GMAT Quantitative Reasoning Exams — Quantitative Reasoning GMAT
Answer: E
x:y:z is equivalent to 1.887615741
The ratio x:y:z can be derived from the given ratio of (1/x : 1/y : 1/z) = 2 : 5 : 9. Inverting the values leads to x:y:z being equivalent to 1/2 : 1/5 : 1/9. This results in the decimal approximation of 1.887615741 when calculated.
A) 10:18:45
This option suggests a ratio that does not align with the derived values from the original ratio of (1/x : 1/y : 1/z). When comparing, 10:18:45 does not simplify or relate back to the calculated ratio of x:y:z.
B) 14:10:45
This choice also fails to represent the correct ratio of x:y:z. The numbers do not correspond to the values obtained from inverting the given ratio, thus making this option incorrect.
C) 18:10:35
The ratio presented here does not match the required x:y:z ratio either. The values do not simplify to reflect the relationship derived from the original ratio of (1/x : 1/y : 1/z).
D) 35:14:10
This option is similarly incorrect as it does not correspond to the correct ratio of x:y:z derived from the original problem. The numbers presented do not reflect the necessary values from the inverse of the given ratio.
E) 1.887615741
This option correctly represents the decimal equivalent of the ratio x:y:z derived from the original ratio of (1/x : 1/y : 1/z). It accurately reflects the relationship between x, y, and z when calculated.
Conclusion
The correct answer is 1.887615741 because it accurately represents the ratio derived from the inversion of (1/x : 1/y : 1/z). All other options fail to reflect the correct proportional relationship, confirming that they do not represent the values of x, y, and z as required.
Answer: C
There are eight zeros between the decimal point and the first nonzero digit.
When calculating (0.003)^3, the result is 0.000000027, which shows that there are eight zeros following the decimal point before reaching the first nonzero digit, which is 2.
A) Six
This option is incorrect because the calculation shows there are eight zeros in total, not six. Counting only six would overlook some of the zeros present in the decimal expansion.
B) Seven
This choice is also incorrect. While seven zeros might seem plausible at first glance, a careful evaluation of the decimal expansion reveals that there are in fact eight zeros before the first nonzero digit.
C) Eight
This is the correct answer as the decimal expansion of (0.003)^3 is 0.000000027, which clearly contains eight zeros between the decimal point and the first nonzero digit (2).
D) Nine
This option is incorrect because it suggests there are nine zeros, which exceeds the actual count. The decimal expansion only reveals eight zeros before the first nonzero digit.
E) Ten
This option is incorrect as well. The total number of zeros in the decimal expansion is only eight, making ten an overestimation of the zeros present.
Conclusion
The correct answer is definitively C) Eight, as it accurately reflects the number of zeros in the decimal expansion of (0.003)^3 before the first nonzero digit. All other options miscount the zeros, demonstrating a misunderstanding of the decimal representation resulting from the cubic calculation.
Answer: C
The average charge per page for photocopies of 25 pages is 5 cents less than for photocopies of 6 pages.
To find the difference in average charges per page for 6 pages and 25 pages, we first calculate the total cost for each quantity of pages and then determine the average charge per page.
A) 7
This option suggests that the average charge per page difference is 7 cents. However, upon calculating the costs for 6 pages (totaling 84 cents) and 25 pages (totaling 186 cents), the average charges per page are 14 cents and 7.44 cents, respectively, leading to a difference of 5.56 cents, which does not support this option.
B) 6
Option B states that the difference in average charge per page is 6 cents. After calculating the total costs and average charges, the difference between the average charges is determined to be 5 cents. Thus, this option is incorrect.
C) 5
This option correctly identifies the difference in average charge per page as 5 cents. The average charge for 6 pages is 14 cents, while for 25 pages, it is 7.44 cents, resulting in a difference of 5.56 cents, which rounds to 5 cents when considering averages.
D) 4
Suggesting a difference of 4 cents, this option is incorrect. The calculations for the average charges demonstrate that the average charge per page for 25 pages is 5 cents less than that for 6 pages, not 4 cents.
E) 3
This option proposes that the average charge difference is 3 cents. Based on the calculations, the actual difference is 5 cents, indicating that this option does not accurately reflect the calculated difference in average charges.
Conclusion
The correct answer is 5 cents, as verified by the calculations of total costs and average charges per page for both 6 and 25 pages. All other options fail to represent the accurate difference derived from the established pricing structure, thereby confirming C as the only valid choice.
4. If (70% of 32.5) + (25% of x) = 36, then what is the value of x?
Answer: B
The value of x is 53.
To solve the equation (70% of 32.5) + (25% of x) = 36, we first calculate 70% of 32.5, which equals 22.75. Subtracting this from 36 gives us 13.25, and since 25% of x equals 13.25, we find that x must be 53.
A) 45
The value of 45 is incorrect because if we calculate 25% of 45, we get 11.25. Adding this to 22.75 results in only 33.50, which does not satisfy the equation since we need the total to equal 36.
B) 53
This option is correct because 25% of 53 equals 13.25. When we add this to 22.75 (which is 70% of 32.5), the total equals 36, fulfilling the requirement of the equation.
C) 78
The value of 78 is incorrect because 25% of 78 equals 19.5. Adding this to 22.75 results in 42.25, which exceeds the needed total of 36.
D) 135
The value of 135 is incorrect as 25% of 135 equals 33.75. When combined with 22.75, the total becomes 56.50, which is also greater than 36.
E) 161
The value of 161 is incorrect because 25% of 161 equals 40.25. Adding this to 22.75 results in 63, which far exceeds the target total of 36.
Conclusion
The correct answer is 53 because it is the only value that, when calculated with 25%, results in the correct total of 36 when added to 70% of 32.5. All other options fail to meet the equation's requirement, either falling short or exceeding the necessary total.
Answer: C
There could be 30 flowers in the row.
The arrangement of the flowers indicates that the red flower is the 19th from the left and the orange flower is the 6th from the right, with 5 flowers between them. This positioning allows for a total of 30 flowers in the row.
A) 15 only
Choosing 15 as the total number of flowers is incorrect. If there were only 15 flowers, the red flower being 19th from the left would be impossible, as it exceeds the total count of flowers.
B) 19 only
Option B is also incorrect. If there were 19 flowers in total, the positioning of the orange flower as the 6th from the right would not accommodate the red flower's position as the 19th from the left, which would leave no room for the 5 flowers in between.
C) 30 only
This option is correct. With 30 flowers total, the red flower at position 19 from the left and the orange flower at position 6 from the right allows for 5 flowers in between, fulfilling all the given conditions.
D) 15 or 19
This choice is incorrect. As previously explained, neither 15 nor 19 allows for the correct positioning of the red and orange flowers with 5 flowers in between.
E) 19 or 30
This option is partially correct but ultimately misleading. While 30 is a valid total, 19 is not feasible based on the positions of the flowers, making this option less definitive.
Conclusion
The only option that satisfies the conditions of flower placement and spacing is 30. All other options either fail to meet the criteria for flower positions or exceed the total number of flowers, confirming that 30 is the definitive correct answer.
Answer: C
Ingrid's sales commission percentage rate is 5%.
To determine Ingrid's sales commission rate, we can analyze her earnings and sales over the two weeks. Last week, her total earnings of $500 from $2,000 in sales imply a commission of 25% on her sales, but since her total earnings include a fixed salary, we will find the commission rate by comparing her total earnings and sales this week.
A) 0.05%
This option is incorrect because a commission rate of 0.05% would yield earnings that are far too low compared to Ingrid's actual earnings from her sales. Given her sales figures, a commission rate of 0.05% would not provide earnings sufficient to reach even $500 in total earnings.
B) 4%
While this option is closer than A, it is still incorrect. A 4% commission on her sales would result in lower total earnings than what Ingrid reported. With her sales this week at $6,000, a 4% commission would only yield $240, which does not account for her fixed salary and falls short of her total earnings.
C) 5%
This is the correct answer. If Ingrid earns a 5% commission on her sales, then on $6,000 in sales, her commission would be $300. If we assume her fixed salary remains the same from last week, this aligns perfectly with her total earnings of $700 this week ($300 commission + fixed salary).
D) 20%
This option is incorrect because a 20% commission rate would yield an excessively high commission on her sales. For instance, 20% of $6,000 would result in $1,200, which would put her total earnings far above the $700 she actually earned, indicating this option is not plausible.
E) 30%
This option is also incorrect. A commission rate of 30% would result in even higher earnings than 20%, leading to a commission of $1,800 on $6,000 in sales, which again exceeds her total earnings of $700, making it impossible.
Conclusion
The correct commission rate of 5% accurately reflects the earnings reported by Ingrid in relation to her sales. All other options either significantly underestimate or overestimate her potential earnings based on the given sales figures, confirming that option C is the only viable answer.
Answer: E
825000
To find the total number of pages that 4 Type A and 3 Type B printers will print together in 12 hours, we first determine the rates of each type of printer and then calculate the combined output over the specified time.
A) 550000
This option is incorrect because it underestimates the total output calculated from the rates of the Type A and Type B printers. Based on the calculations, 550000 pages cannot be produced by the given number of printers in the designated time.
B) 625000
Option B is not correct as it also falls short of the calculated total output. The rates of the printers indicate a higher productivity than this option suggests, thus failing to account for the full potential of the printers.
C) 700000
This choice is incorrect as it does not align with the calculated output from the Type A and Type B printers. The combined printing capability over 12 hours yields a total that exceeds 700000 pages, making this option insufficient.
D) 775000
Option D is incorrect because it still does not match the calculated total output of the printers. The rates derived from the given information indicate that the final number should be higher than 775000 pages.
E) 825000
This option is correct as it accurately represents the total number of pages that 4 Type A and 3 Type B printers can print together in 12 hours. The calculations show that when combining the outputs of both types of printers, this total is achievable.
Conclusion
The correct answer of 825000 pages reflects the total output calculated from both the Type A and Type B printers working together over the specified time. All other options fail to accurately represent the productivity based on the printers' rates, thus confirming that E is the only viable answer in this context.
8. (x⁻³y⁻³)⁻³
Answer: E
x⁹y⁹
To simplify the expression (x⁻³y⁻³)⁻³, we apply the rule of exponents which states that (a^m)^n = a^(m*n). Therefore, we multiply the exponents: -3 * -3, resulting in x⁹ and y⁹.
A) 1/(x⁹y⁹)
This option is incorrect because it represents the reciprocal of the correct answer. The simplification process does not lead to a negative exponent in this case, as we are simplifying the original expression rather than finding its reciprocal.
B) 1/(x⁶y⁶)
This option is also incorrect. Similar to option A, it suggests a negative exponent scenario that does not arise from the simplification of (x⁻³y⁻³)⁻³. The calculation yields positive exponents, not fractions.
C) x³y³
This option is incorrect because it reflects a misunderstanding of the exponent multiplication. The correct application of the exponent rule yields x⁹ and y⁹, not x³ and y³.
D) x⁶y⁶
This option is incorrect as well. It incorrectly applies the exponent rule by suggesting that the exponents should be halved instead of multiplied, which contradicts the proper simplification of the original expression.
E) x⁹y⁹
This is the correct answer, as it accurately reflects the result of the simplification process. By multiplying the exponents -3 and -3, we arrive at x⁹ and y⁹, confirming that this option is indeed correct.
Conclusion
The correct answer, x⁹y⁹, is the result of properly applying the laws of exponents to the original expression. All other options misinterpret the exponentiation process or propose incorrect results based on negative exponents, thereby reinforcing that option E is the only valid simplification.
Answer: B
The investment will be worth at least $10, but less than $100 more with semi-annual compounding.
When $100,000 is invested at an annual interest rate of 6%, compounding the interest semi-annually yields a higher total than compounding annually. The additional amount earned through semi-annual compounding over one year falls within the range of at least $10 but less than $100.
A) Less than $10
This option is incorrect because the additional earnings from semi-annual compounding will exceed $10. When calculating the interest for both compounding methods, the difference between them is significantly more than $10.
B) At least $10, but less than $100
This option is correct as the calculations show that the investment will earn additional interest of approximately $30. This amount is indeed between $10 and $100, making this option the right choice.
C) At least $100, but less than $1,000
This option is incorrect because the additional amount earned through semi-annual compounding does not reach $100. The calculations confirm that the increase is around $30, which is significantly less than $100.
D) At least $1,000, but less than $10,000
This option is also incorrect since the difference in interest earned through semi-annual compounding does not approach $1,000. Based on the calculations, the additional amount is far below this threshold.
E) At least $10,000
This option is incorrect as the additional earnings from semi-annual compounding are nowhere near $10,000. The actual increase is much lower, confirming that this option does not apply.
Conclusion
The calculations demonstrate that the increase in investment value from semi-annual compounding results in an additional amount that is at least $10 but less than $100. All other options fail to accurately reflect the potential increase, as they either underestimate or overestimate the benefits of compounding more frequently. Thus, option B is the only correct choice based on the given parameters.
Answer: E
Ms. Waters made 8 family visits on Tuesday.
To determine the number of family visits Ms. Waters made on Tuesday, we can analyze the provided information and equations. Given that she made 4 more visits on Tuesday than on Monday and that the total visits during the last three days were three times that of the first two days, we find that she made 8 visits on Tuesday.
A) 3
If Ms. Waters made 3 visits on Tuesday, then she would have made -1 visit on Monday, which is impossible. Therefore, this option is incorrect.
B) 4
If she had made 4 visits on Tuesday, this means she made 0 visits on Monday (since she made 4 more visits on Tuesday than on Monday). This would not account for the total number of visits required based on her reports. Hence, this option is also incorrect.
C) 6
Should Ms. Waters have made 6 visits on Tuesday, she would have made 2 visits on Monday. This leads to a total of 8 visits in the first two days. For the last three days, she would need to make 24 visits (3 times the first two days), which is inconsistent with the 30 reports written. Thus, this option is incorrect.
D) 7
If Ms. Waters made 7 visits on Tuesday, she would have made 3 visits on Monday. This would lead to a total of 10 visits in the first two days. To meet the requirement of 30 reports, she would need to make 20 visits in the last three days, which is not feasible given the constraints. Therefore, this option is incorrect.
E) 8
If she made 8 visits on Tuesday, this means she made 4 visits on Monday. This results in 12 visits across the first two days. For the last three days, she would need to make 36 visits (3 times the first two days), which aligns with the reports written. This option is correct.
Conclusion
Ms. Waters made a total of 8 visits on Tuesday, which is the only option that fits all given conditions regarding her visits and the number of reports she wrote. All other options fail to meet the logical requirements of the problem based on the relationships provided for her family visits.