Re M14-15
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16 Sep 2014, 00:53
Official Solution:
If the average (arithmetic mean) weight of 15 items is 8 kilograms and no item weighs more than 10 kilograms, what is the maximum number of items that can weigh 5 kilograms?
A. 10
B. 9
C. 8
D. 7
E. 6
The total weight of the 15 items is \(15*8 = 120\) kilograms.
Let's examine the options.
Could 10 items weigh 5 kilograms each? No, because their total weight would be \(5*10=50\) kilograms. This would require the remaining 5 items to weigh 70 kilograms in total. However, the maximum combined weight for 5 items is \(5*10=50\) kilograms.
Could 9 items weigh 5 kilograms each? No, because their total weight would be \(5*9=45\) kilograms. This would require the remaining 6 items to weigh 75 kilograms in total. However, the maximum combined weight for 6 items is \(6*10=60\) kilograms.
Could 8 items weigh 5 kilograms each? No, because their total weight would be \(5*8=40\) kilograms. This would require the remaining 7 items to weigh 80 kilograms in total. However, the maximum combined weight for 7 items is \(7*10=70\) kilograms.
Could 7 items weigh 5 kilograms each? No, because their total weight would be \(5*7=35\) kilograms. This would require the remaining 8 items to weigh 85 kilograms in total. However, the maximum combined weight for 8 items is \(8*10=80\) kilograms.
Could 6 items weigh 5 kilograms each? Yes. Their total weight would be \(5*6=30\) kilograms, and the remaining 9 items would need to weigh 90 kilograms in total, which is achievable if each of them weighs 10 kilograms.
Answer: E