Math

1. A group of friends spent a total of $36 at a restaurant. They ordered 4 tacos, all the same price, and spent $8 on drinks. There were no other expenses. In which of the following equations does x represent the price, in dollars, of 1 taco?

4 tacos at price x plus $8 for drinks equals $36 total: 4x + 8 = 36.

2. Each card in a box is either red or blue. One card will be randomly selected from the box. The probability that the selected card will be red is 35%. What is the probability that the selected card will be blue?

Since the card is either red or blue, the probabilities must sum to 100%. 100% − 35% = 65%.

3. What is the sum of the polynomials −5x² + 3x − 2 and 8x² − x + 4?

Combine like terms: (−5+8)x² = 3x², (3−1)x = 2x, (−2+4) = 2. Result: 3x² + 2x + 2.

4. Quadrilaterals ABCD and PQRS are congruent such that ABCD ≅ PQRS. The length of BC must be equal to the length of which of the following line segments?

In congruent figures, corresponding sides are equal. B corresponds to Q and C to R, so BC corresponds to QR.

5. What is the decimal form of 7/11?

7 ÷ 11 = 0.636363…, a repeating decimal written as 0.6̄3̄.

6. At a store, 8 notebooks, each the same price, cost a total of $24. At the same store, a binder costs 3 times as much as a notebook. How much do 4 binders cost?

Notebook price = $24 ÷ 8 = $3. Binder = 3 × $3 = $9. Four binders = 4 × $9 = $36.

7. All 25 students in a class were asked to choose their favorite color. Of these students, 4 chose red, 6 chose blue, 10 chose green, and 5 chose yellow. What is the probability that a randomly selected student from this class chose red or blue?

(4 + 6) = 10 students chose red or blue. P = 10/25 = 2/5.

8. (8/27)^(2/3) = ?

Cube root of 8/27 = 2/3. Then (2/3)² = 4/9.

9. A quadrilateral in the standard (x, y) coordinate plane has vertices at (1, 0), (1, 5), (6, 5), and (6, 0). How many right angles does the quadrilateral have?

The vertices form a rectangle with horizontal and vertical sides, so all 4 angles are right angles.

10. Which of the following points is a solution to y < −x − 3?

Test (0, −4): −4 < −0 − 3 = −3. Since −4 < −3 is true, (0, −4) is a solution.

11. Given the function f(x) = 2x² + 5, what is the value of f(−3)?

f(−3) = 2(−3)² + 5 = 2(9) + 5 = 18 + 5 = 23.

12. What is the domain of the set of ordered pairs (−6, −3), (1, 4), (3, −2), (5, 0), and (7, 2)?

The domain is the set of all x-values (first coordinates): {−6, 1, 3, 5, 7}.

13. The equation of a certain line in the standard (x, y) coordinate plane is y = −(3/4)x + 7. Which of the following is the slope of a line parallel to this line?

Parallel lines have the same slope. The slope of the given line is −3/4.

14. Which ordered pair is a solution to both y = x + 2 and y = x²?

Set equal: x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0. So x = 2 (giving y=4) or x = −1 (giving y=1). Of the choices, (−1, 1) is a solution.

15. What is the probability that a number randomly selected from the following set will be a solution to the inequality 2x + 3 ≥ 1? {−5, −4, −3, −2, −1, 0, 1, 2, 3}

2x + 3 ≥ 1 → x ≥ −1. From the set, solutions are {−1, 0, 1, 2, 3} = 5 values. P = 5/9.

16. Two data sets, A and B, are represented by boxplots on the same number line. Data set A: min=5, Q1=10, median=20, Q3=30, max=40. Data set B: min=5, Q1=15, median=25, Q3=35, max=40. One of the following statements about the 2 data sets is true. Which one?

Range of A = 40 − 5 = 35. Range of B = 40 − 5 = 35. The ranges are equal.

17. What value of m is the solution to the equation 5 + 4(m − 3) = 13 − m?

5 + 4m − 12 = 13 − m → 4m − 7 = 13 − m → 5m = 20 → m = 4.

18. Which of the following is a linear equation?

A linear equation has variables raised only to the first power and no products of variables. y = −3x + 7 fits this definition.

19. A solid iron ball has a density of 5.8 grams per cubic centimeter and a volume of 20.0 cubic centimeters. A solid copper ball has a density of 9.6 grams per cubic centimeter. The masses of both balls are equal. The density of each ball is equal to its mass divided by its volume. To the nearest 0.1 cubic centimeters, what is the volume of the copper ball?

Mass of iron ball = 5.8 × 20 = 116 g. Volume of copper = 116 ÷ 9.6 ≈ 12.1 cm³.

20. In the given figure, a transversal intersects two parallel lines. The measure of one angle is 70° and the co-interior (same-side interior) angle measures (5x + 10)°. What is the value of x?

Co-interior angles sum to 180°: 70 + (5x + 10) = 180 → 5x = 100 → x = 20.

21. The table lists the remainder when each of 4 powers of 3 is divided by 10: 3¹ → 3, 3² → 9, 3³ → 7, 3⁴ → 1, then the pattern repeats. What is the remainder when 3¹⁰¹ is divided by 10?

The pattern of remainders (3, 9, 7, 1) repeats every 4 powers. 101 ÷ 4 = 25 remainder 1. So 3¹⁰¹ has the same remainder as 3¹, which is 3.

22. Given that q = 5 and the value of r is 2.4 times the value of q, what is the sum of q and r?

r = 2.4 × 5 = 12. Sum = q + r = 5 + 12 = 17.

23. Each of the following lines in the standard (x, y) coordinate plane passes through 2 points labeled with coordinates. Which of these lines has a slope greater than 2 and a y-intercept less than 5?

A: slope = (6−3)/(1−0) = 3 > 2 ✓, y-intercept = 3 < 5 ✓. B: slope = 2 (not > 2). C: y-intercept = 6 (not < 5). D: slope = 1 (not > 2).

24. A total of 45 students were asked whether they like cats and whether they are athletes. Of the 45 students, 20 like cats. Of the 30 that were athletes, 18 did not like cats. Of the people who were not athletes, how many liked cats?

Athletes who like cats = 30 − 18 = 12. Total who like cats = 20. Non-athletes who like cats = 20 − 12 = 8.

25. What are the x-intercepts of the graph of y = x² + 7x + 12 in the standard (x, y) coordinate plane?

Factor: x² + 7x + 12 = (x + 3)(x + 4) = 0. So x = −3 or x = −4.

26. A car has a maximum speed of 60 miles per hour. What is this car's maximum speed in feet per second? (Note: 1 mile = 5,280 feet)

60 mi/hr × 5,280 ft/mi ÷ 3,600 sec/hr = 316,800 ÷ 3,600 = 88 ft/sec.

27. What is the x-value of the solution to the given system of equations? 3x + y = 10 x − y = 2

Add the equations: 4x = 12 → x = 3.

28. A scatterplot shows hours studied (x) and test scores (y) for 10 students. The data shows a positive linear trend. Which of the following could be the linear equation of the line of best fit?

A positive linear trend means a positive slope. y = 8x + 40 has a reasonable positive slope and a realistic starting score.

29. Sam is practicing free throws. Sam made 8 of the first 20 free throws. Sam made every free throw after the first 20. At the end of practicing, Sam had made 2/3 of all the free throws attempted. How many free throws did Sam attempt after the first 20?

Let n = extra throws. (8 + n)/(20 + n) = 2/3 → 3(8 + n) = 2(20 + n) → 24 + 3n = 40 + 2n → n = 16.

30. A gym charges a fixed membership fee and a fixed monthly fee. The total cost for 3 months is $75 and the total cost for 7 months is $115. What will the gym charge in total for 10 months?

Monthly fee = ($115 − $75) ÷ (7 − 3) = $10/month. Membership = $75 − 3($10) = $45. Total for 10 months = $45 + 10($10) = $145.

31. The graph shows the total cost of 1 coffee and different numbers of muffins at a café. One of the following amounts is the cost of 3 muffins with no coffee. Which one? (The graph shows points: (0, $2.00), (1, $3.50), (2, $5.00), (3, $6.50))

The y-intercept $2.00 is the coffee price. Each muffin costs $1.50. Three muffins without coffee = 3 × $1.50 = $4.50.

32. In the standard (x, y) coordinate plane, the point (5, 13) is on the circle centered at the point (2, 9). What is the radius of the circle?

r = √((5−2)² + (13−9)²) = √(9 + 16) = √25 = 5.

33. The given 8-sided figure is partitioned into 4 congruent squares. The total area of the figure is 400 square inches. Which of the following is closest to the perimeter of the figure in inches?

Each square has area 400 ÷ 4 = 100 sq in, so side = 10 in. The L-shaped 8-sided figure has 10 outer unit-sides, giving perimeter ≈ 10 × 10 = 100 in.

34. When the parabola represented by y = (x + 4)(x − 2) is graphed in the standard (x, y) coordinate plane, what are the coordinates of the vertex?

x-intercepts are −4 and 2. Vertex x = (−4 + 2)/2 = −1. y = (−1 + 4)(−1 − 2) = 3 × (−3) = −9. Vertex: (−1, −9).

35. If 3x − 5 = 16, what is the value of x?

3x = 16 + 5 = 21, so x = 7.

36. A rectangle has length 12 cm and width 7 cm. What is its area?

Area = length × width = 12 × 7 = 84 cm².

37. What is 15% of 80?

15% of 80 = 0.15 × 80 = 12.

38. What is the slope of the line passing through (1, 4) and (3, 10)?

slope = (10 − 4) / (3 − 1) = 6 / 2 = 3.

39. The sum of three consecutive integers is 48. What is the smallest integer?

Let n, n+1, n+2 be the integers. 3n + 3 = 48 → n = 15.

40. What is the area of a triangle with base 10 and height 6?

Area = ½ × base × height = ½ × 10 × 6 = 30.

41. A line has equation 2y = 4x + 8. What is its slope?

Divide both sides by 2: y = 2x + 4. The slope is the coefficient of x, which is 2.

42. If f(x) = x² − 3x + 2, what is f(4)?

f(4) = (4)² − 3(4) + 2 = 16 − 12 + 2 = 6.

43. A car travels 240 miles in 4 hours. What is its average speed?

Speed = distance ÷ time = 240 ÷ 4 = 60 mph.

44. Solve for x: 5/x = 2/3

Cross-multiply: 2x = 15, so x = 7.5.

45. The angles of a triangle measure x°, 2x°, and 3x°. What is the value of x?

Angles of a triangle sum to 180°. x + 2x + 3x = 180 → 6x = 180 → x = 30.

46. What are the solutions of the system y = 2x + 1 and y = −x + 7?

Set equal: 2x + 1 = −x + 7 → 3x = 6 → x = 2, y = 5.

47. Simplify: (2x³y²)(3xy⁴)

Multiply coefficients: 2 × 3 = 6. Add exponents: x³⁺¹ = x⁴, y²⁺⁴ = y⁶. Result: 6x⁴y⁶.

48. A store discounts a $50 item by 20%. What is the sale price?

20% of $50 = $10 discount. $50 − $10 = $40.

49. What is the circumference of a circle with diameter 10? (Use π ≈ 3.14)

C = πd = 3.14 × 10 = 31.4.

50. What is the solution set of the inequality −2x + 4 > 10?

−2x > 6. Dividing by −2 flips the inequality: x < −3.

51. If y varies directly with x and y = 12 when x = 4, what is y when x = 7?

y = kx → 12 = 4k → k = 3. When x = 7: y = 3 × 7 = 21.

52. A data set has values 3, 7, 7, 9, 14. What is the mode?

The mode is the value that appears most often. 7 appears twice; all others appear once.

53. What is the value of 3² + 4² ?

3² = 9, 4² = 16. 9 + 16 = 25. (This is also the Pythagorean triple 3-4-5 squared.)

54. A jar contains 4 red, 5 blue, and 6 green marbles. One is drawn at random. What is the probability it is NOT red?

Total = 15. Not red = 5 + 6 = 11. P(not red) = 11/15.

55. Solve for x: 3(x - 4) = 2x + 1

3x - 12 = 2x + 1 -> x = 13.

56. A rectangle has length 12 cm and width 7 cm. What is its perimeter?

Perimeter = 2(l + w) = 2(12 + 7) = 38 cm.

57. What is 15% of 240?

15% x 240 = 0.15 x 240 = 36.

58. A car travels 180 miles in 3 hours. At the same rate, how far does it travel in 5 hours?

Rate = 180/3 = 60 mph. Distance = 60 x 5 = 300 miles.

59. If f(x) = 2x^2 - 3, what is f(4)?

f(4) = 2(16) - 3 = 32 - 3 = 29.

60. A triangle has base 10 inches and height 8 inches. What is its area?

Area = (1/2) x base x height = (1/2) x 10 x 8 = 40 sq in.

61. What is the slope of the line through (2, 5) and (6, 13)?

Slope = (13 - 5) / (6 - 2) = 8 / 4 = 2.

62. Simplify: (3x^2 y)(4xy^3)

Multiply coefficients: 3 x 4 = 12. Add exponents: x^(2+1) = x^3, y^(1+3) = y^4. Result: 12x^3 y^4.

63. A store marks an item up 40% from its cost of $85. What is the selling price?

Markup = 40% x $85 = $34. Selling price = $85 + $34 = $119.

64. The sum of three consecutive integers is 72. What is the smallest?

Let integers be n, n+1, n+2. Then 3n + 3 = 72 -> n = 23. Integers: 23, 24, 25.

65. A circle has radius 5 cm. What is its area? (Use pi = 3.14)

Area = pi x r^2 = 3.14 x 25 = 78.5 cm^2.

66. Solve the inequality: 2x + 5 > 17

2x + 5 > 17 -> 2x > 12 -> x > 6.

67. A bag has 3 quarters, 5 dimes, and 2 nickels. What is the total value?

3 x $0.25 + 5 x $0.10 + 2 x $0.05 = $0.75 + $0.50 + $0.10 = $1.35.

68. Which equation represents a line parallel to y = 3x - 7?

Parallel lines have equal slopes. The slope of y = 3x - 7 is 3. Only y = 3x + 4 has slope 3.

69. A right triangle has legs 9 and 40. What is the hypotenuse?

c^2 = 9^2 + 40^2 = 81 + 1600 = 1681. sqrt(1681) = 41.

70. If a = 3 and b = -2, what is a^2 - 2ab + b^2?

a^2 - 2ab + b^2 = (a - b)^2 = (3 - (-2))^2 = 5^2 = 25.

71. A data set is: 4, 8, 8, 9, 11, 14. What is the median?

With 6 values the median is the average of the 3rd and 4th: (8 + 9) / 2 = 8.5.

72. Evaluate: -4^2 + (-4)^2

-4^2 = -16 (negation after squaring). (-4)^2 = 16. Sum: -16 + 16 = 0.

73. Notebooks cost $2.50 each. With a 20% discount, how much do 6 notebooks cost?

Full price: 6 x $2.50 = $15. Discount: 20% x $15 = $3. Final: $15 - $3 = $12.

74. Two lines intersect and one angle measures 65 degrees. What is the measure of its vertical angle?

Vertical angles are opposite angles formed by intersecting lines. They are always equal, so the vertical angle is also 65 degrees.