Mathematical Reasoning GED

Question 1

Fix It Fast is an auto repair shop that employs 10 mechanics. Each day, the shop owner randomly picks 1 mechanic to receive a free lunch. What is the probability the shop owner will pick the same mechanic to receive a free lunch 2 days in a row?

Your Answer: Option(s)

Correct Answer: Option(s) B

Rationale

Probability of the shop owner picking the same mechanic 2 days in a row is 1/100.

The probability of the shop owner selecting the same mechanic for a free lunch two days consecutively can be determined by multiplying the individual probabilities of selection each day. Since there are 10 mechanics and only 1 can be chosen each day, the probability of picking the same mechanic twice in a row is 1/10 * 1/10 = 1/100.

A) 1/20
This probability would be accurate if the shop owner was choosing 2 mechanics out of the 10 each day. However, since only 1 mechanic is selected daily, the likelihood of choosing the same mechanic two days in a row is considerably lower.

B) 1/100
Correct! The probability of picking the same mechanic for a free lunch on two consecutive days is calculated by multiplying the probability of selection each day: 1/10 * 1/10 = 1/100.

C) 1/5
This probability would be relevant if the shop owner was choosing 1 mechanic out of 5 for a free lunch daily. However, with 10 mechanics in total, the chance of selecting the same mechanic two days in a row differs and is actually much smaller.

D) 1/10
While 1/10 represents the probability of choosing any single mechanic on a given day, the question specifically asks for the likelihood of selecting the same mechanic consecutively on two separate days. Therefore, the correct calculation involves multiplying the daily probabilities, resulting in a different outcome.

Conclusion
In this scenario, the probability of Fix It Fast's shop owner selecting the identical mechanic for a free lunch on two successive days is 1/100. By applying the basic principle of probability multiplication, where the chances of independent events occurring together are calculated by multiplying their individual probabilities, we arrive at this precise likelihood.

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Question 2

What is the slope of the line represented by the table?

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

The slope of the line represented by the table is -2.

The slope of a line is calculated by determining the ratio of the vertical change (change in y-coordinates) to the horizontal change (change in x-coordinates) between any two points on the line.

A) -4
This choice represents a slope that is different from the correct answer. A slope of -4 would indicate a steeper line compared to a slope of -2, implying a faster rate of change between points on the line.

B) -2.5
Similarly, a slope of -2.5 is not the correct answer. This value would also result in a line with a different steepness compared to a slope of -2, leading to a distinct visual representation on a graph.

D) -0.5
A slope of -0.5 differs from the correct answer. This value would indicate a shallower line compared to a slope of -2, suggesting a slower rate of change between points on the line.

Conclusion
The correct slope of the line represented by the table is -2. This value signifies the consistent rate of change between points on the line, reflecting a specific inclination that aligns with the data points provided in the table. Understanding slope is crucial in analyzing the relationship between variables in linear equations and graphing them accurately for interpretation and prediction purposes.

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Question 3

Which graph shows a line described by 4x - 3y = 12?

Your Answer: Option(s)

Correct Answer: Option(s) D

Rationale

Graph D) M-97D.png

Graph D corresponds to the line described by the equation 4x - 3y = 12. The slope-intercept form of this equation is y = (4/3)x - 4, indicating a slope of 4/3 and a y-intercept at (0, -4).

A) Graph A) M-97A.png
This graph does not represent the line 4x - 3y = 12. The line in this graph has a different slope and does not intersect the y-axis at y = -4.

B) Graph B) M-97B.png
The line in this graph does not align with the equation 4x - 3y = 12. Its slope and y-intercept are inconsistent with the characteristics of the given line.

C) Graph C) M-97C.png
Graph C does not depict the line defined by 4x - 3y = 12. The slope and y-intercept of the line in this graph do not match those of the equation.

Conclusion
Among the provided graphs, only Graph D accurately represents the line corresponding to the equation 4x - 3y = 12. The correct graph exhibits a slope of 4/3 and intersects the y-axis at y = -4, consistent with the properties of the given linear equation. Graphically identifying solutions to linear equations aids in understanding relationships between variables and visualizing mathematical concepts in a tangible way.

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Question 4

An advertisement poster in the window of a shoe store is in the shape of a rectangle. The length of the poster is 9 less than 4 × the width. Which expression represents the length of the poster when w is the width

Your Answer: Option(s)

Correct Answer: Option(s) A

Rationale

4w - 9

To determine the length of the rectangular advertisement poster, we first establish that the length is 9 less than 4 × the width. This relationship is accurately represented by the expression 4w - 9, where 4w signifies four × the width and subtracting 9 accounts for the length being 9 units less than this value.

B) 9 - 4w
This expression incorrectly reverses the order of operations, starting with 9 and then subtracting 4 × the width. However, the given information specifies that the length is dependent on the width, making 4w - 9 the appropriate arrangement to reflect this relationship accurately.

C) 4w + 9
Adding 9 to 4 × the width results in a value greater than the length of the poster, contrary to the provided condition that the length is 9 less than 4 × the width. Thus, 4w + 9 does not correctly represent the length based on the given scenario.

D) 9w - 4
This expression combines the width and constant values in an incorrect manner. The length of the poster is not determined by multiplying the width by 9 and subtracting 4; instead, it should reflect 4 × the width minus 9 to align with the information provided.

Conclusion
The expression 4w - 9 accurately represents the length of the rectangular advertisement poster based on the given conditions where the length is specifically described as 9 less than 4 × the width. By correctly formulating this relationship, the expression captures the necessary adjustment to the width to determine the length effectively within the context of the problem.

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Question 5

The radius of the sphere below is 6 centimeters (cm). What is the volume, in cubic centimeters, of the sphere?

Your Answer: Option(s)

Correct Answer: Option(s) A

Rationale

Volume of the sphere is 904.32 cubic centimeters.

The volume of a sphere is calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. Substituting the given radius of 6 cm into the formula yields a volume of 904.32 cubic centimeters.

A) 904.32
This is the correct answer, as calculated using the formula V = (4/3)πr^3 with r = 6 cm.

B) 150.72
This value is incorrect as it does not match the calculated volume for a sphere with a radius of 6 cm.

C) 25.12
This value is incorrect and significantly lower than the actual volume of the sphere with a radius of 6 cm.

D) 75.36
This value is incorrect and does not align with the calculated volume based on the provided radius of 6 cm.

Conclusion
The volume of a sphere is directly proportional to the cube of its radius, as demonstrated by the formula V = (4/3)πr^3. In this case, the given radius of 6 cm results in a volume of 904.32 cubic centimeters, making option A the correct choice for the volume of the sphere. Remember, when calculating the volume of a sphere, it is essential to use the correct formula and substitute the given values accurately to obtain the precise volume measurement.

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Probability of the shop owner picking the same mechanic 2 days in a row is 1/100.

The slope of the line represented by the table is -2.

Graph D) M-97D.png

4w - 9

Volume of the sphere is 904.32 cubic centimeters.

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