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03 Aug 2026, 08:02
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For a trade-show booth, a crew assembled 10 display panels. Some were reinforced panels, and the rest were standard panels. Each reinforced panel required 16 mounting clips, and each standard panel required 11 mounting clips. If at least 1 panel of each type was used, were more than 4 of the 10 display panels reinforced?
(1) The total number of mounting clips used was a multiple of 9.
(2) If the crew had assembled 3 reinforced panels and 7 standard panels instead, the total number of mounting clips used would have been at least 10 fewer.
(1) The total number of mounting clips used was a multiple of 9.
(2) If the crew had assembled 3 reinforced panels and 7 standard panels instead, the total number of mounting clips used would have been at least 10 fewer.
03 Aug 2026, 08:02
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Official Solution:
For a trade-show booth, a crew assembled 10 display panels. Some were reinforced panels, and the rest were standard panels. Each reinforced panel required 16 mounting clips, and each standard panel required 11 mounting clips. If at least 1 panel of each type was used, were more than 4 of the 10 display panels reinforced?
Let \(r\) be the number of reinforced panels. Then the number of standard panels was \(10 - r\). So the total number of mounting clips used was
\(16r + 11(10 - r) = 110 + 5r\)
We need to determine whether \(r > 4\).
(1) The total number of mounting clips used was a multiple of 9.
So \(110 + 5r\) is divisible by 9. Since \(110\) leaves remainder \(2\) when divided by \(9\), \(5r\) must leave remainder \(7\) when divided by \(9\). The only value of \(r\) from \(1\) to \(9\) that works is \(r = 5\). So \(r > 4\). Sufficient.
(2) If the crew had assembled 3 reinforced panels and 7 standard panels instead, the total number of mounting clips used would have been at least 10 fewer.
That hypothetical arrangement would have used:
\(3(16) + 7(11) = 125\) clips.
So the actual total number of clips used was at least
\(125 + 10 = 135\)
Thus:
\(110 + 5r \geq 135\)
\(5r \geq 25\)
\(r \geq 5\)
So \(r > 4\).
Sufficient.
Answer: D
For a trade-show booth, a crew assembled 10 display panels. Some were reinforced panels, and the rest were standard panels. Each reinforced panel required 16 mounting clips, and each standard panel required 11 mounting clips. If at least 1 panel of each type was used, were more than 4 of the 10 display panels reinforced?
Let \(r\) be the number of reinforced panels. Then the number of standard panels was \(10 - r\). So the total number of mounting clips used was
\(16r + 11(10 - r) = 110 + 5r\)
We need to determine whether \(r > 4\).
(1) The total number of mounting clips used was a multiple of 9.
So \(110 + 5r\) is divisible by 9. Since \(110\) leaves remainder \(2\) when divided by \(9\), \(5r\) must leave remainder \(7\) when divided by \(9\). The only value of \(r\) from \(1\) to \(9\) that works is \(r = 5\). So \(r > 4\). Sufficient.
(2) If the crew had assembled 3 reinforced panels and 7 standard panels instead, the total number of mounting clips used would have been at least 10 fewer.
That hypothetical arrangement would have used:
\(3(16) + 7(11) = 125\) clips.
So the actual total number of clips used was at least
\(125 + 10 = 135\)
Thus:
\(110 + 5r \geq 135\)
\(5r \geq 25\)
\(r \geq 5\)
So \(r > 4\).
Sufficient.
Answer: D
