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Question
1
If x - 2y = 2 and x ^ 2 + 4xy + 4y ^ 2 = 8, what is the value of xy?
Rationale
<h2>xy equals 1/2.</h2>
To find the value of xy, we can manipulate the given equations. Starting from the first equation, x - 2y = 2, we can express x in terms of y, and then substitute into the second equation to solve for xy.
<b>A) 1/2</b>
By substituting x = 2 + 2y into the second equation, we derive that (2 + 2y)^2 + 4(2 + 2y)(y) + 4y^2 = 8 simplifies to yield xy = 1/2. This value satisfies both original equations, confirming it as the solution.
<b>B) 1</b>
If we assume xy = 1, substituting back into the second equation would not satisfy the equation since the derived expressions do not hold true. Therefore, this choice does not match the values obtained from the equations.
<b>C) 2</b>
Assuming xy = 2 leads to a contradiction when substituted back into the original equations, revealing that the values of x and y derived from this assumption would not satisfy the necessary conditions set by the equations.
<b>D) 5/2</b>
If we consider xy = 5/2, substituting this value into either of the equations results in a disparity that cannot be resolved, hence failing to satisfy the original equations. Thus, this choice is not valid.
<b>Conclusion</b>
The correct value of xy, derived through manipulation of the equations provided, is 1/2. This value satisfies both equations and maintains the relationships established in the problem. Therefore, it accurately reflects the solution sought in the question.
To find the value of xy, we can manipulate the given equations. Starting from the first equation, x - 2y = 2, we can express x in terms of y, and then substitute into the second equation to solve for xy.
<b>A) 1/2</b>
By substituting x = 2 + 2y into the second equation, we derive that (2 + 2y)^2 + 4(2 + 2y)(y) + 4y^2 = 8 simplifies to yield xy = 1/2. This value satisfies both original equations, confirming it as the solution.
<b>B) 1</b>
If we assume xy = 1, substituting back into the second equation would not satisfy the equation since the derived expressions do not hold true. Therefore, this choice does not match the values obtained from the equations.
<b>C) 2</b>
Assuming xy = 2 leads to a contradiction when substituted back into the original equations, revealing that the values of x and y derived from this assumption would not satisfy the necessary conditions set by the equations.
<b>D) 5/2</b>
If we consider xy = 5/2, substituting this value into either of the equations results in a disparity that cannot be resolved, hence failing to satisfy the original equations. Thus, this choice is not valid.
<b>Conclusion</b>
The correct value of xy, derived through manipulation of the equations provided, is 1/2. This value satisfies both equations and maintains the relationships established in the problem. Therefore, it accurately reflects the solution sought in the question.
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Question
2
Let M = (3)(89,416.04) and N = 89,412 + 89,414 + 89,419. Which of the following is equal to 5M - 5N?
Rationale
<h2>5M - 5N is equal to 15.6.</h2>
To solve for 5M - 5N, we first calculate M and N. M equals 3 times 89,416.04, resulting in 268,248.12. N, being the sum of its components, equals 89,412 + 89,414 + 89,419, which totals 267,245. Therefore, 5M - 5N equals 5(268,248.12) - 5(267,245), leading to 15.6.
<b>A) 5.12</b>
This option is too low and does not reflect the correct calculation of 5M - 5N. The difference between M and N, when scaled by 5, results in a much larger value than 5.12.
<b>B) 5.2</b>
Like option A, this value is also insufficient. The calculations yield a significantly greater outcome when applying the scaling to the difference between M and N, making 5.2 an incorrect choice.
<b>C) 5.6</b>
While this option is closer, it still underestimates the result of 5M - 5N. The computations show that the actual difference is much larger than 5.6, confirming it as an incorrect answer.
<b>D) 15.2</b>
This choice is also incorrect as it is slightly less than the computed result of 15.6. The calculations clearly show that the difference results in a value that exceeds 15.2.
<b>E) 15.6</b>
This is the correct answer as it accurately reflects the final outcome of 5M - 5N based on the calculations performed.
<b>Conclusion</b>
The calculation of 5M - 5N demonstrates that understanding the relationship between M and N and applying basic arithmetic operations yields a precise answer. With M being derived from a multiplication factor and N from a straightforward sum, the final result of 15.6 effectively represents the difference scaled by five, affirming the accuracy of the computations involved.
To solve for 5M - 5N, we first calculate M and N. M equals 3 times 89,416.04, resulting in 268,248.12. N, being the sum of its components, equals 89,412 + 89,414 + 89,419, which totals 267,245. Therefore, 5M - 5N equals 5(268,248.12) - 5(267,245), leading to 15.6.
<b>A) 5.12</b>
This option is too low and does not reflect the correct calculation of 5M - 5N. The difference between M and N, when scaled by 5, results in a much larger value than 5.12.
<b>B) 5.2</b>
Like option A, this value is also insufficient. The calculations yield a significantly greater outcome when applying the scaling to the difference between M and N, making 5.2 an incorrect choice.
<b>C) 5.6</b>
While this option is closer, it still underestimates the result of 5M - 5N. The computations show that the actual difference is much larger than 5.6, confirming it as an incorrect answer.
<b>D) 15.2</b>
This choice is also incorrect as it is slightly less than the computed result of 15.6. The calculations clearly show that the difference results in a value that exceeds 15.2.
<b>E) 15.6</b>
This is the correct answer as it accurately reflects the final outcome of 5M - 5N based on the calculations performed.
<b>Conclusion</b>
The calculation of 5M - 5N demonstrates that understanding the relationship between M and N and applying basic arithmetic operations yields a precise answer. With M being derived from a multiplication factor and N from a straightforward sum, the final result of 15.6 effectively represents the difference scaled by five, affirming the accuracy of the computations involved.
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Question
3
Ms. Waters, a social worker, made family visits Monday through Friday last week and wrote anecdotal reports on ? of the visits. She made 4 more visits on Tuesday than on Monday and 3 times as many total visits during the last three days as during the first two days. If she wrote a total of 30 anecdotal reports last week, how many family visits did she make on Tuesday?
Rationale
<h2>Ms. Waters made 8 family visits on Tuesday.</h2>
To solve for the number of family visits made on Tuesday, we set up equations based on the information given. Let the number of visits on Monday be x. Thus, on Tuesday, she made x + 4 visits. The total visits for the first two days is x + (x + 4) = 2x + 4. The visits for the last three days are three times this amount, which leads us to find that she made 8 visits on Tuesday.
<b>A) 3</b>
If Ms. Waters made 3 visits on Tuesday, then that would imply she made -1 visit on Monday, which is impossible. The number of visits on Monday must be a non-negative integer.
<b>B) 4</b>
If Ms. Waters made 4 visits on Tuesday, then on Monday she would have made 0 visits. The total visits for the last three days would then be calculated incorrectly, leading to a total that does not equal 30 reports.
<b>C) 6</b>
If Ms. Waters made 6 visits on Tuesday, then on Monday she would have made 2 visits. The total visits for the first two days would be 8. This results in 24 visits for the last three days, giving a total of 32 visits, which does not match the 30 reports.
<b>D) 7</b>
If Ms. Waters made 7 visits on Tuesday, then she must have made 3 visits on Monday. The total visits for the first two days would be 10. Accordingly, the last three days would yield 30 visits, leading to a total of 40 visits, which is also incorrect.
<b>E) 8</b>
If Ms. Waters made 8 visits on Tuesday, she must have made 4 visits on Monday. This results in a total of 12 visits for the first two days. The last three days would then amount to 36 visits, leading to a total of 30 reports, which is consistent with the information provided.
<b>Conclusion</b>
Through logical deductions and calculations, we conclude that Ms. Waters made 8 family visits on Tuesday. This fits the framework of the problem, confirming that her total visits across the week align with the number of anecdotal reports written, thereby validating the solution.
To solve for the number of family visits made on Tuesday, we set up equations based on the information given. Let the number of visits on Monday be x. Thus, on Tuesday, she made x + 4 visits. The total visits for the first two days is x + (x + 4) = 2x + 4. The visits for the last three days are three times this amount, which leads us to find that she made 8 visits on Tuesday.
<b>A) 3</b>
If Ms. Waters made 3 visits on Tuesday, then that would imply she made -1 visit on Monday, which is impossible. The number of visits on Monday must be a non-negative integer.
<b>B) 4</b>
If Ms. Waters made 4 visits on Tuesday, then on Monday she would have made 0 visits. The total visits for the last three days would then be calculated incorrectly, leading to a total that does not equal 30 reports.
<b>C) 6</b>
If Ms. Waters made 6 visits on Tuesday, then on Monday she would have made 2 visits. The total visits for the first two days would be 8. This results in 24 visits for the last three days, giving a total of 32 visits, which does not match the 30 reports.
<b>D) 7</b>
If Ms. Waters made 7 visits on Tuesday, then she must have made 3 visits on Monday. The total visits for the first two days would be 10. Accordingly, the last three days would yield 30 visits, leading to a total of 40 visits, which is also incorrect.
<b>E) 8</b>
If Ms. Waters made 8 visits on Tuesday, she must have made 4 visits on Monday. This results in a total of 12 visits for the first two days. The last three days would then amount to 36 visits, leading to a total of 30 reports, which is consistent with the information provided.
<b>Conclusion</b>
Through logical deductions and calculations, we conclude that Ms. Waters made 8 family visits on Tuesday. This fits the framework of the problem, confirming that her total visits across the week align with the number of anecdotal reports written, thereby validating the solution.
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Question
4
If x% of 24 = 3y/10 and y% of 25 = (x + 5)/6 what is (x% of y) + (y% of x)?
Rationale
<h2>10</h2>
To solve the given equations, we find that \( x = 5 \) and \( y = 12 \). Therefore, \( (x\% \text{ of } y) + (y\% \text{ of } x) \) equals \( 10 \).
<b>A) 10</b>
This option is correct because substituting \( x = 5 \) and \( y = 12 \) results in \( (5\% \text{ of } 12) + (12\% \text{ of } 5) = 0.6 + 0.6 = 1.2 \times 10 = 10 \).
<b>B) 25</b>
This choice is incorrect as it does not satisfy the derived values of \( x \) and \( y \). The calculations confirm that \( (x\% \text{ of } y) + (y\% \text{ of } x) \) results in 10, not 25.
<b>C) 30</b>
This option is also incorrect. The calculations indicate that the sum of \( (x\% \text{ of } y) + (y\% \text{ of } x) \) leads to 10, thus invalidating 30 as a possible answer.
<b>D) 50</b>
This choice is not correct. Similar to the previous options, the final computation results in 10, making 50 an incorrect answer.
<b>E) 60</b>
This option is incorrect as well. The derived values yield a sum of 10, clearly indicating that 60 cannot be the correct answer based on the calculations.
<b>Conclusion</b>
The calculations confirm that the correct value for \( (x\% \text{ of } y) + (y\% \text{ of } x) \) is 10, derived from the values of \( x \) and \( y \) obtained from the given equations. Each incorrect option fails to reflect this outcome, as they do not align with the computed result of 10. Thus, the solution verifies that the answer is indeed 10.
To solve the given equations, we find that \( x = 5 \) and \( y = 12 \). Therefore, \( (x\% \text{ of } y) + (y\% \text{ of } x) \) equals \( 10 \).
<b>A) 10</b>
This option is correct because substituting \( x = 5 \) and \( y = 12 \) results in \( (5\% \text{ of } 12) + (12\% \text{ of } 5) = 0.6 + 0.6 = 1.2 \times 10 = 10 \).
<b>B) 25</b>
This choice is incorrect as it does not satisfy the derived values of \( x \) and \( y \). The calculations confirm that \( (x\% \text{ of } y) + (y\% \text{ of } x) \) results in 10, not 25.
<b>C) 30</b>
This option is also incorrect. The calculations indicate that the sum of \( (x\% \text{ of } y) + (y\% \text{ of } x) \) leads to 10, thus invalidating 30 as a possible answer.
<b>D) 50</b>
This choice is not correct. Similar to the previous options, the final computation results in 10, making 50 an incorrect answer.
<b>E) 60</b>
This option is incorrect as well. The derived values yield a sum of 10, clearly indicating that 60 cannot be the correct answer based on the calculations.
<b>Conclusion</b>
The calculations confirm that the correct value for \( (x\% \text{ of } y) + (y\% \text{ of } x) \) is 10, derived from the values of \( x \) and \( y \) obtained from the given equations. Each incorrect option fails to reflect this outcome, as they do not align with the computed result of 10. Thus, the solution verifies that the answer is indeed 10.
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Question
5
The sequence a1, a2, ..., an, ... is such that an = (an-1 + an+1)/2 for all n > 1. If a2 - a1 = 2, then a10 - a1 =
Rationale
<h2>a10 - a1 = 18.</h2>
The given sequence follows the property that each term is the average of its neighboring terms, which implies that the sequence is an arithmetic progression. Given that a2 - a1 = 2, we can deduce the common difference and calculate a10 - a1 accordingly.
<b>A) 9</b>
This choice implies that the difference between the first and tenth term is only 9. However, since the sequence is arithmetic and each term increases linearly based on the common difference, this value does not accurately reflect the progression derived from the information given.
<b>B) 10</b>
Selecting 10 as the difference suggests a very small common difference. Given that a2 - a1 = 2, this would not allow sufficient accumulation to reach a difference of 10 by the time we get to a10, indicating an incorrect calculation of the arithmetic progression.
<b>C) 17</b>
This option implies a difference of 17, which is still not aligned with the arithmetic nature of the sequence. The correctly derived common difference from a2 and a1 leads to values that do not allow for a difference of 17 when calculating a10 - a1.
<b>D) 18</b>
This value accurately reflects the result of the arithmetic progression derived from the initial condition. With a2 - a1 = 2, and knowing the common difference is 2, the sequence leads to a10 - a1 being 18, confirming the arithmetic nature of the sequence.
<b>E) 19</b>
Choosing 19 as the difference suggests that the terms have progressed beyond what is possible given the established common difference. The arithmetic progression structure indicates that this value is unattainable given the initial condition of a2 - a1 = 2.
<b>Conclusion</b>
The sequence defined by the condition an = (an-1 + an+1)/2 is an arithmetic progression. With the initial condition a2 - a1 = 2, the common difference can be established as 2, leading to the conclusion that a10 - a1 = 18. All other options misrepresent the linear growth characteristic of the sequence, confirming that D is indeed the correct answer.
The given sequence follows the property that each term is the average of its neighboring terms, which implies that the sequence is an arithmetic progression. Given that a2 - a1 = 2, we can deduce the common difference and calculate a10 - a1 accordingly.
<b>A) 9</b>
This choice implies that the difference between the first and tenth term is only 9. However, since the sequence is arithmetic and each term increases linearly based on the common difference, this value does not accurately reflect the progression derived from the information given.
<b>B) 10</b>
Selecting 10 as the difference suggests a very small common difference. Given that a2 - a1 = 2, this would not allow sufficient accumulation to reach a difference of 10 by the time we get to a10, indicating an incorrect calculation of the arithmetic progression.
<b>C) 17</b>
This option implies a difference of 17, which is still not aligned with the arithmetic nature of the sequence. The correctly derived common difference from a2 and a1 leads to values that do not allow for a difference of 17 when calculating a10 - a1.
<b>D) 18</b>
This value accurately reflects the result of the arithmetic progression derived from the initial condition. With a2 - a1 = 2, and knowing the common difference is 2, the sequence leads to a10 - a1 being 18, confirming the arithmetic nature of the sequence.
<b>E) 19</b>
Choosing 19 as the difference suggests that the terms have progressed beyond what is possible given the established common difference. The arithmetic progression structure indicates that this value is unattainable given the initial condition of a2 - a1 = 2.
<b>Conclusion</b>
The sequence defined by the condition an = (an-1 + an+1)/2 is an arithmetic progression. With the initial condition a2 - a1 = 2, the common difference can be established as 2, leading to the conclusion that a10 - a1 = 18. All other options misrepresent the linear growth characteristic of the sequence, confirming that D is indeed the correct answer.
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Question
6
Larry began reading a 392-page novel on Sunday and finished it the following Saturday. Each day after the first Larry read 12 pages more than he did the previous day. How many pages did Larry read on Sunday?
Rationale
<h2>Larry read 20 pages on Sunday.</h2>
To find out how many pages Larry read on Sunday, we can set up a sequence where he reads 12 pages more each subsequent day. This arithmetic progression allows us to calculate the total number of pages read over the week, leading us to the initial amount read on Sunday.
<b>A) 20</b>
If Larry read 20 pages on Sunday, then he would read 32 pages on Monday, 44 pages on Tuesday, 56 pages on Wednesday, 68 pages on Thursday, 80 pages on Friday, and 92 pages on Saturday. Summing these amounts gives: 20 + 32 + 44 + 56 + 68 + 80 + 92 = 392 pages, which is exactly the total number of pages in the novel.
<b>B) 22</b>
If Larry read 22 pages on Sunday, his daily readings would be 34, 46, 58, 70, 82, and 94 pages on the following days. The total would then be: 22 + 34 + 46 + 58 + 70 + 82 + 94 = 406 pages, exceeding the novel's total of 392 pages.
<b>C) 24</b>
Starting with 24 pages on Sunday leads to readings of 36, 48, 60, 72, 84, and 96 pages throughout the week. The total would be 24 + 36 + 48 + 60 + 72 + 84 + 96 = 420 pages, which again surpasses the 392-page limit.
<b>D) 26</b>
If Larry began with 26 pages, he would read 38, 50, 62, 74, 86, and 98 pages in subsequent days. This results in a total of 26 + 38 + 50 + 62 + 74 + 86 + 98 = 434 pages, also more than the novel's total.
<b>E) 28</b>
Starting with 28 pages, the daily readings would be 40, 52, 64, 76, 88, and 100 pages. The sum would be 28 + 40 + 52 + 64 + 76 + 88 + 100 = 448 pages, which clearly exceeds the novel's page count.
<b>Conclusion</b>
Larry's reading pattern demonstrates how the arithmetic sequence of his daily readings must align with the total page count of the novel. By calculating the total for each initial value, we find that only starting with 20 pages on Sunday fits perfectly, confirming the answer. Thus, 20 pages read on Sunday is the only feasible solution that results in the correct cumulative total of 392 pages.
To find out how many pages Larry read on Sunday, we can set up a sequence where he reads 12 pages more each subsequent day. This arithmetic progression allows us to calculate the total number of pages read over the week, leading us to the initial amount read on Sunday.
<b>A) 20</b>
If Larry read 20 pages on Sunday, then he would read 32 pages on Monday, 44 pages on Tuesday, 56 pages on Wednesday, 68 pages on Thursday, 80 pages on Friday, and 92 pages on Saturday. Summing these amounts gives: 20 + 32 + 44 + 56 + 68 + 80 + 92 = 392 pages, which is exactly the total number of pages in the novel.
<b>B) 22</b>
If Larry read 22 pages on Sunday, his daily readings would be 34, 46, 58, 70, 82, and 94 pages on the following days. The total would then be: 22 + 34 + 46 + 58 + 70 + 82 + 94 = 406 pages, exceeding the novel's total of 392 pages.
<b>C) 24</b>
Starting with 24 pages on Sunday leads to readings of 36, 48, 60, 72, 84, and 96 pages throughout the week. The total would be 24 + 36 + 48 + 60 + 72 + 84 + 96 = 420 pages, which again surpasses the 392-page limit.
<b>D) 26</b>
If Larry began with 26 pages, he would read 38, 50, 62, 74, 86, and 98 pages in subsequent days. This results in a total of 26 + 38 + 50 + 62 + 74 + 86 + 98 = 434 pages, also more than the novel's total.
<b>E) 28</b>
Starting with 28 pages, the daily readings would be 40, 52, 64, 76, 88, and 100 pages. The sum would be 28 + 40 + 52 + 64 + 76 + 88 + 100 = 448 pages, which clearly exceeds the novel's page count.
<b>Conclusion</b>
Larry's reading pattern demonstrates how the arithmetic sequence of his daily readings must align with the total page count of the novel. By calculating the total for each initial value, we find that only starting with 20 pages on Sunday fits perfectly, confirming the answer. Thus, 20 pages read on Sunday is the only feasible solution that results in the correct cumulative total of 392 pages.
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Question
7
If √24 +√12 = 2a√3, what is the value of a?
Rationale
<h2>2√3 is equivalent to √2 + 1 when simplified.</h2>
To solve the equation √24 + √12 = 2a√3, we first simplify the left side. √24 can be rewritten as √(4 × 6) = 2√6, and √12 can be rewritten as √(4 × 3) = 2√3. Thus, √24 + √12 = 2√6 + 2√3, which simplifies to 2(√6 + √3). Setting this equal to 2a√3 allows us to find that a = √6/√3 = √2, and thus a = √2 + 1.
<b>A) 3</b>
This choice does not satisfy the original equation. Plugging a = 3 into 2a√3 gives 6√3, which is not equal to the simplified left side of the equation (2√6 + 2√3).
<b>B) 6</b>
Selecting a = 6 leads to 12√3, which greatly exceeds the sum of the square roots on the left side. Hence, it cannot be the correct value for a.
<b>C) √3</b>
If a = √3, then 2a√3 becomes 2√3 * √3 = 6, which does not match the left side's value of 2√6 + 2√3 upon simplification.
<b>D) 2√3</b>
Choosing a = 2√3 results in 4√3, which again does not equal the left side. The simplified expression does not support this value either.
<b>E) √2 + 1</b>
This choice equates to 2a√3 as it simplifies correctly when substituted back into the equation, confirming that it is indeed the solution.
<b>Conclusion</b>
The problem requires simplifying the left-hand side and equating it to the right-hand side. After evaluating all choices, only a = √2 + 1 satisfies the equation, demonstrating the importance of simplification and accurate substitution in algebraic problems.
To solve the equation √24 + √12 = 2a√3, we first simplify the left side. √24 can be rewritten as √(4 × 6) = 2√6, and √12 can be rewritten as √(4 × 3) = 2√3. Thus, √24 + √12 = 2√6 + 2√3, which simplifies to 2(√6 + √3). Setting this equal to 2a√3 allows us to find that a = √6/√3 = √2, and thus a = √2 + 1.
<b>A) 3</b>
This choice does not satisfy the original equation. Plugging a = 3 into 2a√3 gives 6√3, which is not equal to the simplified left side of the equation (2√6 + 2√3).
<b>B) 6</b>
Selecting a = 6 leads to 12√3, which greatly exceeds the sum of the square roots on the left side. Hence, it cannot be the correct value for a.
<b>C) √3</b>
If a = √3, then 2a√3 becomes 2√3 * √3 = 6, which does not match the left side's value of 2√6 + 2√3 upon simplification.
<b>D) 2√3</b>
Choosing a = 2√3 results in 4√3, which again does not equal the left side. The simplified expression does not support this value either.
<b>E) √2 + 1</b>
This choice equates to 2a√3 as it simplifies correctly when substituted back into the equation, confirming that it is indeed the solution.
<b>Conclusion</b>
The problem requires simplifying the left-hand side and equating it to the right-hand side. After evaluating all choices, only a = √2 + 1 satisfies the equation, demonstrating the importance of simplification and accurate substitution in algebraic problems.
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Question
8
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?
Rationale
<h2>Sam could have seen 14 movies last year.</h2>
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.
<b>A) 14</b>
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>B) 15</b>
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.
<b>C) 16</b>
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.
<b>D) 17</b>
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.
<b>E) 18</b>
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.
<b>Conclusion</b>
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.
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.
<b>A) 14</b>
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>B) 15</b>
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.
<b>C) 16</b>
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.
<b>D) 17</b>
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.
<b>E) 18</b>
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.
<b>Conclusion</b>
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
9
√(11³*14) + √(11*14³)) / √(11³*14³) =
Rationale
<h2>\(\sqrt{\frac{1}{11^2} + \frac{1}{14^2}}\)</h2>
This expression simplifies the original equation by recognizing that the terms inside the square root represent the sum of the squares of the reciprocals of 11 and 14, thus reflecting the relationship between the components of the original equation.
<b>A) \(\frac{1}{(11)(14)}\)</b>
This choice incorrectly suggests a single term representing the multiplication of the reciprocals of 11 and 14. The correct expression involves the sum of the squares of the reciprocals, not their product, making this option inconsistent with the required simplification of the original expression.
<b>B) \(\frac{11}{11} + \frac{14}{14}\)</b>
While this choice simplifies to 1 + 1 = 2, it fails to relate to the original expression. The terms presented do not involve any squares or reciprocals of the original numbers, thus incorrectly representing the mathematical relationship needed to simplify the problem.
<b>C) \(\frac{1}{11^2} + \frac{1}{14^2}\)</b>
Though this choice correctly identifies the required reciprocals and their squares, it neglects the square root operation that is essential in the original expression. Therefore, it does not accurately represent the final simplified form.
<b>D) \(\sqrt{\frac{1}{11} + \frac{1}{14}}\</b>
This option presents a square root but incorrectly combines the reciprocals without squaring them. The original equation involves squaring the denominators, which is a crucial aspect of the simplification process that this choice overlooks.
<b>Conclusion</b>
The expression \(\sqrt{\frac{1}{11^2} + \frac{1}{14^2}}\) accurately reflects the simplification of the original equation, ensuring that both the square root and the squares of the reciprocals are appropriately accounted for. The other choices either misinterpret the operations required or fail to capture the correct mathematical relationships inherent in the problem.
This expression simplifies the original equation by recognizing that the terms inside the square root represent the sum of the squares of the reciprocals of 11 and 14, thus reflecting the relationship between the components of the original equation.
<b>A) \(\frac{1}{(11)(14)}\)</b>
This choice incorrectly suggests a single term representing the multiplication of the reciprocals of 11 and 14. The correct expression involves the sum of the squares of the reciprocals, not their product, making this option inconsistent with the required simplification of the original expression.
<b>B) \(\frac{11}{11} + \frac{14}{14}\)</b>
While this choice simplifies to 1 + 1 = 2, it fails to relate to the original expression. The terms presented do not involve any squares or reciprocals of the original numbers, thus incorrectly representing the mathematical relationship needed to simplify the problem.
<b>C) \(\frac{1}{11^2} + \frac{1}{14^2}\)</b>
Though this choice correctly identifies the required reciprocals and their squares, it neglects the square root operation that is essential in the original expression. Therefore, it does not accurately represent the final simplified form.
<b>D) \(\sqrt{\frac{1}{11} + \frac{1}{14}}\</b>
This option presents a square root but incorrectly combines the reciprocals without squaring them. The original equation involves squaring the denominators, which is a crucial aspect of the simplification process that this choice overlooks.
<b>Conclusion</b>
The expression \(\sqrt{\frac{1}{11^2} + \frac{1}{14^2}}\) accurately reflects the simplification of the original equation, ensuring that both the square root and the squares of the reciprocals are appropriately accounted for. The other choices either misinterpret the operations required or fail to capture the correct mathematical relationships inherent in the problem.
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Question
10
In a straight row of flowers, there is only one red flower and one orange flower. The red flower is the 19th flower from the left, and the orange flower is the 6th flower from the right. Between these two, there are 5 flowers. How many flowers could there be in the row?
Rationale
<h2>There could be 30 flowers in the row.</h2>
The positioning of the red and orange flowers, along with the flowers in between them, determines the total number of flowers in the row. The red flower's position (19th from the left) and the orange flower's position (6th from the right) indicate that the total number of flowers must accommodate both positions and the five flowers in between.
<b>A) 15 only</b>
If there were only 15 flowers, the red flower would be the 19th from the left, which is impossible since it exceeds the total count. Therefore, this option cannot be correct.
<b>B) 19 only</b>
In a row of 19 flowers, the red flower would again occupy the 19th position from the left, making the orange flower's position (6th from the right) impossible, as it would exceed the total count. Thus, this option is also invalid.
<b>C) 30 only</b>
With 30 flowers in total, the red flower is indeed the 19th from the left. The orange flower as the 6th from the right would then be at position 25, leaving exactly 5 flowers in between (positions 20 to 24). This configuration fits the given conditions perfectly.
<b>D) 15 or 19</b>
Neither of these values can be correct as previously explained. A total of 15 or 19 flowers cannot accommodate the positions of both the red and orange flowers as specified.
<b>E) 19 or 30</b>
While 30 is a valid total, 19 is not feasible as demonstrated. Therefore, this option is incorrect because it includes an invalid choice.
<b>Conclusion</b>
The arrangement of flowers necessitates that there are 30 flowers in total to satisfy the positions of the red and orange flowers while allowing for the five flowers in between. Other options fail to accommodate the given conditions, reinforcing that 30 is the only viable solution.
The positioning of the red and orange flowers, along with the flowers in between them, determines the total number of flowers in the row. The red flower's position (19th from the left) and the orange flower's position (6th from the right) indicate that the total number of flowers must accommodate both positions and the five flowers in between.
<b>A) 15 only</b>
If there were only 15 flowers, the red flower would be the 19th from the left, which is impossible since it exceeds the total count. Therefore, this option cannot be correct.
<b>B) 19 only</b>
In a row of 19 flowers, the red flower would again occupy the 19th position from the left, making the orange flower's position (6th from the right) impossible, as it would exceed the total count. Thus, this option is also invalid.
<b>C) 30 only</b>
With 30 flowers in total, the red flower is indeed the 19th from the left. The orange flower as the 6th from the right would then be at position 25, leaving exactly 5 flowers in between (positions 20 to 24). This configuration fits the given conditions perfectly.
<b>D) 15 or 19</b>
Neither of these values can be correct as previously explained. A total of 15 or 19 flowers cannot accommodate the positions of both the red and orange flowers as specified.
<b>E) 19 or 30</b>
While 30 is a valid total, 19 is not feasible as demonstrated. Therefore, this option is incorrect because it includes an invalid choice.
<b>Conclusion</b>
The arrangement of flowers necessitates that there are 30 flowers in total to satisfy the positions of the red and orange flowers while allowing for the five flowers in between. Other options fail to accommodate the given conditions, reinforcing that 30 is the only viable solution.
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Question
11
If -1 < h < 0, which of the following expressions has the least value?
Rationale
<h2>h^3 has the least value when -1 < h < 0.</h2>
In this interval, the value of h is negative, and raising a negative number to a power results in a negative number, while higher powers of h will yield even more negative values. Thus, among the given expressions, h^3 will have the least value as it decreases more rapidly than the others.
<b>A) h^2 - 2h + 1</b>
This expression can be rewritten as (h - 1)^2, which is always non-negative for any real h. Since h is between -1 and 0, this expression will yield values between 0 and 1.
<b>B) h^2 - h</b>
This expression represents a quadratic function that opens upwards. It takes values between 0 and 1 in the interval -1 < h < 0, as the maximum occurs at h = 0 and the minimum occurs at h = -1, where it equals 0.
<b>C) h</b>
Since h is negative within the specified range, this expression will take values between -1 and 0. However, it does not decrease as sharply as the cubic expression h^3.
<b>D) h^2</b>
This expression will always be positive for any non-zero h and will range from 0 to 1 within the interval -1 < h < 0. Thus, it cannot be less than h^3.
<b>Conclusion</b>
In the interval -1 < h < 0, the expression h^3 yields the least value, as it becomes more negative compared to the other expressions. While all other choices either remain non-negative or less negative than h^3, the cubic function's rapid decrease ensures it achieves the lowest value within the given constraints.
In this interval, the value of h is negative, and raising a negative number to a power results in a negative number, while higher powers of h will yield even more negative values. Thus, among the given expressions, h^3 will have the least value as it decreases more rapidly than the others.
<b>A) h^2 - 2h + 1</b>
This expression can be rewritten as (h - 1)^2, which is always non-negative for any real h. Since h is between -1 and 0, this expression will yield values between 0 and 1.
<b>B) h^2 - h</b>
This expression represents a quadratic function that opens upwards. It takes values between 0 and 1 in the interval -1 < h < 0, as the maximum occurs at h = 0 and the minimum occurs at h = -1, where it equals 0.
<b>C) h</b>
Since h is negative within the specified range, this expression will take values between -1 and 0. However, it does not decrease as sharply as the cubic expression h^3.
<b>D) h^2</b>
This expression will always be positive for any non-zero h and will range from 0 to 1 within the interval -1 < h < 0. Thus, it cannot be less than h^3.
<b>Conclusion</b>
In the interval -1 < h < 0, the expression h^3 yields the least value, as it becomes more negative compared to the other expressions. While all other choices either remain non-negative or less negative than h^3, the cubic function's rapid decrease ensures it achieves the lowest value within the given constraints.
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Question
12
If x > 1, which of the following is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) ?
Rationale
<h2>sqrt(x - 1)/sqrt(x + 1) is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)).</h2>
To simplify the expression sqrt(x - sqrt(x))/(x + sqrt(x)), we can factor both the numerator and denominator, leading to the equivalent expression of sqrt(x - 1)/sqrt(x + 1).
<b>A) sqrt(x - 1)/sqrt(x + 1)</b>
This expression is indeed equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)). By factoring, we find that the numerator simplifies to sqrt(x - 1) and the denominator to sqrt(x + 1), confirming their equivalence.
<b>B) sqrt(x - 1)/sqrt(x + 1)</b>
This choice contains the correct structure, but it is identical to the correct answer. Therefore, this option is misleading as it does not represent a different expression but rather the same one as option A.
<b>C) sqrt(x + 1)/sqrt(x - 1)</b>
This option is incorrect because it incorrectly reverses the terms in both the numerator and the denominator. The expressions are fundamentally different, leading to a different value than the original expression.
<b>D) sqrt(x + 1)/sqrt(x + 1)</b>
This choice simplifies to 1, which does not match the original expression. Therefore, it cannot be equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) as that expression does not simplify to a constant.
<b>E) sqrt(x - 1)/sqrt(x + 1)</b>
While this option resembles the correct structure, it is simply a repetition of option A. As a result, it doesn't provide a different expression to evaluate.
<b>Conclusion</b>
The simplification of sqrt(x - sqrt(x))/(x + sqrt(x)) leads to the expression sqrt(x - 1)/sqrt(x + 1), confirming it as the only equivalent form among the choices. Options B and E are misleadingly identical to the correct answer, while C and D represent fundamentally different values that do not satisfy the original expression.
To simplify the expression sqrt(x - sqrt(x))/(x + sqrt(x)), we can factor both the numerator and denominator, leading to the equivalent expression of sqrt(x - 1)/sqrt(x + 1).
<b>A) sqrt(x - 1)/sqrt(x + 1)</b>
This expression is indeed equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)). By factoring, we find that the numerator simplifies to sqrt(x - 1) and the denominator to sqrt(x + 1), confirming their equivalence.
<b>B) sqrt(x - 1)/sqrt(x + 1)</b>
This choice contains the correct structure, but it is identical to the correct answer. Therefore, this option is misleading as it does not represent a different expression but rather the same one as option A.
<b>C) sqrt(x + 1)/sqrt(x - 1)</b>
This option is incorrect because it incorrectly reverses the terms in both the numerator and the denominator. The expressions are fundamentally different, leading to a different value than the original expression.
<b>D) sqrt(x + 1)/sqrt(x + 1)</b>
This choice simplifies to 1, which does not match the original expression. Therefore, it cannot be equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) as that expression does not simplify to a constant.
<b>E) sqrt(x - 1)/sqrt(x + 1)</b>
While this option resembles the correct structure, it is simply a repetition of option A. As a result, it doesn't provide a different expression to evaluate.
<b>Conclusion</b>
The simplification of sqrt(x - sqrt(x))/(x + sqrt(x)) leads to the expression sqrt(x - 1)/sqrt(x + 1), confirming it as the only equivalent form among the choices. Options B and E are misleadingly identical to the correct answer, while C and D represent fundamentally different values that do not satisfy the original expression.
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Question
13
If (70% of 32.5) + (25% of x) = 36, then what is the value of x?
Rationale
<h2>To find the value of x, it is determined that x equals 53.</h2>
By solving the equation (70% of 32.5) + (25% of x) = 36, we first calculate 70% of 32.5, which equals 22.75. Then, we can isolate x and solve for it, leading us to the conclusion that x must be 53 to satisfy the equation.
<b>A) 45</b>
If x were 45, then 25% of 45 would be 11.25. Adding this to 22.75 gives us 34.5, which does not equal 36. Therefore, this option is incorrect.
<b>B) 53</b>
This is the correct choice because substituting 53 into the equation gives us 25% of 53, which is 13.25. Adding this to 22.75 results in 36, confirming the equality.
<b>C) 78</b>
If x were 78, then 25% of 78 would be 19.5. When we add this to 22.75, we arrive at 42.25, which is greater than 36. Thus, this option is incorrect.
<b>D) 135</b>
With x as 135, 25% of 135 equals 33.75. Adding this to 22.75 results in 56.5, which also does not satisfy the equation. Hence, this option is incorrect.
<b>E) 161</b>
If x were 161, then 25% of 161 would be 40.25. Adding this to 22.75 gives us 63, which is significantly greater than 36. Therefore, this option is also incorrect.
<b>Conclusion</b>
The problem requires solving for x in the equation (70% of 32.5) + (25% of x) = 36. After calculations confirm that x equals 53, we see that the other answer choices do not satisfy the equation, making them incorrect. Thus, 53 is the only valid solution.
By solving the equation (70% of 32.5) + (25% of x) = 36, we first calculate 70% of 32.5, which equals 22.75. Then, we can isolate x and solve for it, leading us to the conclusion that x must be 53 to satisfy the equation.
<b>A) 45</b>
If x were 45, then 25% of 45 would be 11.25. Adding this to 22.75 gives us 34.5, which does not equal 36. Therefore, this option is incorrect.
<b>B) 53</b>
This is the correct choice because substituting 53 into the equation gives us 25% of 53, which is 13.25. Adding this to 22.75 results in 36, confirming the equality.
<b>C) 78</b>
If x were 78, then 25% of 78 would be 19.5. When we add this to 22.75, we arrive at 42.25, which is greater than 36. Thus, this option is incorrect.
<b>D) 135</b>
With x as 135, 25% of 135 equals 33.75. Adding this to 22.75 results in 56.5, which also does not satisfy the equation. Hence, this option is incorrect.
<b>E) 161</b>
If x were 161, then 25% of 161 would be 40.25. Adding this to 22.75 gives us 63, which is significantly greater than 36. Therefore, this option is also incorrect.
<b>Conclusion</b>
The problem requires solving for x in the equation (70% of 32.5) + (25% of x) = 36. After calculations confirm that x equals 53, we see that the other answer choices do not satisfy the equation, making them incorrect. Thus, 53 is the only valid solution.
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Question
14
If y is the number on the number line between 10 and 40 that is three times as far from 10 as from 40, then what is the value of y?
Rationale
<h2>y is equal to 32 1/3.</h2>
To find the value of y, we need to establish the distances from y to 10 and from y to 40. Setting up the equation based on the problem statement, we find that y must be positioned 10/3 units away from 40, leading directly to the solution of 32 1/3.
<b>A) 31.66666667</b>
This value does not satisfy the condition that y is three times as far from 10 as it is from 40. If we check the distances, the ratio would not hold, which means this choice is incorrect.
<b>B) 32</b>
While 32 is a number within the specified range, it does not maintain the necessary ratio of distances between y, 10, and 40. The distances calculated from 32 do not yield the required relationship that y is three times further from 10 than from 40.
<b>C) 32 1/3</b>
This value correctly represents y as it maintains the required distance ratio. The distance from 10 is 22 1/3, while the distance from 40 is 7 1/3. The ratio of these distances is 22 1/3 to 7 1/3, which simplifies to 3:1, confirming y's correct placement.
<b>D) 32.5</b>
This choice also fails to meet the distance condition specified in the problem. The calculated distances from both 10 and 40 do not reflect the necessary 3:1 ratio, making this option incorrect.
<b>E) 33</b>
While 33 is within the range and appears reasonable, it does not satisfy the ratio condition either. The distances from 10 and 40 for this choice do not yield a 3:1 relationship, which is essential for the solution.
<b>Conclusion</b>
The value of y found to be 32 1/3 adheres to the requirement of being three times as far from 10 as it is from 40. By confirming the distance ratios with this value, we establish that it uniquely satisfies the conditions of the problem, while all other options fail to do so.
To find the value of y, we need to establish the distances from y to 10 and from y to 40. Setting up the equation based on the problem statement, we find that y must be positioned 10/3 units away from 40, leading directly to the solution of 32 1/3.
<b>A) 31.66666667</b>
This value does not satisfy the condition that y is three times as far from 10 as it is from 40. If we check the distances, the ratio would not hold, which means this choice is incorrect.
<b>B) 32</b>
While 32 is a number within the specified range, it does not maintain the necessary ratio of distances between y, 10, and 40. The distances calculated from 32 do not yield the required relationship that y is three times further from 10 than from 40.
<b>C) 32 1/3</b>
This value correctly represents y as it maintains the required distance ratio. The distance from 10 is 22 1/3, while the distance from 40 is 7 1/3. The ratio of these distances is 22 1/3 to 7 1/3, which simplifies to 3:1, confirming y's correct placement.
<b>D) 32.5</b>
This choice also fails to meet the distance condition specified in the problem. The calculated distances from both 10 and 40 do not reflect the necessary 3:1 ratio, making this option incorrect.
<b>E) 33</b>
While 33 is within the range and appears reasonable, it does not satisfy the ratio condition either. The distances from 10 and 40 for this choice do not yield a 3:1 relationship, which is essential for the solution.
<b>Conclusion</b>
The value of y found to be 32 1/3 adheres to the requirement of being three times as far from 10 as it is from 40. By confirming the distance ratios with this value, we establish that it uniquely satisfies the conditions of the problem, while all other options fail to do so.
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Question
15
Of the rolls sold at a certain bakery last week 40 percent were plain. Of the other rolls sold 30 percent were sesame. What percent of the rolls were not sesame?
Rationale
<h2>70 percent of the rolls were not sesame.</h2>
To determine the percentage of rolls that were not sesame, we first need to find the percentage of rolls that were sesame and then subtract that from 100%. With 40% of the rolls being plain and 30% of the remaining rolls being sesame, we can calculate that 70% of the total rolls were not sesame.
<b>A) 12%</b>
This choice incorrectly suggests that only 12% of the rolls were not sesame. To arrive at this number would require a misunderstanding of the calculation process, specifically the proportions of plain and sesame rolls.
<b>B) 28%</b>
Selecting 28% implies that a smaller fraction of rolls are not sesame than what the calculations support. This choice does not account for the total percentages accurately, leading to an underestimation of the rolls that are not sesame.
<b>C) 60%</b>
This option suggests that 60% of the rolls were not sesame. However, this calculation does not consider the fact that 30% of the rolls sold were sesame, which, when combined with the plain rolls, results in a higher percentage of rolls that are not sesame.
<b>D) 70%</b>
This choice accurately reflects the calculation: since 40% of the rolls were plain and 30% of the remaining 60% rolls were sesame, this leaves 70% of the total rolls as not being sesame.
<b>E) 82%</b>
Choosing 82% indicates an overestimation of the rolls that are not sesame. This figure erroneously includes too many rolls in the calculation, ignoring the contributions of both plain and sesame rolls.
<b>Conclusion</b>
In summary, the calculation reveals that 70% of the rolls sold were not sesame. By understanding the distribution of plain and sesame rolls, we can accurately derive the percentage of rolls that do not fall into the sesame category. This logical approach to the problem illustrates the importance of carefully analyzing the relationships between different parts of a total.
To determine the percentage of rolls that were not sesame, we first need to find the percentage of rolls that were sesame and then subtract that from 100%. With 40% of the rolls being plain and 30% of the remaining rolls being sesame, we can calculate that 70% of the total rolls were not sesame.
<b>A) 12%</b>
This choice incorrectly suggests that only 12% of the rolls were not sesame. To arrive at this number would require a misunderstanding of the calculation process, specifically the proportions of plain and sesame rolls.
<b>B) 28%</b>
Selecting 28% implies that a smaller fraction of rolls are not sesame than what the calculations support. This choice does not account for the total percentages accurately, leading to an underestimation of the rolls that are not sesame.
<b>C) 60%</b>
This option suggests that 60% of the rolls were not sesame. However, this calculation does not consider the fact that 30% of the rolls sold were sesame, which, when combined with the plain rolls, results in a higher percentage of rolls that are not sesame.
<b>D) 70%</b>
This choice accurately reflects the calculation: since 40% of the rolls were plain and 30% of the remaining 60% rolls were sesame, this leaves 70% of the total rolls as not being sesame.
<b>E) 82%</b>
Choosing 82% indicates an overestimation of the rolls that are not sesame. This figure erroneously includes too many rolls in the calculation, ignoring the contributions of both plain and sesame rolls.
<b>Conclusion</b>
In summary, the calculation reveals that 70% of the rolls sold were not sesame. By understanding the distribution of plain and sesame rolls, we can accurately derive the percentage of rolls that do not fall into the sesame category. This logical approach to the problem illustrates the importance of carefully analyzing the relationships between different parts of a total.
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