = \sqrt{x - 1}$, what is the domain of $g$?
    • $x > 1$
    • $x \geq 1$
    • $x > 0$
    • $x \geq 0$

    Correct answer: $x \geq 1$

    Correct answer: $x \geq 1$. Explanation: Consider the values of $x$ that make the expression under the square root non-negative.

36. What is the slope of the line represented by the function f(x) = −3x + 7?
    • 7
    • -3
    • 3
    • -7

    Correct answer: -3

    Correct answer: −3. Explanation: Identify the coefficient of x in the linear function.

37. If h(x) = 5 − 2x², what is the maximum value of h(x)?
    • 5
    • 3
    • 0
    • -5

    Correct answer: 5

    Correct answer: 5. Explanation: Analyze the vertex of the parabola represented by h(x).

38. A set of data is normally distributed with a mean of 50 and a standard deviation of 5. What is the probability that a randomly selected data point is less than 45?
    • 0.1587
    • 0.3413
    • 0.4772
    • 0.8413

    Correct answer: 0.1587

    Correct answer: 0.1587. Explanation: This is 1 standard deviation below the mean in a normal distribution.

39. What is the range of the data set $\{3, 7, 7, 10, 15, 18\}$?
    • 15
    • 18
    • 21
    • 25

    Correct answer: 15

    Correct answer: 15. Explanation: Range is the difference between the highest and lowest values.

40. A box contains 5 red, 4 blue, and 3 green balls. If two balls are picked at random without replacement, what is the probability that both are green?
    • $\frac{1}{6}$
    • $\frac{1}{12}$
    • $\frac{1}{22}$
    • $\frac{1}{44}$

    Correct answer: $\frac{1}{22}$

    Correct answer: $\frac{1}{22}$. Explanation: Calculate using the probability formula for without replacement.

41. In a survey, 70\% of respondents chose chocolate as their favorite flavor. If 5 people are surveyed, what is the probability that at least 3 will choose chocolate?
    • 0.8369
    • 0.7443
    • 0.6823
    • 0.5798

    Correct answer: 0.8369

    Correct answer: 0.8369. Explanation: This is a binomial probability problem.

42. A standard six-sided die is rolled twice. What is the probability of rolling a sum of 8?
    • $\frac{1}{6}$
    • $\frac{5}{36}$
    • $\frac{1}{12}$
    • $\frac{1}{9}$

    Correct answer: $\frac{5}{36}$

    Correct answer: $\frac{5}{36}$. Explanation: Count the number of successful outcomes over the total possible outcomes.

43. If the interquartile range of a data set is 12, and the first quartile is 15, what is the third quartile?
    • 27
    • 3
    • 12
    • 15

    Correct answer: 27

    Correct answer: 27. Explanation: The third quartile is the first quartile plus the interquartile range.

44. A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?
    • $\frac{1}{32}$
    • $\frac{5}{32}$
    • $\frac{10}{32}$
    • $\frac{15}{32}$

    Correct answer: $\frac{10}{32}$

    Correct answer: $\frac{10}{32}$. Explanation: Use binomial probability formula for exactly 3 successes.

45. If the mean of a data set is 20 and the standard deviation is 4, what percentage of the data is expected to lie between 16 and 24?
    • 68\%
    • 75\%
    • 95\%
    • 99.7\%

    Correct answer: 68\%

    Correct answer: 68\%. Explanation: This is within one standard deviation of the mean in a normal distribution.

46. In a standard deck of 52 cards, what is the probability of drawing an ace or a heart?
    • $\frac{1}{13}$
    • $\frac{4}{13}$
    • $\frac{1}{4}$
    • $\frac{16}{52}$

    Correct answer: $\frac{4}{13}$

    Correct answer: $\frac{4}{13}$. Explanation: Calculate the probability considering the overlap between aces and hearts.

47. In a probability experiment, the outcomes are equally likely. If the experiment has 15 possible outcomes, what is the probability of any single outcome occurring?
    • $\frac{1}{5}$
    • $\frac{1}{10}$
    • $\frac{1}{15}$
    • $\frac{1}{20}$

    Correct answer: $\frac{1}{15}$

    Correct answer: $\frac{1}{15}$. Explanation: The probability of a single outcome in a uniform probability model is $\frac{1}{\text{number of outcomes}}$.

48. A set of data has a bell-shaped distribution with a mean of 70 and a standard deviation of 10. According to the empirical rule, what percentage of the data falls between 50 and 90?
    • 68\%
    • 95\%
    • 99.7\%
    • 100\%

    Correct answer: 95\%

    Correct answer: 95\%. Explanation: The empirical rule states that approximately 95\% of the data in a normal distribution lies within two standard deviations of the mean.

49. If two dice are rolled, what is the probability that the sum of the numbers on the dice will be greater than 10?
    • $\frac{1}{12}$
    • $\frac{1}{6}$
    • $\frac{1}{4}$
    • $\frac{5}{36}$

    Correct answer: $\frac{1}{12}$

    Correct answer: $\frac{1}{12}$. Explanation: Only the sums 11 and 12 are greater than 10, which can be achieved with a few specific combinations of dice rolls.

50. What is the mode of the data set $\{22, 27, 22, 30, 22, 31, 30, 35\}$?
    • 22
    • 27
    • 30
    • There is no mode

    Correct answer: 22

    Correct answer: 22. Explanation: The mode is the number that appears most frequently in the data set.

51. What is the median of the data set $\{12, 17, 23, 29, 34, 42, 45, 47, 53\}$?
    • 29
    • 34
    • 42
    • 45

    Correct answer: 34

    Correct answer: 34. Explanation: The median is the middle number when the data set is ordered from least to greatest.

52. If the odds in favor of an event occurring are 3 to 2, what is the probability of the event occurring?
    • $\frac{2}{5}$
    • $\frac{3}{5}$
    • $\frac{2}{3}$
    • $\frac{3}{2}$

    Correct answer: $\frac{3}{5}$

    Correct answer: $\frac{3}{5}$. Explanation: Odds of 3 to 2 mean 3 favorable outcomes for every 2 unfavorable ones, out of a total of 5 outcomes.

53. If a regular hexagon has a side length of 8 cm, what is its perimeter?
    • $24 \text{ cm}$
    • $32 \text{ cm}$
    • $40 \text{ cm}$
    • $48 \text{ cm}$

    Correct answer: $48 \text{ cm}$

    Correct answer: $48 \text{ cm}$. Explanation: The perimeter of a regular hexagon is six times its side length. Thus, $6 \times 8 = 48 \text{ cm}$.

54. What is the volume of a cube with a surface area of $150 \text{ cm}^2$?
    • $125 \text{ cm}^3$
    • $216 \text{ cm}^3$
    • $343 \text{ cm}^3$
    • $512 \text{ cm}^3$

    Correct answer: $125 \text{ cm}^3$

    Correct answer: $125 \text{ cm}^3$. Explanation: First, find the length of a side using the surface area formula. Then calculate the volume.

55. What is the area of a rhombus with diagonals of length 10 cm and 24 cm?
    • $120 \text{ cm}^2$
    • $240 \text{ cm}^2$
    • $360 \text{ cm}^2$
    • $480 \text{ cm}^2$

    Correct answer: $120 \text{ cm}^2$

    Correct answer: $120 \text{ cm}^2$. Explanation: The area of a rhombus is $\frac{1}{2} \times d_1 \times d_2$. Substituting the diagonals, $\frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^2$.

56. If a right triangle has one angle measuring 30 degrees, what is the length of the hypotenuse if the opposite side is 6 cm?
    • 6 $\text{cm}$
    • 12 $\text{cm}$
    • 6$\sqrt{3}$ $\text{cm}$
    • 12$\sqrt{3}$ $\text{cm}$

    Correct answer: 12 $\text{cm}$

    Correct answer: 12 $\text{cm}$. Explanation: Apply the properties of 30-60-90 triangles.

57. What is the sum of the interior angles of a pentagon?
    • $360^\circ$
    • $540^\circ$
    • $720^\circ$
    • $900^\circ$

    Correct answer: $540^\circ$

    Correct answer: $540^\circ$. Explanation: The sum of the interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$. For a pentagon, $n = 5$, so the sum is $3 \times 180^\circ = 540^\circ$.

58. If the base of a right pyramid is a square with side length 6 cm and the height of the pyramid is 9 cm, what is the volume?
    • $54 \text{ cm}^3$
    • $108 \text{ cm}^3$
    • $162 \text{ cm}^3$
    • $216 \text{ cm}^3$

    Correct answer: $108 \text{ cm}^3$

    Correct answer: $108 \text{ cm}^3$. Explanation: The volume of a pyramid is $\frac{1}{3} \times \text{base area} \times \text{height}$. The base area is $6^2 = 36 \text{ cm}^2$. Thus, $\frac{1}{3} \times 36 \times 9 = 108 \text{ cm}^3$.

59. A right circular cone has a base radius of 6 cm and a height of 8 cm. What is the volume of the cone?
    • $96\pi \text{ cm}^3$
    • $144\pi \text{ cm}^3$
    • $192\pi \text{ cm}^3$
    • $288\pi \text{ cm}^3$

    Correct answer: $96\pi \text{ cm}^3$

    Correct answer: $96\pi \text{ cm}^3$. Explanation: The volume of a cone is $\frac{1}{3}\pi r^2h$. Substituting $r = 6$ and $h = 8$, the volume is $\frac{1}{3}\pi \times 6^2 \times 8 = 96\pi \text{ cm}^3$.

60. What is the volume of a cylinder with a radius of 3 cm and a height of 10 cm?
    • 90 $\text{cm}^3$
    • 180 $\text{cm}^3$
    • 270 $\text{cm}^3$
    • 282.6 $\text{cm}^3$

    Correct answer: 282.6 $\text{cm}^3$

    Correct answer: 282.6 $\text{cm}^3$. Explanation: Use the formula for the volume of a cylinder.

61. What is the value of the absolute value expression |3 - 11|?
    • -8
    • 14
    • -14
    • 8

    Correct answer: 8

    The value is 8. Absolute value measures distance from zero on the number line, so it is never negative. First evaluate inside the bars: 3 - 11 = -8. The absolute value of -8 is 8 because -8 is eight units from zero.

62. A student computes -7 + 4 and writes -11. Which statement best describes the student's error and the correct result?
    • The student added the absolute values instead of subtracting; the correct sum is -3
    • The student should have gotten -3 by adding the absolute values
    • The student should have gotten 11 because a negative plus a positive is positive
    • The student is correct because two numbers always add their absolute values; the sum is -11

    Correct answer: The student added the absolute values instead of subtracting; the correct sum is -3

    The correct sum is -3, and the student added the absolute values (7 + 4) instead of subtracting them. When adding integers with different signs, subtract the smaller absolute value from the larger (7 - 4 = 3) and keep the sign of the number with the larger absolute value, which is negative here. So -7 + 4 = -3.

63. Which list correctly orders the fractions 2/3, 3/5, and 5/8 from least to greatest?
    • 3/5, 5/8, 2/3
    • 3/5, 2/3, 5/8
    • 5/8, 3/5, 2/3
    • 2/3, 5/8, 3/5

    Correct answer: 3/5, 5/8, 2/3

    The correct order from least to greatest is 3/5, 5/8, 2/3. Rewriting each over a common denominator of 120 gives 2/3 = 80/120, 3/5 = 72/120, and 5/8 = 75/120. Comparing numerators, 72 < 75 < 80, so 3/5 < 5/8 < 2/3.

64. Convert the percent 7.5% to a decimal.
    • 0.75
    • 0.075
    • 7.5
    • 0.0075

    Correct answer: 0.075

    7.5% equals 0.075. To convert a percent to a decimal, divide by 100, which moves the decimal point two places to the left: 7.5 becomes 0.075. Writing 0.75 moves the point only one place, which would correspond to 75%.

65. What is the decimal equivalent of the fraction 5/8?
    • 0.58
    • 0.85
    • 0.375
    • 0.625

    Correct answer: 0.625

    5/8 equals 0.625. To convert a fraction to a decimal, divide the numerator by the denominator: 5 divided by 8 is 0.625. The value 0.375 is the decimal for 3/8, a common mix-up with the complement of 5/8.

66. A jacket costs $80, and a store applies a 35% discount. How much is the discount in dollars?
    • $45
    • $28
    • $24
    • $52

    Correct answer: $28

    The discount is $28. To find a percent of a number, convert the percent to a decimal and multiply: 35% = 0.35, and 0.35 times 80 = 28. The value $52 is the sale price after the discount, not the discount amount itself.

67. What is the greatest common factor of 48 and 60?
    • 24
    • 6
    • 12
    • 4

    Correct answer: 12

    The greatest common factor is 12. Using prime factorization, 48 = 2 to the 4th power times 3, and 60 = 2 squared times 3 times 5. The GCF takes the lowest power of each shared prime: 2 squared times 3 = 4 times 3 = 12. Choosing 24 takes too high a power of 2 that 60 does not contain.

68. What is the least common multiple of 9 and 15?
    • 45
    • 90
    • 3
    • 135

    Correct answer: 45

    The least common multiple is 45. Factoring gives 9 = 3 squared and 15 = 3 times 5. The LCM takes the highest power of each prime present: 3 squared times 5 = 9 times 5 = 45. Multiplying 9 times 15 = 135 overcounts the shared factor of 3.

69. Solve the proportion 4/9 = x/63 for x.
    • 36
    • 28
    • 21
    • 14

    Correct answer: 28

    The value of x is 28. To solve a proportion, cross-multiply: 4 times 63 = 9 times x, so 252 = 9x and x = 28. Alternatively, 63 divided by 9 is 7, and 4 times 7 = 28.

70. A recipe that serves 4 people uses 6 cups of flour. Using the same ratio, how many cups of flour are needed to serve 10 people?
    • 20
    • 12
    • 15
    • 18

    Correct answer: 15

    15 cups are needed. Set up the proportion 6 cups over 4 people equals x cups over 10 people. Cross-multiplying gives 4x = 60, so x = 15. The unit rate is 6 divided by 4 = 1.5 cups per person, and 1.5 times 10 = 15.

71. The mass of a hydrogen atom is about 0.00000000000000000000000000167 kg. Which expression writes this number correctly in scientific notation?
    • 1.67 x 10 to the 27 power
    • 1.67 x 10 to the -27 power
    • 16.7 x 10 to the -28 power
    • 0.167 x 10 to the -26 power

    Correct answer: 1.67 x 10 to the -27 power

    The correct form is 1.67 x 10 to the -27 power. Scientific notation requires the coefficient to be at least 1 and less than 10, and the exponent counts how many places the decimal moves. For a number this small, the exponent is negative. Writing 16.7 times a power of ten is invalid because 16.7 is not between 1 and 10.

72. Evaluate the expression 6 + 2 x (5 - 1) squared.
    • 38
    • 256
    • 26
    • 68

    Correct answer: 38

    The value is 38. Following the order of operations, work inside parentheses first: 5 - 1 = 4. Then apply the exponent: 4 squared = 16. Next multiply: 2 times 16 = 32. Finally add: 6 + 32 = 38. Getting 68 comes from adding 6 + 2 before multiplying, which violates the order of operations.

73. A population grows from 250 to 320. What is the percent increase, rounded to the nearest whole percent?
    • 28%
    • 22%
    • 70%
    • 32%

    Correct answer: 28%

    The percent increase is 28%. The percent increase formula is the change in value divided by the original value, times 100. The change is 320 - 250 = 70, and 70 divided by 250 = 0.28, which is 28%. Dividing 70 by the new value 320 instead of the original 250 incorrectly gives about 22%.

74. What is the prime factorization of 180?
    • 2 cubed x 3 x 5
    • 2 x 3 squared x 5 squared
    • 2 squared x 3 x 15
    • 2 squared x 3 squared x 5

    Correct answer: 2 squared x 3 squared x 5

    The prime factorization of 180 is 2 squared x 3 squared x 5. Dividing step by step: 180 = 2 x 90, 90 = 2 x 45, 45 = 3 x 15, and 15 = 3 x 5. Collecting the primes gives two 2s, two 3s, and one 5. The option containing 15 is not a valid prime factorization because 15 is composite.

75. Which of the following numbers is irrational?
    • 3/8
    • $\sqrt{7}$
    • $\sqrt{16}$
    • 0.272727 repeating

    Correct answer: $\sqrt{7}$

    $\sqrt{7}$ is irrational. A rational number can be written as a ratio of two integers and its decimal form either terminates or repeats; an irrational number cannot. $\sqrt{7}$ is about 2.6457513... with no repeating pattern, so it is irrational. $\sqrt{16}$ equals 4, an integer, and 0.272727 repeating is rational because it can be written as $\frac{3}{11}$.

76. A teacher wants students to see why -3 - 5 equals -8 rather than -2. Which number-line model best represents this subtraction?
    • Start at -3 and move 5 units in the positive direction, landing on 2
    • Start at -5 and move 3 units right, landing on -2
    • Start at -3 and move 5 units further in the negative direction, landing on -8
    • Start at 3 and move 5 units left, landing on -2

    Correct answer: Start at -3 and move 5 units further in the negative direction, landing on -8

    The model that starts at -3 and moves 5 units in the negative direction to land on -8 is correct. Subtracting a positive number means moving left (more negative) on the number line, so -3 - 5 = -8. Moving in the positive direction would model adding 5, which gives the common wrong answer of 2.

77. Which fraction is equivalent to the repeating decimal 0.4 with the 4 repeating?
    • 4/10
    • 4/99
    • 4/9
    • 2/5

    Correct answer: 4/9

    The repeating decimal 0.444... equals 4/9. Let x = 0.444...; then 10x = 4.444..., and subtracting gives 9x = 4, so x = 4/9. The fraction 4/10 equals the terminating decimal 0.4, which is not the same as the repeating value.

78. A coat originally priced at $120 is marked down to $90. What is the percent decrease?
    • 15%
    • 33%
    • 25%
    • 30%

    Correct answer: 25%

    The percent decrease is 25%. Using the percent change formula, the decrease is 120 - 90 = 30, and 30 divided by the original price 120 = 0.25, or 25%. Dividing by the new price of 90 would incorrectly give about 33%.

79. Which value of the absolute value expressions is the greatest: |-9|, |6|, |-7|, or |0|?
    • |-7|
    • |-9|
    • |0|
    • |6|

    Correct answer: |-9|

    |-9| is the greatest at a value of 9. Absolute value gives distance from zero, so |-9| = 9, |6| = 6, |-7| = 7, and |0| = 0. Since 9 is the largest, |-9| is greatest. A common error is treating -9 as the smallest because the number itself is most negative, but the bars remove the sign.

80. In a parking lot, the ratio of cars to trucks is 5 to 2. If there are 35 vehicles total, how many are trucks?
    • 25
    • 14
    • 10
    • 7

    Correct answer: 10

    There are 10 trucks. The ratio 5 to 2 means the parts sum to 5 + 2 = 7 equal shares. Dividing 35 vehicles by 7 gives 5 vehicles per share, and trucks get 2 shares, so 2 times 5 = 10. The 25 represents the number of cars, not trucks.

81. Which expression equals the product of (3 x 10 to the 4 power) and (2 x 10 to the 5 power), written in scientific notation?
    • 5 x 10 to the 9 power
    • 6 x 10 to the 1 power
    • 6 x 10 to the 9 power
    • 6 x 10 to the 20 power

    Correct answer: 6 x 10 to the 9 power

    The product is 6 x 10 to the 9 power. Multiply the coefficients (3 times 2 = 6) and add the exponents of the powers of ten (4 + 5 = 9). The result, 6 x 10 to the 9 power, already has a coefficient between 1 and 10, so no further adjustment is needed. Multiplying the exponents instead of adding them gives the wrong power of 20.

82. Which choice shows 5/12, 0.4, and 43% ordered from least to greatest?
    • 43%, 0.4, 5/12
    • 5/12, 0.4, 43%
    • 0.4, 43%, 5/12
    • 0.4, 5/12, 43%

    Correct answer: 0.4, 5/12, 43%

    The correct order is 0.4, 5/12, 43%. Convert everything to decimals: 5/12 is about 0.4167, 0.4 stays 0.4, and 43% equals 0.43. Comparing 0.40 < 0.4167 < 0.43 gives the order. The most reliable comparison method is converting fractions and percents to a common decimal form first.

83. A class found that 18 is 24% of the number of pages in a book. Approximately how many pages are in the book?
    • 75
    • 60
    • 432
    • 43

    Correct answer: 75

    The book has 75 pages. When a part and its percent are known, divide the part by the decimal form of the percent: 18 divided by 0.24 = 75. Multiplying 18 by 0.24 instead would shrink the value rather than recover the whole, giving the incorrect 4.32.

84. Which of the following sets contains only rational numbers?
    • $\{\pi, 3, -1\}$
    • $\{\sqrt{5}, \frac{1}{2}, 4\}$
    • $\{\sqrt{2}, 0.5, \frac{7}{4}\}$
    • $\{-3, 0.5, \frac{7}{4}, 0.\overline{6}\}$

    Correct answer: $\{-3, 0.5, \frac{7}{4}, 0.\overline{6}\}$

    The set $\{-3, 0.5, \frac{7}{4}, 0.\overline{6}\}$ contains only rational numbers. Every element can be written as a ratio of two integers: $-3 = \frac{-3}{1}$, $0.5 = \frac{1}{2}$, $\frac{7}{4}$ is already a fraction, and $0.\overline{6}$ equals $\frac{2}{3}$. The other sets each include an irrational number such as $\sqrt{2}$, $\sqrt{5}$, or $\pi$, whose decimals never terminate or repeat.

85. A student simplifies 12 divided by 2 + 1 as 12 divided by 3 to get 4. Which statement correctly identifies the issue?
    • The student is correct because addition always comes before division
    • Without parentheses, division is done before addition, so the value is 6 + 1 = 7
    • The expression has no defined value because operations are ambiguous
    • The expression equals 4 only if a calculator is used

    Correct answer: Without parentheses, division is done before addition, so the value is 6 + 1 = 7

    Without parentheses, the value is 7, computed as 12 divided by 2 first (giving 6), then plus 1. The order of operations performs multiplication and division before addition and subtraction, so the student wrongly added 2 + 1 first. Writing the expression as (12 divided by 2) + 1 makes the intended grouping explicit.

86. Which number has exactly three distinct prime factors?
    • 12
    • 49
    • 30
    • 16

    Correct answer: 30

    30 has exactly three distinct prime factors. Its prime factorization is 2 x 3 x 5, which lists three different primes. By contrast, 16 = 2 to the 4th power (one prime), 12 = 2 squared x 3 (two primes), and 49 = 7 squared (one prime), so none of those has three distinct prime factors.

87. A student is asked to use the distributive property to rewrite 4(2x + 5). Which expression is the correct result?
    • 2x + 20
    • 8x + 20
    • 6x + 9
    • 8x + 5

    Correct answer: 8x + 20

    The correct result is 8x + 20. The distributive property says you multiply the factor outside the parentheses by each term inside, so 4 times 2x is 8x and 4 times 5 is 20. A common mistake is distributing only to the first term and writing 8x + 5, which leaves the 5 un-multiplied.

88. Simplify the expression 7a + 3b - 2a + 5b by combining like terms.
    • 13ab
    • 5a + 2b
    • 9a + 2b
    • 5a + 8b

    Correct answer: 5a + 8b

    The simplified expression is 5a + 8b. Like terms share the same variable, so combine the a-terms (7a - 2a = 5a) and the b-terms (3b + 5b = 8b) separately. You cannot merge 5a and 8b into a single term because a and b are different variables, which rules out an answer like 13ab.

89. Solve the linear equation 3x - 7 = 11 for x.
    • X = 6
    • X = 18
    • X = 12
    • X = 4

    Correct answer: X = 6

    The solution is x = 6. Add 7 to both sides to get 3x = 18, then divide both sides by 3 to find x = 6. Forgetting to add 7 before dividing, or dividing 11 by 3, leads to incorrect values.

90. Solve the two-step equation (x / 4) + 6 = 10 for x.
    • X = 64
    • X = 4
    • X = 1
    • X = 16

    Correct answer: X = 16

    The solution is x = 16. First subtract 6 from both sides to get x / 4 = 4, then multiply both sides by 4 to get x = 16. Undoing the operations in reverse order of how they were applied is the key idea behind solving two-step equations.

91. Which expression is the fully simplified form of 3(2x - 4) + 5x?
    • 11x - 12
    • 11x - 4
    • 6x - 12
    • 6x - 4 + 5x

    Correct answer: 11x - 12

    The simplified form is 11x - 12. First distribute the 3 to get 6x - 12, then combine the like terms 6x and 5x to get 11x, leaving 11x - 12. Stopping at 6x - 12 + 5x or only distributing to one term gives an incomplete or wrong answer.

92. Solve the system of equations: x + y = 12 and x - y = 4. What is the value of x?
    • X = 16
    • X = 8
    • X = 4
    • X = 6

    Correct answer: X = 8

    The value of x is 8. Adding the two equations eliminates y because +y and -y cancel, giving 2x = 16, so x = 8 (and y = 4). The elimination method works cleanly here because the y-coefficients are already opposites.

93. Solve the inequality 2x + 3 < 11 for x.
    • X < 7
    • X > 4
    • X < 4
    • X < 8

    Correct answer: X < 4

    The solution is x < 4. Subtract 3 from both sides to get 2x < 8, then divide both sides by 2 to get x < 4. Because you divide by a positive number, the inequality direction stays the same.

94. Solve the quadratic equation $x^2 - 7x + 12 = 0$ by factoring. What are the solutions?
    • X = 3 and x = 4
    • X = 2 and x = 6
    • X = -3 and x = -4
    • X = 1 and x = 12

    Correct answer: X = 3 and x = 4

    The solutions are x = 3 and x = 4. The trinomial factors as (x - 3)(x - 4) = 0 because -3 and -4 multiply to 12 and add to -7. Setting each factor equal to zero gives x = 3 and x = 4.

95. Which equation correctly represents this situation: 'A number n decreased by 8 equals three times the number'?
    • 8 - n = 3n
    • N - 8 = 3n
    • N - 8 = n + 3
    • 3n - 8 = n

    Correct answer: N - 8 = 3n

    The correct equation is n - 8 = 3n. 'A number decreased by 8' translates to n - 8, and 'three times the number' translates to 3n, and 'equals' joins them. Reversing the subtraction to 8 - n changes the meaning, since 'decreased by' subtracts 8 from n, not n from 8.

96. Using the laws of exponents, simplify $x^5 \times x^3$.
    • $x^8$
    • $x^{15}$
    • $2x^8$
    • $x^2$

    Correct answer: $x^8$

    The result is $x^8$. When multiplying powers with the same base, you add the exponents (the product rule), so 5 + 3 = 8. Multiplying the exponents to get $x^{15}$ is a common error that confuses the product rule with the power-of-a-power rule.

97. Use the quadratic formula to find the solutions of $x^2 + 2x - 8 = 0$.
    • X = -2 and x = 4
    • X = 2 and x = -4
    • X = 2 and x = 4
    • X = -1 and x = 8

    Correct answer: X = 2 and x = -4

    The solutions are $x = 2$ and $x = -4$. With $a = 1$, $b = 2$, $c = -8$, the discriminant is $2^2 - 4(1)(-8) = 4 + 32 = 36$, and its square root is 6, so $x = \frac{-2 \pm 6}{2}$, giving 2 and -4. These match the factors $(x - 2)(x + 4)$.

98. On a number line, which description correctly graphs the inequality x >= -2?
    • A closed (filled) circle at -2 with shading extending to the left
    • An open circle at -2 with shading extending to the right
    • An open circle at -2 with shading extending to the left
    • A closed (filled) circle at -2 with shading extending to the right

    Correct answer: A closed (filled) circle at -2 with shading extending to the right

    The correct graph is a closed (filled) circle at -2 with shading to the right. The 'greater than or equal to' symbol includes -2 itself, so the circle is filled, and 'greater than' means values larger than -2, which lie to the right on a number line. An open circle would be used only for strict inequalities like greater-than without the equal-to.

99. Factor the polynomial $x^2 - 9$ completely.
    • $(x - 3)(x + 3)$
    • $(x + 3)(x + 3)$
    • $(x - 9)(x + 1)$
    • $(x - 3)(x - 3)$

    Correct answer: $(x - 3)(x + 3)$

    The factored form is $(x - 3)(x + 3)$. This is a difference of two squares, $x^2 - 9 = x^2 - 3^2$, which always factors as $(x - a)(x + a)$. Squaring a single binomial like $(x - 3)(x - 3)$ would produce $x^2 - 6x + 9$, not $x^2 - 9$.

100. A rectangle has a length that is 3 meters more than its width w, and its perimeter is 26 meters. Which equation models this situation?
     • W(w + 3) = 26
     • W + (w + 3) = 26
     • 2w + (w + 3) = 26
     • 2w + 2(w + 3) = 26

     Correct answer: 2w + 2(w + 3) = 26

     The correct model is 2w + 2(w + 3) = 26. Perimeter of a rectangle is 2 times length plus 2 times width; here width is w and length is w + 3, so the perimeter is 2w + 2(w + 3). The equation w + (w + 3) = 26 mistakenly uses only one length and one width instead of doubling each.

101. Simplify the expression $\frac{6x^3}{2x}$ using the laws of exponents.
     • $4x^2$
     • $3x^3$
     • $3x^4$
     • $3x^2$

     Correct answer: $3x^2$

     The result is $3x^2$. Divide the coefficients (6 divided by 2 = 3) and subtract the exponents of $x$ (3 - 1 = 2) using the quotient rule, giving $3x^2$. Subtracting exponents, not dividing them, is the rule when dividing powers with the same base.

102. Solve the multi-step equation 5(x - 2) = 3x + 4 for x.
     • X = 1
     • X = 3
     • X = -3
     • X = 7

     Correct answer: X = 7

     The solution is x = 7. Distribute to get 5x - 10 = 3x + 4, subtract 3x from both sides to get 2x - 10 = 4, add 10 to get 2x = 14, then divide by 2 to find x = 7. Collecting variable terms on one side and constants on the other is the standard strategy.

103. A student writes that $(x + 4)^2$ equals $x^2 + 16$. What is the error in the student's work?
     • The student should have written $x^2 - 16$
     • The student should have written $x^2 + 4$
     • The student forgot the middle term $8x$, since $(x + 4)^2 = x^2 + 8x + 16$
     • There is no error; the work is correct

     Correct answer: The student forgot the middle term $8x$, since $(x + 4)^2 = x^2 + 8x + 16$

     The error is that the student omitted the middle term $8x$. Squaring a binomial means $(x + 4)(x + 4)$, which by the distributive property gives $x^2 + 4x + 4x + 16 = x^2 + 8x + 16$. The student incorrectly squared each term separately, a frequent algebra misconception.

104. Which of these is a correct example of using the distributive property to expand -2(3x - 7)?
     • -6x - 7
     • 6x - 14
     • -6x + 14
     • -6x - 14

     Correct answer: -6x + 14

     The correct expansion is -6x + 14. Distributing -2 to each term gives -2 times 3x = -6x and -2 times -7 = +14 (a negative times a negative is positive). Writing -6x - 14 incorrectly keeps the second product negative.

105. Solve the inequality -3x > 12 for x.
     • X < 4
     • X > -4
     • X < -4
     • X > 4

     Correct answer: X < -4

     The solution is x < -4. Dividing both sides by -3 isolates x, but dividing or multiplying an inequality by a negative number reverses the inequality sign, so 'greater than' becomes 'less than.' Forgetting to flip the sign is the most common error and would wrongly give x > -4.

106. Solve the system using substitution: y = 2x and 3x + y = 15. What is the value of x?
     • X = 5
     • X = 3
     • X = 6
     • X = 15

     Correct answer: X = 3

     The value of x is 3. Substitute y = 2x into the second equation to get 3x + 2x = 15, which simplifies to 5x = 15, so x = 3 (and y = 6). Substitution works well here because one equation is already solved for y.

107. Solve the quadratic equation $x^2 - 5x = 0$ by factoring. What are the solutions?
     • X = -5 and x = 5
     • X = 0 and x = -5
     • X = 0 and x = 5
     • X = 5 only

     Correct answer: X = 0 and x = 5

     The solutions are x = 0 and x = 5. Factor out the common factor x to get x(x - 5) = 0, then set each factor equal to zero: x = 0 or x - 5 = 0. Dividing both sides by x at the start would lose the valid solution x = 0.

108. Using the laws of exponents, simplify $(x^4)^3$.
     • $x^{12}$
     • $x^{64}$
     • $3x^4$
     • $x^7$

     Correct answer: $x^{12}$

     The result is $x^{12}$. The power-of-a-power rule says to multiply the exponents, so 4 times 3 = 12. Adding the exponents to get $x^7$ confuses this rule with the product rule used when multiplying like bases.

109. A phone plan charges a flat fee of $20 per month plus $0.10 per minute. Which equation gives the total monthly cost C for m minutes?
     • C = 0.10 + 20m
     • C = 20 + 0.10m
     • C = 20m + 0.10
     • C = 20 + 10m

     Correct answer: C = 20 + 0.10m

     The correct equation is C = 20 + 0.10m. The flat fee of $20 does not depend on minutes, so it is the constant, while the per-minute charge of $0.10 is multiplied by the number of minutes m. Placing the variable on the wrong number, as in 0.10 + 20m, reverses the roles of the rate and the fixed fee.

110. Which expression results from combining like terms in $4x^2 + 3x - x^2 + 5x$?
     • $4x^2 + 8x$
     • $3x^2 + 2x$
     • $11x^3$
     • $3x^2 + 8x$

     Correct answer: $3x^2 + 8x$

     The result is $3x^2 + 8x$. The $x^2$ terms combine as $4x^2 - x^2 = 3x^2$, and the $x$ terms combine as $3x + 5x = 8x$. Terms with $x^2$ and terms with $x$ are not like terms, so they cannot be merged together.

111. Solve the two-step equation 2x - 5 = 9 for x.
     • X = 4
     • X = 7
     • X = 2
     • X = 14

     Correct answer: X = 7

     The solution is x = 7. Add 5 to both sides to get 2x = 14, then divide both sides by 2 to get x = 7. Reversing the operations (undo subtraction, then undo multiplication) is the core technique for two-step equations.

112. Use the quadratic formula to find the solutions of $2x^2 - 4x - 6 = 0$.
     • X = -3 and x = 1
     • X = 3 and x = -1
     • X = 2 and x = -1
     • X = 3 and x = 1

     Correct answer: X = 3 and x = -1

     The solutions are $x = 3$ and $x = -1$. With $a = 2$, $b = -4$, $c = -6$, the discriminant is $(-4)^2 - 4(2)(-6) = 16 + 48 = 64$, whose square root is 8, so $x = \frac{4 \pm 8}{4}$, giving 3 and -1. Dividing the whole equation by 2 first gives $x^2 - 2x - 3 = 0$, which factors to the same roots.

113. Factor the trinomial $x^2 + 6x + 9$ completely.
     • $(x + 3)(x - 3)$
     • $(x + 3)^2$
     • $(x + 9)(x + 1)$
     • $(x + 6)(x + 3)$

     Correct answer: $(x + 3)^2$

     The factored form is $(x + 3)^2$, written as $(x + 3)(x + 3)$. This is a perfect-square trinomial because 9 is 3 squared and 6 is twice 3. The difference-of-squares form $(x + 3)(x - 3)$ would instead produce $x^2 - 9$.

114. A teacher wants students to see that 3(x + 2) and 3x + 6 are equivalent. Which representation best demonstrates this equivalence visually?
     • A number line with the point x = 2 marked
     • A bar graph comparing the values 3 and 6
     • An area model showing a rectangle of width 3 split into a part of area 3x and a part of area 6
     • A pie chart divided into three equal sections

     Correct answer: An area model showing a rectangle of width 3 split into a part of area 3x and a part of area 6

     The best representation is an area model showing a rectangle of width 3 split into regions of area 3x and 6. A rectangle with width 3 and length (x + 2) has total area 3(x + 2), and partitioning the length into x and 2 shows the two sub-areas 3x and 6, making the distributive property concrete. Number lines, bar graphs, and pie charts do not illustrate the distribution of multiplication over addition.

115. Solve the compound process: simplify and solve 4(x + 1) - 2x = 14 for x.
     • X = 5
     • X = 6
     • X = 7
     • X = 3

     Correct answer: X = 5

     The solution is x = 5. Distribute to get 4x + 4 - 2x = 14, combine like terms to get 2x + 4 = 14, subtract 4 to get 2x = 10, then divide by 2 to find x = 5. Simplifying the left side before solving keeps the equation manageable.

116. A student solves -2x + 5 < 9 and writes x < -2. Identify the student's error.
     • The student did not reverse the inequality sign when dividing by -2; the correct answer is x > -2
     • The student should have gotten x < 2
     • There is no error; x < -2 is correct
     • The student should have added 5 instead of subtracting it

     Correct answer: The student did not reverse the inequality sign when dividing by -2; the correct answer is x > -2

     The error is failing to reverse the inequality sign when dividing by a negative number. Subtracting 5 gives -2x < 4, and dividing both sides by -2 requires flipping the sign, producing x > -2. The student kept the original direction, a classic mistake when working with inequalities.

117. Which expression is equivalent to $5^0 \times 5^2$ using the laws of exponents?
     • 0
     • 25
     • 10
     • 5

     Correct answer: 25

     The expression equals 25. Any nonzero base raised to the zero power equals 1, so $5^0 = 1$, and $5^2 = 25$, giving $1 \times 25 = 25$. Treating $5^0$ as 0 is a common error that would wrongly make the whole product 0.

118. Two consecutive even integers have a sum of 46. Which equation can be used to find the smaller integer n, and what is that integer?
     • N + (n + 1) = 46, giving n = 22
     • 2n = 46, giving n = 23
     • N + (n + 2) = 46, giving n = 22
     • N + (n + 2) = 46, giving n = 24

     Correct answer: N + (n + 2) = 46, giving n = 22

     The correct equation is n + (n + 2) = 46, and the smaller integer is 22. Consecutive even integers differ by 2, so the next even integer after n is n + 2; solving 2n + 2 = 46 gives 2n = 44 and n = 22 (the integers are 22 and 24). Using n + 1 would model consecutive integers, not consecutive even integers.

119. A circular garden has a radius of 7 meters. Using 3.14 for pi, which value is closest to the area of the garden?
     • 307.72 square meters
     • 21.98 square meters
     • 43.96 square meters
     • 153.86 square meters

     Correct answer: 153.86 square meters

     The area is about 153.86 square meters. The area of a circle is found with A = pi times r squared, so A = 3.14 times 7 times 7 = 3.14 times 49, which is about 153.86 square meters. The value near 43.96 comes from mistakenly using the circumference formula (2 times pi times r) instead of the area formula.

120. A circular fountain has a diameter of 14 feet. Using 3.14 for pi, approximately how far would a person walk in going exactly once around the edge of the fountain?
     • 43.96 feet
     • 21.98 feet
     • 153.86 feet
     • 87.92 feet

     Correct answer: 43.96 feet

     The distance around is about 43.96 feet. Circumference is C = pi times diameter, so C = 3.14 times 14, which is about 43.96 feet. Using the radius (7) in C = pi times d by mistake, or confusing circumference with area, produces the other choices; walking once around a circle measures its circumference, not its area.

121. A triangle has a base of 12 centimeters and a height of 5 centimeters. What is its area?
     • 34 square centimeters
     • 17 square centimeters
     • 30 square centimeters
     • 60 square centimeters

     Correct answer: 30 square centimeters

     The area is 30 square centimeters. The area of a triangle is one-half times base times height, so A = (1/2) times 12 times 5 = (1/2) times 60 = 30 square centimeters. The value 60 results from forgetting the one-half factor, which is the most common error with the triangle area formula.

122. A trapezoid has parallel sides (bases) of 8 inches and 14 inches and a height of 6 inches. What is its area?
     • 48 square inches
     • 66 square inches
     • 42 square inches
     • 132 square inches

     Correct answer: 66 square inches

     The area is 66 square inches. The area of a trapezoid is one-half times the sum of the two parallel bases times the height: A = (1/2) times (8 + 14) times 6 = (1/2) times 22 times 6 = 11 times 6 = 66 square inches. Skipping the one-half factor gives 132, a frequent mistake.

123. A rectangular swimming pool is 15 meters long and 9 meters wide. What is the perimeter of the pool?
     • 24 meters
     • 69 meters
     • 48 meters
     • 135 meters

     Correct answer: 48 meters

     The perimeter is 48 meters. The perimeter of a rectangle is 2 times length plus 2 times width: P = 2 times 15 + 2 times 9 = 30 + 18 = 48 meters. The value 135 is the area (length times width), and 24 is just length plus width without doubling.

124. A right triangle has legs measuring 9 cm and 12 cm. What is the length of the hypotenuse?
     • 21 cm
     • 15 cm
     • 225 cm
     • 10.5 cm

     Correct answer: 15 cm

     The hypotenuse is 15 cm. By the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the legs: $c^2 = 9^2 + 12^2 = 81 + 144 = 225$, so $c = \sqrt{225} = 15$ cm. The value 225 is $c^2$, not $c$, so students must remember to take the square root.

125. Which of the following sets of three whole numbers is a Pythagorean triple?
     • 9, 10, 14
     • 8, 15, 17
     • 5, 6, 8
     • 6, 7, 8

     Correct answer: 8, 15, 17

     The set 8, 15, 17 is a Pythagorean triple. A Pythagorean triple is three positive integers a, b, c where a squared plus b squared equals c squared; here 8 squared plus 15 squared = 64 + 225 = 289, and 17 squared = 289, so the equation holds. None of the other sets satisfy a squared plus b squared = c squared, so they cannot form a right triangle with integer sides.

126. What is the distance between the points (2, 3) and (8, 11) in the coordinate plane?
     • 6
     • 14
     • 8
     • 10

     Correct answer: 10

     The distance is 10. The distance formula gives $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(8 - 2)^2 + (11 - 3)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. The value 14 comes from simply adding the horizontal and vertical changes (6 + 8) instead of using the Pythagorean-based distance formula.

127. In triangle ABC, angle A measures 47 degrees and angle B measures 68 degrees. What is the measure of angle C?
     • 65 degrees
     • 135 degrees
     • 45 degrees
     • 115 degrees

     Correct answer: 65 degrees

     Angle C measures 65 degrees. The interior angles of any triangle sum to 180 degrees, so angle C = 180 minus 47 minus 68 = 180 minus 115 = 65 degrees. The value 115 is the sum of the two known angles, not the missing angle.

128. Two angles are complementary. If one angle measures 35 degrees, what is the measure of the other angle?
     • 65 degrees
     • 325 degrees
     • 55 degrees
     • 145 degrees

     Correct answer: 55 degrees

     The other angle measures 55 degrees. Complementary angles add to 90 degrees, so the missing angle is 90 minus 35 = 55 degrees. Supplementary angles (not complementary) add to 180 degrees, which would give 145; mixing up these two terms is the usual error.

129. Angle P and angle Q are supplementary. If angle P measures 110 degrees, what is the measure of angle Q?
     • 90 degrees
     • 20 degrees
     • 250 degrees
     • 70 degrees

     Correct answer: 70 degrees

     Angle Q measures 70 degrees. Supplementary angles have measures that add up to 180 degrees, so angle Q = 180 minus 110 = 70 degrees. Using 90 degrees (the complementary sum) would give 20 degrees, which confuses supplementary with complementary angles.

130. An angle measures 124 degrees. How is this angle best classified?
     • Right
     • Acute
     • Straight
     • Obtuse

     Correct answer: Obtuse

     A 124-degree angle is obtuse. An obtuse angle measures more than 90 degrees but less than 180 degrees, and 124 falls in that range. An acute angle is less than 90 degrees, a right angle is exactly 90 degrees, and a straight angle is exactly 180 degrees, so none of those fit a 124-degree angle.

131. Triangle ABC is similar to triangle DEF. In triangle ABC, side AB = 6 cm corresponds to side DE = 9 cm in triangle DEF. If side BC = 8 cm, what is the length of the corresponding side EF?
     • 11 cm
     • 12 cm
     • 14 cm
     • 5.33 cm

     Correct answer: 12 cm

     Side EF is 12 cm. In similar figures corresponding sides are proportional, so the scale factor from ABC to DEF is 9 divided by 6 = 1.5, and EF = 8 times 1.5 = 12 cm. Adding the difference of 3 cm to get 11 (instead of multiplying by the scale factor) is a common proportional-reasoning error.

132. A teacher asks students to explain the difference between congruent figures and similar figures. Which statement is correct?
     • Congruent and similar both mean the figures are identical in size and shape
     • Congruent figures have the same shape and size, while similar figures have the same shape but not necessarily the same size
     • Congruent figures have the same shape but different size, while similar figures are identical in every way
     • Similar figures must have the same area, while congruent figures need not

     Correct answer: Congruent figures have the same shape and size, while similar figures have the same shape but not necessarily the same size

     Congruent figures have the same shape and size, while similar figures have the same shape but not necessarily the same size. Congruence requires corresponding sides and angles to be equal, so the figures match exactly; similarity requires only equal corresponding angles and proportional corresponding sides, so similar figures can be scaled versions of each other. All congruent figures are similar (with a scale factor of 1), but not all similar figures are congruent.

133. In which quadrant of the coordinate plane does the point (-4, 7) lie?
     • Quadrant II
     • Quadrant IV
     • Quadrant I
     • Quadrant III

     Correct answer: Quadrant II

     The point (-4, 7) lies in Quadrant II. The quadrants are numbered counterclockwise starting from the upper right: Quadrant I has positive x and positive y, Quadrant II has negative x and positive y, Quadrant III has both negative, and Quadrant IV has positive x and negative y. Since the x-coordinate is negative and the y-coordinate is positive, the point falls in Quadrant II.

134. A right circular cylinder has a radius of 4 cm and a height of 10 cm. In terms of pi, what is its volume?
     • 160 pi cubic centimeters
     • 40 pi cubic centimeters
     • 80 pi cubic centimeters
     • 400 pi cubic centimeters

     Correct answer: 160 pi cubic centimeters

     The volume is 160 pi cubic centimeters. The volume of a cylinder is pi times r squared times h, so V = pi times 4 squared times 10 = pi times 16 times 10 = 160 pi cubic centimeters. Using the radius once instead of squaring it (4 times 10 = 40) gives the incorrect 40 pi.

135. A cone has a base radius of 3 cm and a height of 12 cm. In terms of pi, what is its volume?
     • 324 pi cubic centimeters
     • 12 pi cubic centimeters
     • 36 pi cubic centimeters
     • 108 pi cubic centimeters

     Correct answer: 36 pi cubic centimeters

     The volume is 36 pi cubic centimeters. The volume of a cone is one-third times pi times r squared times h, so V = (1/3) times pi times 3 squared times 12 = (1/3) times pi times 9 times 12 = (1/3) times 108 pi = 36 pi cubic centimeters. Forgetting the one-third factor gives 108 pi, which is the volume of a cylinder with the same base and height.

136. A rectangular prism (box) measures 5 cm by 7 cm by 9 cm. What is its volume?
     • 21 cubic centimeters
     • 142 cubic centimeters
     • 315 cubic centimeters
     • 630 cubic centimeters

     Correct answer: 315 cubic centimeters

     The volume is 315 cubic centimeters. The volume of a rectangular prism is length times width times height, so V = 5 times 7 times 9 = 315 cubic centimeters. The value 142 corresponds to adding paired face products for surface area rather than multiplying all three dimensions for volume.

137. A closed cylindrical can has a radius of 3 cm and a height of 8 cm. In terms of pi, what is its total surface area (including both circular ends)?
     • 48 pi square centimeters
     • 57 pi square centimeters
     • 144 pi square centimeters
     • 66 pi square centimeters

     Correct answer: 66 pi square centimeters

     The total surface area is 66 pi square centimeters. The surface area of a closed cylinder is the lateral area plus the two circles: 2 times pi times r times h plus 2 times pi times r squared = (2 times pi times 3 times 8) plus (2 times pi times 9) = 48 pi plus 18 pi = 66 pi square centimeters. Leaving off the two end caps gives only the lateral area of 48 pi.

138. A student needs to convert 250 centimeters to meters. Which result is correct, and what is the reasoning?
     • 2.5 meters, because there are 100 centimeters in 1 meter so divide by 100
     • 25 meters, because there are 10 centimeters in 1 meter
     • 0.25 meters, because divide by 1,000
     • 25,000 meters, because multiply by 100

     Correct answer: 2.5 meters, because there are 100 centimeters in 1 meter so divide by 100

     The correct result is 2.5 meters, because there are 100 centimeters in 1 meter, so 250 divided by 100 = 2.5 meters. When converting from a smaller unit (centimeters) to a larger unit (meters), the number of units decreases, so you divide. Multiplying by 100 incorrectly enlarges the value, and using 10 centimeters per meter reflects a wrong conversion factor.

139. A teacher shows a student who computed the area of a triangle with base 10 and height 6 and got 60 square units. Which feedback best identifies the student's error?
     • The student should have squared the base before multiplying
     • The student multiplied base times height but forgot to multiply by one-half, so the area should be 30 square units
     • The student should have added the base and height instead of multiplying them
     • The student used the perimeter formula instead of the area formula

     Correct answer: The student multiplied base times height but forgot to multiply by one-half, so the area should be 30 square units

     The best feedback is that the student multiplied base times height but forgot to multiply by one-half, so the area should be 30 square units. The triangle area formula is one-half times base times height; computing 10 times 6 = 60 finds the area of a rectangle (or parallelogram) with those dimensions, which is exactly double the triangle's area. Identifying the missing one-half factor pinpoints the conceptual mistake.

140. A school cafeteria offers a lunch combo where a student picks one of 4 entrees, one of 3 side dishes, and one of 2 drinks. Using the fundamental counting principle, how many different lunch combos are possible?
     • 48
     • 24
     • 9
     • 14

     Correct answer: 24

     There are 24 possible combos. The fundamental counting principle states that when independent choices are made in sequence, you multiply the number of options for each stage, so 4 entrees times 3 sides times 2 drinks equals 24. Adding the numbers (4 plus 3 plus 2 equals 9) is the common error, but addition counts choosing only one item, not building a complete combo.

141. A spinner is divided into 6 equal sections numbered 1 through 6, and a fair coin is flipped. What is the probability of the spinner landing on 4 AND the coin showing heads?
     • 7 out of 12
     • 1 out of 12
     • 1 out of 3
     • 1 out of 8

     Correct answer: 1 out of 12

     The probability is 1 out of 12. For independent events, the probability that both occur is the product of their individual probabilities, so (1/6) times (1/2) equals 1/12. The events are independent because the spinner result has no effect on the coin result; adding the probabilities would be incorrect because that finds the chance of one OR the other, not both together.

142. During a fundraiser, a student records the daily donations in dollars: 14, 9, 22, 9, 31, and 18. What is the range of this data set?
     • 22
     • 13
     • 31
     • 9

     Correct answer: 22

     The range is 22. To calculate the range of a data set, subtract the smallest value from the largest value: the maximum is 31 and the minimum is 9, so 31 minus 9 equals 22. The range measures total spread, so 31 (the maximum alone) and 9 (the minimum alone) are not the answer.

143. A student flips a coin 50 times and records 30 heads. Which statement correctly compares the experimental and theoretical probability of getting heads?
     • Experimental probability is 0.5 and theoretical probability is 0.6
     • Experimental probability is 0.6 and theoretical probability is 0.5
     • Both the experimental and theoretical probabilities are 0.5
     • Both the experimental and theoretical probabilities are 0.6

     Correct answer: Experimental probability is 0.6 and theoretical probability is 0.5

     The experimental probability is 0.6 and the theoretical probability is 0.5. Experimental probability comes from observed results, so it is 30 heads divided by 50 flips, which equals 0.6. Theoretical probability comes from the model of a fair coin, so it is 1 out of 2, or 0.5. The two differ here only because of natural variation in a finite number of trials.

144. A teacher records quiz scores of 6, 8, 8, 10, and 13. What are the mean, median, and mode of this data set?
     • Mean 8, median 9, mode 8
     • Mean 9, median 9, mode 10
     • Mean 8, median 8, mode 13
     • Mean 9, median 8, mode 8

     Correct answer: Mean 9, median 8, mode 8

     The mean is 9, the median is 8, and the mode is 8. The mean is the sum (45) divided by the count (5), giving 9. The median is the middle value of the ordered list 6, 8, 8, 10, 13, which is 8. The mode is the most frequently occurring value, which is 8 because it appears twice. Confusing these measures is the common student error: the mean balances the values while the median locates the center position.

145. A coach must choose 3 of her 8 players to fill the distinct roles of captain, co-captain, and scorekeeper. How many different assignments are possible?
     • 56
     • 512
     • 24
     • 336

     Correct answer: 336

     There are 336 possible assignments. Because the three roles are distinct, order matters, so this is a permutation: 8 times 7 times 6 equals 336. The key difference between permutations and combinations is that permutations count arrangements where order matters, while combinations (which would give 56 here) count selections where order does not matter, such as simply picking 3 players for an undefined group.

146. A student measures the daily high temperatures (in degrees) as 4, 6, 9, and 11. The mean is 7.5. What is the mean absolute deviation (MAD) of this data set?
     • 5
     • 7.5
     • 10
     • 2.5

     Correct answer: 2.5

     The mean absolute deviation is 2.5. To find the MAD, take the absolute distance of each value from the mean (3.5, 1.5, 1.5, 3.5), then average those distances: their sum is 10, divided by 4 values equals 2.5. The MAD describes the typical distance of data points from the mean; the value 10 is only the sum of the deviations before averaging.

147. A teacher shows students a scatter plot of hours studied versus test score, where the points rise from lower left to upper right. Which conclusion best describes the relationship shown?
     • There is a positive correlation between hours studied and test score
     • There is a negative correlation between hours studied and test score
     • Studying more hours causes a guaranteed score increase for every student
     • There is no correlation between the two variables

     Correct answer: There is a positive correlation between hours studied and test score

     There is a positive correlation between hours studied and test score. In a scatter plot, points that trend upward from left to right indicate a positive correlation, meaning the two quantities tend to increase together; a downward trend would indicate negative correlation. The claim of a guaranteed cause-and-effect increase is wrong because correlation shown in a scatter plot does not by itself establish causation.

148. What is the average rate of change of the function $f(x) = x^2 + 1$ over the interval $[1, 4]$?
     • 3
     • 5
     • 6
     • 15

     Correct answer: 5

     Correct answer: 5. The average rate of change over an interval equals $\frac{f(b) - f(a)}{b - a}$. Here $f(4) = 4^2 + 1 = 17$ and $f(1) = 1^2 + 1 = 2$, so the rate is $\frac{17 - 2}{4 - 1} = \frac{15}{3} = 5$.

149. A circular pizza is cut into equal slices, and each slice forms a sector with a central angle of $45^\circ$. If the pizza has a radius of 12 inches, what is the area of one slice? (Use $\pi$ in the answer.)
     • $18\pi \text{ in}^2$
     • $9\pi \text{ in}^2$
     • $36\pi \text{ in}^2$
     • $144\pi \text{ in}^2$

     Correct answer: $18\pi \text{ in}^2$

     Correct answer: $18\pi \text{ in}^2$. The area of a sector is the fraction of the full circle given by its central angle: $\frac{45}{360} = \frac{1}{8}$ of the whole circle. The full circle's area is $\pi r^2 = \pi (12)^2 = 144\pi$. One slice is $\frac{1}{8} \times 144\pi = 18\pi \text{ in}^2$.

150. A bag holds 12 marbles: 5 are red, 4 are blue, and 3 are yellow. One marble is drawn at random. What is the probability that the marble is red OR yellow?
     • 1 out of 12
     • 5 out of 36
     • 2 out of 3
     • 1 out of 4

     Correct answer: 2 out of 3

     The probability is 2 out of 3. Because a single marble cannot be both red and yellow, these are mutually exclusive events, so the addition rule applies: add the separate probabilities, P(red) = 5/12 and P(yellow) = 3/12, to get 8/12, which simplifies to 2/3. Multiplying the probabilities (5/12 times 3/12) is the common error, but multiplication finds the chance of two events both happening in sequence, not the chance of one OR the other on a single draw.

151. A small company reports these annual salaries in thousands of dollars: 38, 41, 42, 44, and 250. Which measure of center best represents a typical salary for this data set, and why?
     • The mean, because it uses every value in the data set
     • The median, because the very high outlier pulls the mean upward and makes it unrepresentative
     • The range, because it shows the full spread of the salaries
     • The mode, because it is the value that occurs most often

     Correct answer: The median, because the very high outlier pulls the mean upward and makes it unrepresentative

     The median is the best measure of center here because the single very high salary of 250 is an outlier that pulls the mean upward to an unrepresentative value. The mean of this data set is 83 (415 divided by 5), which is larger than every salary except the outlier, so it does not describe a typical salary; the median of 42 sits among the clustered values and resists distortion from extreme points. The range is a measure of spread, not center, and the mode fails because no value repeats.