GED-MATH Sample Questions & Answers
Algebraic problem solving carries the larger share, working through expressions, equations and functions, while quantitative problem solving with numbers and quantities makes up the rest of these two reasoning domains.
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- Question 1Advanced
Algebraic Problem Solving · Quadratic Equations and Functions
A small business is analyzing its weekly profit. The company's weekly revenue is modeled by the function R(x) = -2x² + 80x, and its weekly costs are modeled by C(x) = 10x + 150, where x is the number of units sold.
The profit function, P(x), is found by subtracting the cost from the revenue, P(x) = R(x) - C(x). The business breaks even when the profit is zero. What is the minimum number of units the company must sell to start making a profit (i.e., for P(x) > 0)?
Show answer & explanation
Correct answer: D
First, define the profit function: P(x) = R(x) - C(x) = (-2x² + 80x) - (10x + 150) = -2x² + 70x - 150. To find the break-even points, set P(x) = 0: -2x² + 70x - 150 = 0. Divide by -2 to simplify: x² - 35x + 75 = 0. Use the quadratic formula where a=1, b=-35, c=75. x = [35 ± √((-35)² - 4175)] / 2 = [35 ± √(1225 - 300)] / 2 = [35 ± √925] / 2. √925 is approx 30.4. So, x = (35 ± 30.4) / 2. The two break-even points are x ≈ (35 - 30.4)/2 = 4.6/2 = 2.3 and x ≈ (35 + 30.4)/2 = 65.4/2 = 32.7. The profit is positive between these two points. To start making a profit, they must sell more than 2.3 units. Since units must be whole, they must sell 3 units.
- Question 2Intermediate
Algebraic Problem Solving · Midpoint Formula
A coordinate plane shows two locations: Town A is at (-4, 5) and Town B is at (8, -4). A new cell tower needs to be built exactly halfway between the two towns. What are the coordinates of the cell tower's location?
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Correct answer: A
To find the location exactly halfway between two points, use the midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2).
Given points: (-4, 5) and (8, -4).
x-coordinate: (-4 + 8) / 2 = 4 / 2 = 2.
y-coordinate: (5 + (-4)) / 2 = 1 / 2 = 0.5.
The midpoint coordinates are (2, 0.5). - Question 3Intermediate
Algebraic Problem Solving · Exponential Functions
The population of a town is 25,000 and is growing at a rate of 3% per year. Which expression represents the town's population after 't' years?
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Correct answer: C
The formula for exponential growth is P(t) = P₀(1 + r)ᵗ, where P₀ is the initial amount, r is the growth rate as a decimal, and t is time. The initial population is 25,000. The rate is 3%, or 0.03. The growth factor is (1 + 0.03) = 1.03. Therefore, the expression is 25000(1.03)ᵗ.
- Question 4Beginner
Quantitative Problem Solving · Unit Rates and Conversions
A scientist is tracking the temperature of a chemical reaction. The initial temperature was -8°C. After 10 minutes, the temperature rose to 27°C. What was the average rate of temperature change per minute?
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Correct answer: B
To find the rate of change, calculate the total change in temperature and divide by the total time.
Total change = Final temperature - Initial temperature = 27°C - (-8°C) = 27 + 8 = 35°C.
Time = 10 minutes.
Rate of change = 35°C / 10 minutes = 3.5°C per minute. - Question 5Intermediate
Quantitative Problem Solving · Data Interpretation and Percentages
A survey of 150 students was conducted to find their favorite type of movie. The results are shown in the pie chart below. How many more students chose Action movies than chose Comedy movies?
pie title Favorite Movie Type "Comedy" : 20 "Action" : 32 "Drama" : 18 "Sci-Fi" : 24 "Other" : 6Show answer & explanation
Correct answer: B
First, calculate the number of students for each category:
Action: 32% of 150 = 0.32 * 150 = 48 students.
Comedy: 20% of 150 = 0.20 * 150 = 30 students.
Next, find the difference: 48 (Action) - 30 (Comedy) = 18 students.
Alternatively, find the difference in percentage first: 32% - 20% = 12%. Then calculate 12% of 150: 0.12 * 150 = 18 students. - Question 6Beginner
Algebraic Problem Solving · Slope of a Line
What is the slope of a line that passes through the points (3, -6) and (-1, 2)?
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Correct answer: C
The slope formula is m = (y₂ - y₁) / (x₂ - x₁).
Let (x₁, y₁) = (3, -6) and (x₂, y₂) = (-1, 2).
m = (2 - (-6)) / (-1 - 3)
m = (2 + 6) / (-4)
m = 8 / -4
m = -2 - Question 7Advanced
Algebraic Problem Solving · Solving Quadratic Equations
A rectangular swimming pool is 50 feet long and 25 feet wide. It is surrounded by a concrete deck of uniform width. If the total area of the pool and deck together is 2100 square feet, what is the width of the deck?
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Correct answer: A
Let 'x' be the uniform width of the deck. The total length of the pool and deck is (50 + 2x), and the total width is (25 + 2x). The total area is length * width = 2100.
(50 + 2x)(25 + 2x) = 2100
1250 + 100x + 50x + 4x² = 2100
4x² + 150x + 1250 = 2100
4x² + 150x - 850 = 0
Divide by 2: 2x² + 75x - 425 = 0
Using the quadratic formula or by testing the answers, let's test x=5:
Total Length = 50 + 2(5) = 60 feet
Total Width = 25 + 2(5) = 35 feet
Total Area = 60 * 35 = 2100 square feet. This matches the given information, so the width is 5 feet. - Question 8Advanced
Quantitative Problem Solving · Pythagorean Theorem
A contractor needs to build a wheelchair ramp to a doorway that is 4 feet off the ground. For safety, the angle of elevation of the ramp must be no more than 5 degrees, which means the ramp must have a length of at least 11.5 feet for every 1 foot of height. What is the minimum horizontal distance (run) the ramp must cover along the ground? Use the Pythagorean theorem a² + b² = c².
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Correct answer: A
- Calculate the minimum ramp length (hypotenuse): 4 feet (height) * 11.5 feet/foot = 46 feet.
- This forms a right triangle where the height (a) is 4 ft, the ramp length (c) is 46 ft, and the horizontal distance (b) is unknown.
- Use the Pythagorean theorem: a² + b² = c² => 4² + b² = 46².
- 16 + b² = 2116.
- b² = 2116 - 16 = 2100.
- b = √2100 ≈ 45.83 feet. The minimum horizontal distance is approximately 45.8 feet.
- Question 9Advanced
Algebraic Problem Solving · Polynomials and Geometric Formulas
A city planner is designing a new public park shaped like a trapezoid. The two parallel bases measure (3x + 2) meters and (5x - 4) meters. The height of the trapezoid is (x + 5) meters. Which expression represents the area of the park?
Show answer & explanation
Correct answer: B
The formula for the area of a trapezoid is A = ½h(b₁ + b₂).
- Add the bases: b₁ + b₂ = (3x + 2) + (5x - 4) = 8x - 2.
- Multiply by half the height: A = ½(x + 5)(8x - 2).
- It's easier to multiply by (8x - 2) first, then multiply by ½. Or, multiply ½ by (8x-2) first: (4x - 1).
- Now multiply (x + 5)(4x - 1).
- Using FOIL: (x * 4x) + (x * -1) + (5 * 4x) + (5 * -1) = 4x² - x + 20x - 5.
- Combine like terms: 4x² + 19x - 5.
- Question 10Beginner
Algebraic Problem Solving · Linear Functions
The sum of the interior angles of a polygon can be found using the formula S = 180(n-2), where 'n' is the number of sides. This formula represents a linear relationship between the number of sides and the sum of the angles.
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Correct answer: A
The formula S = 180(n-2) can be rewritten as S = 180n - 360. This is in the form of a linear equation y = mx + b, where S is y, n is x, the slope (m) is 180, and the y-intercept (b) is -360. Because it can be written in this form, it represents a linear relationship.
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