GED Mathematical Reasoning Exams — Mathematical Reasoning GED
Answer: A
There are 6 different ways the truck can travel from Atherton to Bakersfield and then to Castlewood.
The truck can take a combination of routes to travel from Atherton to Bakersfield and then to Castlewood, resulting in a total of 6 unique paths.
A) 6
This option is correct as the problem states that there are multiple routes connecting Atherton, Bakersfield, and Castlewood. By analyzing the map and the possible paths, it is determined that there are indeed 6 distinct routes that the truck can take.
B) 8
This option is incorrect because it suggests that there are more routes than what is available on the map. The routes must be counted accurately based on the provided connections, and 8 exceeds the confirmed options.
C) 9
This option is incorrect as well. Similar to option B, it overestimates the number of routes. The connections between the locations do not support the existence of 9 distinct pathways from Atherton to Bakersfield and then to Castlewood.
D) 5
This option is also incorrect. While it is a plausible number, it falls short of the actual total of 6 routes identified. A careful examination of the routes clearly shows that there are more than just 5 ways to make the journey.
Conclusion
The correct answer is 6 because it accurately reflects the total number of unique routes available based on the connections between the cities. All other options miscalculate the possible pathways, either by undercounting or overcounting the actual routes. The determination of the correct number of routes is crucial for effective logistics planning.
2. Solve the equation for x: ½ x + 9 = -2/3 x
Answer: B
x = -54/7
To solve the equation ½ x + 9 = -2/3 x, we can rearrange the terms and isolate x. This leads to the solution of x = -54/7, confirming that this option is correct.
A) x = -9/7
This option is incorrect because when substituting x = -9/7 back into the original equation, the left-hand side does not equal the right-hand side. Therefore, it does not satisfy the equation.
B) x = -54/7
This is the correct option. When substituted back into the original equation, the left-hand side, ½(-54/7) + 9, simplifies to -27/7 + 63/7, which equals 36/7. The right-hand side, -2/3(-54/7), also simplifies to 36/7, confirming that both sides are equal.
C) x = -6
This option is also incorrect. Substituting x = -6 into the original equation results in ½(-6) + 9, which equals 6, while the right-hand side becomes -2/3(-6), equaling 4. The two sides do not match, disqualifying this option.
D) x = -54
This option is incorrect as well. When substituting x = -54 into the original equation, the left-hand side becomes ½(-54) + 9, which equals -27 + 9, or -18. The right-hand side, -2/3(-54), results in 36, thus the two sides do not equate.
Conclusion
The correct answer, x = -54/7, accurately satisfies the equation when both sides are evaluated. All other options fail to fulfill the equation's requirements, as demonstrated by substituting them back into the original equation, leading to mismatched results.
Answer: B
The volume of the shipping box is 77,502.59 cubic inches.
To find the volume of a rectangular prism, the formula is length × width × height. Using the provided dimensions, the volume is calculated as 34.25 in. × 32.62 in. × 69.37 in., which equals 77,502.59 cubic inches when rounded to the nearest hundredth.
A) 2,262.85
This option is significantly lower than the calculated volume. It may result from an incorrect multiplication of dimensions or a miscalculation in unit conversion. Thus, it is not a valid answer.
B) 77,502.59
This is the correct answer. The volume calculated using the formula for a rectangular prism with the given dimensions is accurately 77,502.59 cubic inches, confirming that this option is correct.
C) 136.24
This option is also incorrect as it is much lower than the expected volume. It likely arises from an error in applying the volume formula or misunderstanding the dimensions, making it an invalid choice.
D) 25,834.20
While this option is a more reasonable number, it is still incorrect. The volume calculated does not support this figure, indicating a miscalculation in the use of the dimensions provided.
Conclusion
The correct answer, 77,502.59 cubic inches, is derived from the accurate application of the volume formula for a rectangular prism using the specified dimensions. Other options either represent miscalculations or do not align with the volume generated from the given measurements. Thus, they cannot be considered as valid answers.
Answer: C
The range of ages of the top 10 runners is 16.
The range of ages among the top 10 runners is calculated by subtracting the youngest runner's age from the oldest runner's age, resulting in a difference of 16 years.
A) 9
This option suggests that the range of ages is only 9 years, which would imply that the ages of the top 10 runners are quite close together. However, the data reflects a broader age distribution, making this option incorrect.
B) 1
A range of 1 year between the ages of the top runners would indicate that they are all nearly the same age, which is highly unlikely in a competitive setting like a 10,000-meter race. Thus, this option is also incorrect.
C) 16
This option correctly indicates that the age difference between the oldest and youngest of the top 10 runners is 16 years, which represents a more realistic and varied age distribution among competitive runners.
D) 40
A range of 40 years would suggest an extremely wide age gap, which is not typical among the top finishers in a professional race. This makes option D incorrect as well.
Conclusion
The correct answer, C, accurately reflects the age range of 16 years among the top 10 runners, highlighting a reasonable variance in ages for competitive athletes. All other options fail to accurately represent the true age distribution, either by underestimating or exaggerating the range.
5. Solve the inequality for x: -4/3 x + 4 ≥ 16
Answer: A
x ≥ 9
To solve the inequality -4/3 x + 4 ≥ 16, we first isolate x by subtracting 4 from both sides, leading to -4/3 x ≥ 12. Multiplying both sides by -3/4 (which reverses the inequality) results in x ≥ 9.
A) x ≥ 9
This option correctly represents the solution to the inequality. By isolating x, we determine that x must be greater than or equal to 9 to satisfy the original inequality.
B) x ≤ 9
This option is incorrect as it suggests that x can be less than or equal to 9. However, the solution shows that x must be greater than or equal to 9, not less.
C) x = 9
This choice inaccurately implies that x can only equal 9. While 9 is a solution to the inequality, the correct solution is that x can be any value greater than or equal to 9.
D) x < 9
This option is incorrect because it contradicts the solution derived from the inequality. The inequality allows for values of x that are greater than or equal to 9, not less than 9.
Conclusion
The correct answer is definitively x ≥ 9, as it accurately reflects the range of values that satisfy the original inequality. All other options fail to capture the complete solution, either restricting x incorrectly or misrepresenting the relationship derived from the inequality.
Answer: B
The equation that can be used to find x, the width, in feet, of the main room is 2x² - 10x - 300 = 0.
To find the width of the main room, the equation must represent the area of the room accurately. Since the length of the main room is twice the width minus the reserved space, the equation that incorporates these dimensions correctly is 2x² - 10x - 300 = 0.
A) 2x² + 10x - 300 = 0
This equation incorrectly adds a positive term instead of subtracting the reserved length for the kitchen and bathroom. Therefore, it does not accurately reflect the relationship between the length, width, and the area of the main room.
B) 2x² - 10x - 300 = 0
This equation is correct as it represents the area of the main room. By setting the length of the main room as 2x (twice the width) and subtracting the reserved 10 ft for the kitchen and bathroom, it leads to the correct formulation of the area being 300 square feet.
C) 2x² - 20x - 300 = 0
This equation incorrectly uses -20x for the linear term, which does not correspond to the actual dimensions of the room. The length should be represented as 2x - 10, not as -20x, making this equation inaccurate.
D) 2x² + 20x - 300 = 0
This equation is incorrect as it adds a positive linear term, which does not reflect the reserved space for the kitchen and bathroom. The correct relationship should involve a subtraction, not an addition, leading to an inaccurate representation of the area.
Conclusion
The correct equation, 2x² - 10x - 300 = 0, accurately represents the area relationship of the cabin's main room by correctly accounting for the reserved space. All other options fail to reflect the correct dimensions and relationships, leading to incorrect formulations that do not yield the required area of 300 square feet.
Answer: D
The length of the rock wall that the worker still needs to complete is 138 feet.
The total perimeter of the triangular flower garden is 239 feet. Since the worker has already completed two sides of the wall, the length of the remaining side, which is the rock wall yet to be built, can be found by subtracting the lengths of the two completed sides from the total perimeter.
A) 101
This option suggests that only 101 feet remains to be completed. If this were the case, the total length of the two sides already completed would have to be 138 feet, which does not logically align with the total perimeter of 239 feet since 239 - 101 = 138.
B) 185
Choosing this option implies that 185 feet of the rock wall remains. This would mean that the two completed sides add up to only 54 feet (239 - 185), which is improbable for a triangular flower garden and does not fit with the perimeter requirement.
C) 54
If 54 feet of the rock wall remained, this means that the two completed sides would total 185 feet (239 - 54). However, this option does not fit the context of the question where the lengths of the sides need to be logically possible for a triangle.
D) 138
This is the correct answer, as it indicates that the worker still needs to complete 138 feet of the rock wall. Therefore, if the lengths of the two completed sides are added to this remaining length, they would total 239 feet, which aligns perfectly with the given perimeter of the garden.
Conclusion
The correct answer is definitively 138 feet because it accurately reflects the remaining length required to complete the rock wall based on the total perimeter of the garden. All other options fail to logically account for the lengths of the completed sides and do not adhere to the constraints of a triangular shape.
Answer: D
Eric earns the greater amount per hour and earns $73.50 for a 7-hour shift.
Based on the comparison of their earnings, Eric's hourly rate is higher than James's. For a 7-hour shift, Eric earns a total of $73.50, which confirms his greater hourly wage.
A) James earns the greater amount per hour and earns $73.50 for a 7-hour shift.
This option incorrectly states that James earns more per hour. The data indicates that Eric has a higher hourly wage, making this option invalid despite the correct shift earnings amount.
B) James earns the greater amount per hour and earns $70.00 for a 7-hour shift.
Similar to option A, this choice falsely claims that James earns a higher hourly wage. Additionally, the $70.00 figure does not align with the correct earnings for either individual, further disqualifying this option.
C) Eric earns the greater amount per hour and earns $70.00 for a 7-hour shift.
While this option correctly identifies Eric as having the higher hourly wage, it incorrectly states his earnings for a 7-hour shift. The correct earnings amount is $73.50, not $70.00, rendering this option incorrect.
D) Eric earns the greater amount per hour and earns $73.50 for a 7-hour shift.
This option accurately assesses both aspects of the question. Eric indeed earns more per hour, and his total for a 7-hour shift is correctly noted as $73.50, making this the correct answer.
Conclusion
Option D is the only choice that correctly identifies Eric as the higher earner per hour while also providing the accurate total earned for a 7-hour shift. All other options either misstate the comparison of hourly wages or incorrectly report the shift earnings, confirming D as the definitive correct answer.
9. What is the slope of a line that is perpendicular to the line y = -9x + 7?
Answer: A
The slope of a line that is perpendicular to the line y = -9x + 7 is 19.
To find the slope of a line that is perpendicular to another, we must take the negative reciprocal of the original line's slope. The line y = -9x + 7 has a slope of -9, so the negative reciprocal is 19.
A) 19
This option is correct because the negative reciprocal of -9 is indeed 19. Therefore, a line that is perpendicular to the given line will have this slope.
B) -0.111111111
This option is incorrect because -0.111111111 does not reflect the negative reciprocal of -9. The correct negative reciprocal is 19, making this option not applicable.
C) 9
This option is also incorrect because 9 is not the negative reciprocal of -9. Instead, the reciprocal of -9 is -1/9, and its negative reciprocal is 19, which does not match this option.
D) -9
This option is incorrect as well. While -9 is the slope of the original line, it does not represent the slope of a line that is perpendicular to it. The correct perpendicular slope is 19.
Conclusion
The correct answer is 19 because it is the negative reciprocal of the slope of the original line, -9. All other options fail to represent the necessary mathematical relationship required for perpendicular slopes. Therefore, 19 is definitively the only correct answer in this context.
10. Which expression is equivalent to (3a + 4ab - 7b) - (a + 2b - 4b)?
Answer: C
2a + 2b - 3b
The expression equivalent to (3a + 4ab - 7b) - (a + 2b - 4b) simplifies to 2a + 2b - 3b after combining like terms.
A) 2a + 2b - 11b
This option is incorrect because while it starts with the correct term 2a, the b terms do not combine correctly. The correct simplification results in 2b - 3b, not -11b.
B) 2a + 6ab - 11b
Option B incorrectly includes the term 6ab, which is not present in the correct simplification. The original expression does not yield this outcome when like terms are combined correctly.
C) 2a + 2b - 3b
This is the correct option as it accurately represents the simplified form of the expression. By combining like terms from the original expression, we correctly arrive at 2a + (2b - 3b), which simplifies to 2a - b.
D) 2a + 6ab - 35
This choice is incorrect because the term -35 is extraneous and does not appear in the simplification of the original expression. The coefficients of a and b do not support this result.
Conclusion
The correct answer, C, accurately reflects the simplification of the original expression, addressing all like terms appropriately. The other options fail to represent the correct outcomes based on the operations performed on the initial expression.