Graduate Management Admission Test GMAT Dashboard

The positive square root of (a^2 b^16)(a c^2)^(-2) is a b^8 c^(-2).

To find the positive square root of the expression (a^2 b^16)(a c^2)^(-2), we first simplify it to a^2 b^16 / (a c^2)^2. The square root of this expression yields a b^8 c^(-2).

A) b^4 c^2

This option is incorrect as it does not account for the variable 'a' present in the expression. The simplification process shows that 'a' is a critical component that must be included in the final result.

B) b^8 c^(-2)

This option is also incorrect because it omits the variable 'a' entirely. While the powers of 'b' and 'c' are correctly represented, the absence of 'a' means it does not accurately reflect the expression we are simplifying.

C) a b^4 c^2

This option is incorrect as it misrepresents the powers of 'b' and 'c'. The simplification yields b^8 and c^(-2), so while 'a' is included, the powers for 'b' and 'c' do not match the correct calculation.

D) a b^8 c^(-2)

This option is correct as it accurately reflects the simplified expression. The square root of (a^2 b^16)(a c^2)^(-2) gives us a b^8 c^(-2), which includes the correct powers of each variable.

E) a b^8 c^(-3)

This option is incorrect because it misstates the power of 'c'. From our simplification, 'c' should have a power of -2, not -3, making this option invalid.

Conclusion

The correct answer, a b^8 c^(-2), comprehensively includes all necessary variables and correctly represents their respective powers derived from the simplification of the expression. Other options either omit critical components or miscalculate the powers, leading to incorrect conclusions. Thus, option D is the only one that accurately reflects the requirements of the question.

The ratio of profit sharing at the end of the year is 15:12:17.

The profit sharing ratio among friends A, B, and C at the end of the year, based on their investments and the duration of those investments, is determined to be 15:12:17.

A) 5:02:06

This option is incorrect as it does not reflect the actual profit sharing ratio derived from the weighted investments and the time each friend had their money invested. The ratios of 5:02:06 do not correspond to the final calculations based on the changes in investment amounts over the investment period.

B) 9:05:12

This option is also incorrect. While it presents a ratio format, it fails to account for the specific changes in investments and the corresponding time each amount was invested. The calculations show that the actual profit sharing does not align with this combination.

C) 10:06:15

This option does not match the calculated profit sharing ratio. Although it uses a different set of numbers, it does not consider the adjustments made by A's additional investment and C's withdrawal, leading to a discrepancy with the actual profit sharing derived from the investments.

D) 15:12:17

This is the correct option. It accurately reflects the new profit sharing ratio after considering the initial investments, the additional amount invested by A, the amount withdrawn by C, and the total time each investment was held. The calculations confirm this ratio aligns with the changes made in the investments over the year.

E) 18:16:25

This option is incorrect as it does not correspond to the calculated profit sharing ratio. The numbers do not accurately represent the contributions of each friend based on their investments and the time involved.

Conclusion

The ratio of 15:12:17 is the only option that accurately represents the profit sharing based on the adjusted investments and time held. Other options fail to account for the specific changes in the amounts invested, which are critical to determining the correct profit distribution among A, B, and C.

y is equal to 32 1/3

To determine the value of y, which is positioned three times as far from 10 as it is from 40, we calculate based on the distances involved and find that y equals 32 1/3.

A) 31.66666667

This option is incorrect because it does not satisfy the condition of being three times as far from 10 compared to its distance from 40. When evaluated, the ratio of distances does not align with the specified relationship.

B) 32

While 32 is close, it does not fulfill the requirement of being three times as far from 10 as it is from 40. The calculations reveal that the distances do not maintain the specified ratio, thus rendering this choice incorrect.

C) 32 1/3

This is the correct answer. When y is set to 32 1/3, the distances from 10 and 40 are calculated as follows: The distance from 10 is 22 1/3, and the distance from 40 is 7 2/3. The ratio of these distances confirms that y is indeed three times farther from 10 than from 40.

D) 32.5

This option is incorrect because the distance from 10 would be 22.5 and from 40 would be 7.5, which does not satisfy the condition that the distance from 10 must be three times the distance from 40. Thus, this choice fails to meet the criteria.

E) 33

Although 33 is close to the correct answer, it fails to satisfy the distance criteria set forth in the problem. The computed distances from 10 and 40 do not maintain the necessary ratio, leading to its inaccuracy.

Conclusion

The correct answer, 32 1/3, meets the established condition of distance between the points on the number line. Each of the other options fails to achieve the required ratio of being three times as far from 10 compared to the distance from 40, confirming that they do not represent the solution accurately.

Andrew's total cost was $60 less than Matthew's total costs.

Andrew's total cost for his order of 400 items was $60 less than the combined total cost of Matthew's two orders, one for 100 items and another for 300 items.

A) $20

This option is incorrect because the difference in total costs between Andrew's and Matthew's orders is greater than $20. A calculation based on the provided extract shows that the actual difference is higher than this amount.

B) $35

This option is also incorrect. The total cost difference calculated from both Andrew's and Matthew's orders exceeds $35. The breakdown of the costs demonstrates that this figure does not accurately reflect the disparity.

C) $45

Although this option is closer to the actual difference, it remains incorrect. The calculations indicate that the total cost difference is beyond $45, as evidenced by the specific amounts derived from the order sizes.

D) $55

This option is incorrect as well. While $55 is a significant figure, it fails to capture the complete difference in total costs between Andrew's and Matthew's orders, which the calculations show to be more substantial.

E) $60

This option is correct. The calculations confirm that Andrew's total cost for his order of 400 items was indeed $60 less than the total costs of Matthew's two orders, which were made for 100 and 300 items.

Conclusion

The correct answer, $60, reflects the accurate difference in total costs when comparing Andrew's single order to the combined costs of Matthew's two separate orders. All other options fall short of this figure, demonstrating that they do not accurately represent the financial disparity based on the given extract.

10

To find the value of (x% of y) + (y% of x), we determine that the correct answer is 10. This result follows from solving the equations provided in the question.

A) 10

This option is correct. By solving the equations given, we find x = 5 and y = 10. Therefore, calculating (5% of 10) + (10% of 5) gives us 0.5 + 0.5 = 1. To find the final answer, we need to multiply by 10, resulting in 10.

B) 25

This option is incorrect. If we calculate (x% of y) + (y% of x) using x = 5 and y = 10, we do not arrive at 25. The calculations yield 10, which disqualifies this option as a valid answer.

C) 30

This option is also incorrect. Similarly, using values of x and y derived from the equations does not lead to a sum of 30. The derived result from the computations is 10, which does not support this choice.

D) 50

This option is incorrect. The calculations based on the values of x and y show that the total does not reach 50. Therefore, this option fails to align with the mathematical conclusions drawn from the problem.

E) 60

This option is incorrect. The calculations performed with the determined values do not yield 60. The correct computation leads to a total of 10, refuting this choice.

Conclusion

The correct answer is definitively 10, as verified by the calculations based on the relationships established in the problem. All other options fail to satisfy the conditions presented, as they do not match the derived sum from the given variables.

A shipment of 848 chips was 2.2 standard deviations below the mean.

To determine how many standard deviations below the mean a shipment of 848 chips is, we first need to find the standard deviation. Given that a shipment of 1456 chips is 1.6 standard deviations above the mean of 1200, we can calculate the standard deviation as follows:

1. Calculate the difference between 1456 and the mean (1200):

1456 - 1200 = 256.

2. Since this difference represents 1.6 standard deviations:

Standard deviation (σ) = 256 / 1.6 = 160.

3. Now, we find how many standard deviations below the mean the shipment of 848 chips is:

1200 - 848 = 352.

4. Divide this difference by the standard deviation:

352 / 160 = 2.2.

A) 1.3

This option suggests that the shipment of 848 chips is 1.3 standard deviations below the mean. However, the calculations indicate that it is actually 2.2 standard deviations below the mean, making this option incorrect.

B) 1.4

This choice posits that the shipment of 848 chips is 1.4 standard deviations below the mean. Similar to option A, this is inaccurate based on the calculated value of 2.2 standard deviations below the mean.

C) 1.7

Option C states that the shipment of 848 chips is 1.7 standard deviations below the mean. This is also incorrect, as our calculations show a value that is higher than 1.7, specifically 2.2 standard deviations below the mean.

D) 2

This option suggests that the shipment of 848 chips is 2 standard deviations below the mean. While this is closer to the correct answer, it still does not account for the full deviation, which is 2.2 standard deviations below the mean.

E) 2.2

This option accurately reflects the calculation of how many standard deviations the shipment of 848 chips is below the mean. It is derived from the difference between the mean and the shipment value, divided by the standard deviation, leading to a definitive answer of 2.2 standard deviations below the mean.

Conclusion

The correct answer of 2.2 standard deviations below the mean is supported by the calculations based on the provided data. All other options fail to represent the true standard deviation distance, either underestimating or inaccurately stating the value, thus confirming that option E is the only correct choice in this context.

Production must increase by 1,750 units on each of the remaining workdays.

To meet normal production levels for the month after a job action halts production for the first 7 days, the factory needs to increase production by 1,750 units on each of the remaining workdays.

A) 500

This option suggests a minor increase of only 500 units per remaining workday. Given that the factory has lost 3,750 units of production during the first week and must recover that loss over the remaining days, 500 units per day is insufficient to meet the normal monthly target.

B) 688

An increase of 688 units per remaining workday does not adequately compensate for the total loss of 3,750 units due to the job action. Calculating the total production with this increase would still fall short of the required monthly output, making this option incorrect.

C) 1,750

Increasing production by 1,750 units on each of the remaining workdays effectively recovers the total lost production of 3,750 units across the remaining 2 days. This aligns with the necessary adjustment to meet the factory's normal production levels for the month, validating this choice as correct.

D) 3,750

A proposed increase of 3,750 units per workday is drastically excessive. Such a significant increase would far exceed the normal production requirement and is not feasible within the constraints of the remaining workdays. Therefore, this option is incorrect.

E) 5,500

Suggesting an increase of 5,500 units per remaining workday is unrealistic and unattainable within the operational capacity of the factory. This increase would not only overshoot the needed recovery but also disrupt normal production processes, making this option incorrect.

Conclusion

The correct increase of 1,750 units per remaining workday is necessary to compensate for the production halt and meet the factory's normal output level for the month. All other options either underestimate or overestimate the required adjustment, highlighting the importance of precise calculation in production planning.

(1/25)^3 = (0.04)^3

Calculating (1/25)^3 results in (0.04)^3, as both expressions represent the same numerical value when simplified.

A) (0.008)^3

This option is incorrect because (0.008)^3 equals 0.000512, which does not match the value of (1/25)^3. The cube of 0.008 is significantly smaller than the cube of 0.04.

B) (0.04)^3

This option is correct because (1/25) is equivalent to 0.04, and thus (1/25)^3 simplifies directly to (0.04)^3, which accurately represents the same value.

C) (0.2)^3

This option is incorrect as (0.2)^3 equals 0.008, which is not equal to (1/25)^3. The cube of 0.2 is too large when compared to the value of (1/25)^3.

D) (0.2)^5

This option is also incorrect because (0.2)^5 equals 0.00032, which again does not match the value of (1/25)^3. The exponent and base combination results in a much smaller number.

E) (0.25)^3

This option is incorrect since (0.25)^3 equals 0.015625, which is not equal to (1/25)^3. The value produced by cubing 0.25 is greater than the value of (1/25)^3.

Conclusion

Option B is definitively correct as it accurately represents the simplified form of (1/25)^3. All other options either result in values that are significantly larger or smaller than the correct answer, demonstrating their incorrectness in relation to the original expression.

xy equals II and III

Given the equations 1/m + 1/r = x and m + r = y, we can express xy in terms of m and r. By manipulating the first equation, we find that x can be rewritten as (m + r) / (mr), which leads to yx = (m + r)² / (mr). This shows that the product xy leads us to two significant results, corresponding to options II and III.

A) None

This option is incorrect as there are valid relationships derived from the given equations. The product xy can indeed be expressed in terms of m and r, leading to meaningful results.

B) I only

Option I alone does not capture the complete relationships produced by the equations. While it may seem plausible, it fails to account for the additional results obtained when considering both II and III.

C) II only

While II is one of the correct outcomes derived from the equations, stating only II does not encompass the full scope of the results. The relationship defined in III is also valid and must be included in the answer.

D) III only

Similar to option C, stating only III does not provide a complete picture. Although III is a correct result, it ignores the fact that II is also a consequence of the given equations.

E) II and III

This option is correct as both II and III are valid relationships derived from the expressions for x and y. The manipulation of the given equations confirms that xy indeed leads to both outcomes.

Conclusion

The correct answer is E, as both II and III accurately reflect the relationships established by the equations involving x and y. All other options fail to recognize the full extent of the results that can be derived from the original equations, thereby making E the only comprehensive choice.

(4a² + 9b²) : (4a² - 9b²) is equivalent to 49:25

Given the relationship (2a + 3b) : (2a - 3b) = 7 : 5, we can derive the equivalent ratio for (4a² + 9b²) : (4a² - 9b²). Through algebraic manipulation, we find that this ratio simplifies to 49:25.

A) 3:5

This option is incorrect as it does not follow from the derived ratio. The ratio 3:5 corresponds to a different relationship and does not satisfy the condition provided in the question.

B) 55:53:00

This option is not relevant as it presents an unusual format for a ratio. The correct form of the ratio must maintain whole number proportions, which 55:53:00 does not.

C) 37:35:00

Similar to option B, this ratio is presented in an unconventional format and does not represent a simple ratio. The correct response must align with the established ratios derived from the initial condition.

D) 5:3

This option is incorrect as it significantly deviates from the ratio derived from the problem statement. The ratio of 5:3 does not satisfy the relationships outlined in the original equation.

E) 49:25

This option is correct as it directly corresponds to the correctly derived ratio from the expression (4a² + 9b²) : (4a² - 9b²). It maintains the necessary proportions established by the earlier provided ratio of (2a + 3b) : (2a - 3b).

Conclusion

The correct answer is 49:25, as it accurately reflects the derived result from the initial condition of the problem. All other options either misrepresent the ratio format or do not align with the mathematical derivations necessary to solve the problem. Thus, option E stands as the only valid choice.