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Question 1
The product of which of the following pairs of numbers is closest to 5,678?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
6 and 950 produce a product closest to 5,678.
Multiplying 6 by 950 gives 5,700, which is only 22 away from 5,678, making it the closest product among the provided options.
A) 5 and 990
Calculating the product of 5 and 990 results in 4,950. This value is 728 less than 5,678, placing it further from the target compared to other options.
B) 6 and 900
The product of 6 and 900 is 5,400. This result is 278 less than 5,678, which makes it closer than option A but still not as close as option C.
C) 6 and 950
As mentioned, multiplying 6 by 950 yields 5,700, which is just 22 away from 5,678. This is the closest product of all the choices, confirming it as the correct answer.
D) 7 and 800
The product of 7 and 800 is 5,600. This result is 78 less than 5,678, making it a reasonable option, but still further from the target compared to option C.
E) 7 and 850
Calculating the product of 7 and 850 results in 5,950. This value is 272 greater than 5,678, placing it further away from the desired number than any other option.
Conclusion
Among the pairs of numbers provided, the multiplication of 6 and 950 results in a product of 5,700, the closest to the target of 5,678. Other combinations yield products that are either significantly higher or lower, making option C the optimal choice for this question.
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Question 2
Chairs are to be arranged in a school gymnasium so that the number of chairs in each row will be 8 greater than the number of rows. If a total of 660 chairs are to be arranged in this fashion, how many rows of chairs will there be?
Your Answer: Option(s)
Correct Answer: Option(s) B
Rationale
There will be 22 rows of chairs.
To find the number of rows, we can set up an equation based on the problem's description. Let the number of rows be represented by \( r \). Then the number of chairs in each row will be \( r + 8 \). The total number of chairs is given by the equation \( r \times (r + 8) = 660 \). Solving this quadratic equation yields \( r = 22 \).
A) 12
If there are 12 rows, then the number of chairs per row would be \( 12 + 8 = 20 \), leading to a total of \( 12 \times 20 = 240 \) chairs. This total is significantly less than 660, making this option incorrect.
B) 22
With 22 rows, the number of chairs in each row would be \( 22 + 8 = 30 \). Multiplying gives a total of \( 22 \times 30 = 660 \) chairs, which matches the requirement perfectly, confirming this choice as correct.
C) 26
For 26 rows, the number of chairs per row would be \( 26 + 8 = 34 \). The total would be \( 26 \times 34 = 884 \) chairs, which exceeds 660, thus making this option invalid.
D) 30
If there are 30 rows, then the chairs per row would be \( 30 + 8 = 38 \), resulting in \( 30 \times 38 = 1140 \) chairs. This is also much greater than 660, ruling this option out.
E) 55
With 55 rows, the chairs per row become \( 55 + 8 = 63 \). The total would then be \( 55 \times 63 = 3465 \) chairs, which is far too high compared to 660, making this choice incorrect.
Conclusion
Through the setup of a quadratic equation based on the relationship between rows and chairs, we determined that 22 rows lead to the correct arrangement of 660 chairs. Other options either result in totals that are too low or too high, affirming the uniqueness of the solution found.
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Question 3
The list shown gives the number of movies that each of six of Sam's friends saw last year: 12, 17, 9, 11, 12, 14. Sam's sister Lucy saw 7 fewer movies last year than Sam did. The range of the numbers of movies that Sam, Lucy, and each of the six friends saw last year was 10. Which of the following could be the number of movies that Sam saw last year?
Your Answer: Option(s)
Correct Answer: Option(s) A
Rationale
Sam could have seen 14 movies last year.
Given that the range of the number of movies seen by Sam, Lucy, and his six friends is 10, the maximum and minimum values in the combined dataset must be 10 units apart. If Sam saw 14 movies, Lucy would have seen 7 fewer, totaling 7 movies, which fits within the range established by the friends' counts.
A) 14
If Sam saw 14 movies, then Lucy would have seen 7 movies (14 - 7). The highest number of movies seen by Sam's friends is 17 and the lowest is 9. The range (17 - 7 = 10) fits perfectly, confirming that this scenario maintains the required range of 10.
B) 15
If Sam saw 15 movies, Lucy would have seen 8 movies (15 - 7). The highest number of movies seen by friends is still 17. The range would then be 17 - 8 = 9, which does not satisfy the condition of a range of 10.
C) 16
If Sam saw 16 movies, Lucy would have seen 9 movies (16 - 7). The range would be 17 - 9 = 8, again failing to meet the required range of 10.
D) 17
If Sam saw 17 movies, Lucy would have seen 10 movies (17 - 7). The range would be 17 - 10 = 7, which does not meet the requirement of a range of 10 either.
E) 18
If Sam saw 18 movies, Lucy would have seen 11 movies (18 - 7). The range would be 17 - 11 = 6, which is also insufficient to fulfill the 10-unit range requirement.
Conclusion
The only scenario that fits the parameters provided is that Sam saw 14 movies, leading to Lucy watching 7 movies. This configuration results in a range of 10 when compared to the highest and lowest movie counts among all individuals involved. Hence, option A is the only valid answer.
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Question 4
Amin's salary was decreased by 20% in 2015, but he got an increase of 50% in 2016. Suri's salary was increased by 50% in 2015 and again by 6.67% in 2016. If Amin's salary was 10% less than Suri's salary in 2016, by what percent was Amin's salary more or less than Suri's salary in 2014?
Your Answer: Option(s)
Correct Answer: Option(s) D
Rationale
Amin's salary was 20% more than Suri's salary in 2014.
To determine the relationship between Amin's and Suri's salaries in 2014, we analyze their respective salary changes. After calculating the effects of the percentage increases and decreases over the years, we find that Amin's 2014 salary was indeed 20% more than that of Suri.
A) 20% less
This option suggests that Amin's salary in 2014 was 20% lower than Suri's salary. However, the calculations show that Amin's salary was higher, not lower, after accounting for the changes in salaries over the two years.
B) 0
Choosing this option implies that Amin's salary in 2014 was equal to Suri's salary. However, the calculations indicate a difference, with Amin earning more than Suri. Therefore, this option does not reflect the relationship established by the salary adjustments.
C) 10% more
This choice states that Amin's salary was only 10% higher than Suri's in 2014. While it is true that Amin's salary increased due to the subsequent raises, the actual calculations reveal that the difference was greater than 10%, specifically 20%.
D) 20% more
This is the correct answer, as the calculations show that after considering the percentage changes to both Amin's and Suri's salaries over the years, Amin's salary in 2014 was indeed 20% more than Suri's salary.
E) 25% more
This option suggests a larger difference than what is calculated. The analysis reveals that Amin's salary was not 25% more than Suri's in 2014, but rather a precise increase of 20%, making this option incorrect.
Conclusion
Through careful calculation of the percentage changes to their salaries, we find that Amin's salary in 2014 was 20% more than Suri's salary. Each salary adjustment was crucial to determining the final comparison, highlighting the importance of accurately applying percentage changes in financial scenarios.
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Question 5
Two weeks ago a certain sweater was offered at a sale price that was 20 percent less than the retail price, and last week it was offered at a clearance price that was 30 percent less than the sale price. The clearance price of the sweater was what percent less than the retail price?
Your Answer: Option(s)
Correct Answer: Option(s) B
Rationale
The clearance price of the sweater was 44% less than the retail price.
To determine how much less the clearance price is compared to the retail price, we first calculate the sale price and then the clearance price based on the percentages provided. The final calculation shows a reduction of 44% from the retail price.
A) 40%
A reduction of 40% would imply that the clearance price is only slightly lower than what we calculated. However, based on the calculations, the actual percentage decrease is greater, specifically at 44%, meaning this option does not accurately reflect the difference between the clearance and retail prices.
B) 44%
This is the correct choice, as the calculations show that the clearance price, after both discounts, results in a total decrease of 44% from the original retail price. This option correctly represents the relationship between the clearance price and the retail price.
C) 50%
A 50% reduction would suggest that the clearance price is half of the retail price, which is not the case in this scenario. The actual calculations demonstrate a smaller percentage decrease when considering both the sale and clearance discounts.
D) 56%
This option implies an even larger decrease than what was calculated. A reduction of 56% would mean the clearance price is significantly lower than the calculated 44%, which does not align with the provided discount percentages.
E) 0.6
This option does not represent a percentage decrease and is irrelevant in the context of the question. The question specifically asks for a percentage less than the retail price, and 0.6 does not fit this requirement.
Conclusion
The calculations confirm that the clearance price is 44% less than the retail price after applying the sequential discounts of 20% and 30%. Each incorrect option either misrepresents the percentage decrease or fails to provide a valid response within the context of the question, emphasizing the accuracy of the 44% reduction.
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