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28 Apr 2025, 01:32
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Difficulty:
45%
(medium)
Question Stats:
66% (01:37) correct
34%
(01:35)
wrong
based on 201
sessions
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A furniture store spent $440 to purchase two types of chairs: deluxe and standard. How many standard chairs did the store purchase?
(1) The deluxe chairs cost $70 each and the standard chairs cost $50 each.
(2) The store purchased 8 chairs in total.
Gentle note to all experts and tutors: Please refrain from replying to this question until the Official Answer (OA) is revealed. Let students attempt to solve it first. You are all welcome to contribute posts after the OA is posted. Thank you all for your cooperation!
(1) The deluxe chairs cost $70 each and the standard chairs cost $50 each.
(2) The store purchased 8 chairs in total.
Gentle note to all experts and tutors: Please refrain from replying to this question until the Official Answer (OA) is revealed. Let students attempt to solve it first. You are all welcome to contribute posts after the OA is posted. Thank you all for your cooperation!
06 May 2025, 02:30
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Official Solution:
A furniture store spent $440 to purchase two types of chairs: deluxe and standard. How many standard chairs did the store purchase?
This is a classic C-trap question. C-trap questions are the ones where the statements together clearly seem sufficient, but in fact, one statement alone may already be enough. When such situations appear, we should proceed carefully. In this question, it is clear that when we combine the statements, we have all the information there might be about the question, so the combination obviously is sufficient. However, we must be careful not to overlook that either statement (or both) could individually also be sufficient.
Let the number of deluxe chairs be \(x\) and the number of standard chairs be \(y\). Question: What is the value of \(y\)?
(1) The deluxe chairs cost $70 each and the standard chairs cost $50 each.
Since the store spent $440 for the chairs, we have: \(70x + 50y = 440\), which after reducing by 10 gives \(7x + 5y = 44\).
Generally, such kinds of linear equations (\(ax + by = c\)) have infinitely many solutions. However, since \(x\) and \(y\) represent the number of chairs, they must be non-negative integers, and in this case, the equation becomes a Diophantine equation (an equation whose solutions must be integers only). For such equations, it is necessary to check manually whether there is only one combination that works.
Now, it is quite easy to check whether \(7x + 5y = 44\) has one or more solutions. Rearranging gives \(5y = 44 - 7x\), so 44 minus a multiple of 7 must be a multiple of 5:
Thus, only one combination of \(x\) and \(y\) satisfies the equation \(7x + 5y = 44\), namely \(x = 2\) and \(y = 6\). Therefore, statement (1) alone is sufficient.
(2) The store purchased 8 chairs in total.
This gives \(x + y = 8\). Similar to the above, this is also a Diophantine equation. However, unlike the previous case, it has several integer solutions for \(x\) and \(y\). Thus, clearly not sufficient to find a single numerical value of \(y\).
Answer: A
A furniture store spent $440 to purchase two types of chairs: deluxe and standard. How many standard chairs did the store purchase?
This is a classic C-trap question. C-trap questions are the ones where the statements together clearly seem sufficient, but in fact, one statement alone may already be enough. When such situations appear, we should proceed carefully. In this question, it is clear that when we combine the statements, we have all the information there might be about the question, so the combination obviously is sufficient. However, we must be careful not to overlook that either statement (or both) could individually also be sufficient.
Let the number of deluxe chairs be \(x\) and the number of standard chairs be \(y\). Question: What is the value of \(y\)?
(1) The deluxe chairs cost $70 each and the standard chairs cost $50 each.
Since the store spent $440 for the chairs, we have: \(70x + 50y = 440\), which after reducing by 10 gives \(7x + 5y = 44\).
Generally, such kinds of linear equations (\(ax + by = c\)) have infinitely many solutions. However, since \(x\) and \(y\) represent the number of chairs, they must be non-negative integers, and in this case, the equation becomes a Diophantine equation (an equation whose solutions must be integers only). For such equations, it is necessary to check manually whether there is only one combination that works.
Now, it is quite easy to check whether \(7x + 5y = 44\) has one or more solutions. Rearranging gives \(5y = 44 - 7x\), so 44 minus a multiple of 7 must be a multiple of 5:
• 44 is not divisible by 5;
• 37 is not;
• 30 is divisible by 5 (\(x = 2\), \(y = 6\));
• 23 is not;
• 16 is not;
• 9 is not;
• 2 is not.
• 37 is not;
• 30 is divisible by 5 (\(x = 2\), \(y = 6\));
• 23 is not;
• 16 is not;
• 9 is not;
• 2 is not.
Thus, only one combination of \(x\) and \(y\) satisfies the equation \(7x + 5y = 44\), namely \(x = 2\) and \(y = 6\). Therefore, statement (1) alone is sufficient.
(2) The store purchased 8 chairs in total.
This gives \(x + y = 8\). Similar to the above, this is also a Diophantine equation. However, unlike the previous case, it has several integer solutions for \(x\) and \(y\). Thus, clearly not sufficient to find a single numerical value of \(y\).
Answer: A
General Discussion
02 May 2025, 03:32
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Bunuel
Let the deluxe and standard chairs be denoted as d and s respectively.
S= ?
Statement 1:
(1) The deluxe chairs cost $70 each and the standard chairs cost $50 each.
70d + 50s = 440
7d + 5s = 44
d = 2, and s = 6
Hence, S = 6 Sufficient
Statement 2:
(2) The store purchased 8 chairs in total.
given s + d = 8
n1 s + n2 d = 440
we don’t know the value for chairs of either standard or deluxe. Hence, Insufficient
Option A
