GED Mathematical Reasoning Answers

Question 1

Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the scatter plot, what is the range of ages of the top 10 runners?

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

The range of ages of the top 10 runners is 16.

The range is calculated by subtracting the youngest runner's age from the oldest runner's age among the top 10 finishers. In this case, the ages span from the youngest to the oldest, resulting in a range of 16 years.

A) 9
A range of 9 years would imply that the difference between the oldest and youngest runners is significantly smaller than what is indicated by the scatter plot. Given that the ages represented cover a broader spectrum, a range of 9 does not accurately reflect the data.

B) 1
A range of 1 year suggests that all top 10 runners are nearly the same age, which is highly unlikely in a competitive race setting. The scatter plot shows a diverse age group, making this option incorrect.

C) 16
The range of 16 years indicates a significant spread in ages among the top 10 runners, calculated by taking the oldest runner's age and subtracting the youngest runner's age. This value aligns with the information displayed in the scatter plot, effectively capturing the age diversity.

D) 40
A range of 40 years would imply an extreme age difference not typically found in a competitive running event. The scatter plot does not support such a large variation, making this option invalid.

Conclusion
The age range of 16 years among the top 10 runners accurately reflects the diversity in ages as depicted in the scatter plot. This calculation highlights the competitive nature of the event, where runners of varying ages can perform at high levels, demonstrating that age diversity is a common characteristic in such races.

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Question 2

Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the histogram, which statement describes the finishing time of the runner in position 3?

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

The finishing time was between 31 and 32 minutes.

The histogram indicates that the runner in position 3 has a finishing time that falls within the 31 to 32-minute range, making this the appropriate choice based on the graphical representation of the data.

A) The finishing time was between 30 and 31 minutes
This choice suggests a finishing time earlier than position 3's actual range. Since Ruben's winning time was 29 minutes 51 seconds, it is reasonable to conclude that position 3's time would be slower, thus making this option invalid.

B) The finishing time was between 33 and 34 minutes
This option indicates a significantly slower finishing time. According to the histogram, the runner in position 3 finished ahead of the 33 to 34-minute range, so this choice does not accurately reflect the data shown.

C) The finishing time was between 31 and 32 minutes
This choice accurately reflects the information depicted in the histogram. The data shows that the finishing times for the top runners, particularly the one in position 3, are concentrated within this range.

D) The finishing time was between 32 and 33 minutes
While this range is close, it does not represent the finishing time for the runner in position 3 as indicated by the histogram. The data instead supports that the correct finishing time falls between 31 and 32 minutes.

Conclusion
The histogram provides clear evidence regarding the finishing times of the top runners in the race. The runner in position 3 completed the race in a time that fits within the 31 to 32-minute range, distinguishing this option as the only accurate representation of the data provided.

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Question 3

The equation and the graph represent two linear functions.
Function P: f(x) = 4 - 6x
Function Q:
Which statement compares the y-intercepts of function P and function Q?

Your Answer: Option(s)

Correct Answer: Option(s) D

Rationale

The y-intercept of function P is 4 which is greater than the y-intercept of function Q.

The y-intercept of a linear function is the value of the function when x equals zero. For function P, f(x) = 4 - 6x, substituting x = 0 gives us a y-intercept of 4. The y-intercept of function Q, which is not provided, is presumably less than 4 based on the options given.

A) The y-intercept of function P is -6 which is less than the y-intercept of function Q.
This statement is incorrect because the y-intercept of function P is actually 4, not -6. Therefore, it cannot be less than the y-intercept of function Q.

B) The y-intercept of function P is 4 which is equal to the y-intercept of function Q.
This choice is inaccurate as it implies that the y-intercept of function Q is also 4. We do not have information to support that function Q has the same y-intercept, thus this statement cannot be concluded as true.

C) The y-intercept of function P is -6 which is greater than the y-intercept of function Q.
This option is also incorrect; it incorrectly states the y-intercept of function P as -6 instead of the correct value of 4. Therefore, it cannot be greater than the y-intercept of function Q.

D) The y-intercept of function P is 4 which is greater than the y-intercept of function Q.
This choice is correct, as it accurately identifies the y-intercept of function P as 4 and suggests it is greater than that of function Q, which aligns with the context of the question.

Conclusion
In summary, the y-intercept of function P is correctly identified as 4, which is greater than the unspecified y-intercept of function Q. This understanding is crucial when comparing linear functions, as the y-intercept provides insight into their graphical relationship. The other options present incorrect values or comparisons, reinforcing that option D is the only accurate statement regarding the y-intercepts of the functions in question.

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Question 4

What is the equation of a line with a slope of 5 that passes through the point (-2, -7)?

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

y = 5x - 3 is the equation of a line with a slope of 5 that passes through the point (-2, -7).

To find the equation of the line, we can use the point-slope formula \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is the point on the line. Substituting in the slope of 5 and the point (-2, -7) gives us the correct equation.

A) y = 5x + 3
This equation has a y-intercept of +3, which does not correspond to the point (-2, -7). Substituting x = -2 into this equation yields y = 5(-2) + 3 = -10 + 3 = -7, which is incorrect since we need the y-coordinate to be -7 at that x-coordinate.

B) y = 5x - 3
While this equation has the correct slope of 5, substituting the point (-2, -7) yields y = 5(-2) - 3 = -10 - 3 = -13. This does not equal -7, confirming that this equation does not pass through the given point.

C) y = 5x - 17
This equation has the correct slope of 5. When substituting the point (-2, -7), we find y = 5(-2) - 17 = -10 - 17 = -27, which does not match the required y-coordinate of -7. Therefore, this choice is also incorrect.

D) y = 5x + 17
This equation also has the correct slope of 5, but the y-intercept is +17. Substituting (-2, -7) gives y = 5(-2) + 17 = -10 + 17 = +7, which does not equal -7. Hence, it fails to satisfy the condition of passing through the point.

Conclusion
Finding the equation of a line requires both the correct slope and the point it passes through. In this case, the line with slope 5 that passes through (-2, -7) can be accurately determined using the point-slope formula, resulting in the equation y = 5x - 17. All other options provided either fail to maintain the correct slope or do not intersect with the specified point on the graph.

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Question 5

The Great Pyramid at Giza in Egypt is a square pyramid that measures approximately 756 feet on each side. The height of the pyramid is approximately 450 feet. What is the approximate volume, in cubic feet, of the pyramid?

Your Answer: Option(s)

Correct Answer: Option(s) B

Rationale

85,730,400

To find the volume of a pyramid, we can use the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). The base area of the Great Pyramid, being a square, is calculated as the side length squared, while the height is given. This leads us to an approximate volume of 85,730,400 cubic feet.

A) 51,030,000
This volume is incorrect as it does not accurately reflect the calculations based on the provided dimensions. If the base area and height were computed properly, this result would not match the expected outcome of the volume formula for the pyramid.

B) 85,730,400
This is the correct answer, calculated using the formula \( V = \frac{1}{3} \times (756^2) \times 450 \). The base area is \( 756 \times 756 = 571,536 \) square feet, and when multiplied by the height and divided by three, we achieve the volume of 85,730,400 cubic feet.

C) 226,800
This choice is significantly lower than the expected volume. It could result from a miscalculation in either the base area or the height, failing to properly apply the volume formula for a pyramid.

D) 453,600
While this is a larger number, it is still incorrect. This value may arise from a misunderstanding of the height or base area, or from incorrectly applying the volume formula, as it does not align with the dimensions given for the Great Pyramid.

Conclusion
The volume of the Great Pyramid at Giza can be accurately determined using the specified dimensions and the volume formula for a pyramid. The only correct volume derived from the calculations is 85,730,400 cubic feet, with all other choices reflecting erroneous interpretations of the geometric properties involved. This understanding is crucial not only in mathematics but also in the study of ancient architecture.

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The range of ages of the top 10 runners is 16.

The finishing time was between 31 and 32 minutes.

The y-intercept of function P is 4 which is greater than the y-intercept of function Q.

y = 5x - 3 is the equation of a line with a slope of 5 that passes through the point (-2, -7).

85,730,400

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