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06 Aug 2026, 05:54
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At a climbing gym, 4 junior climbers and 3 instructors were weighed for harness fitting. If the average (arithmetic mean) weight of all 7 people was 55 kilograms and the heaviest instructor weighed 80 kilograms, was the average (arithmetic mean) weight of the 4 junior climbers less than 53 kilograms?
(1) The average (arithmetic mean) weight of the other two instructors was 56 kilograms.
(2) The three lightest people in the group were junior climbers, each weighing 47 kilograms.
(1) The average (arithmetic mean) weight of the other two instructors was 56 kilograms.
(2) The three lightest people in the group were junior climbers, each weighing 47 kilograms.
06 Aug 2026, 05:54
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Official Solution:
At a climbing gym, 4 junior climbers and 3 instructors were weighed for harness fitting. If the average (arithmetic mean) weight of all 7 people was 55 kilograms and the heaviest instructor weighed 80 kilograms, was the average (arithmetic mean) weight of the 4 junior climbers less than 53 kilograms?
The total weight of all 7 people was:
\(7 * 55 = 385\) kilograms
The question asks whether the total weight of the 4 junior climbers was less than:
\(4 * 53 = 212\) kilograms
(1) The average weight of the other two instructors was 56 kilograms.
So the total weight of those two instructors was:
\(2 * 56 = 112\) kilograms
The heaviest instructor weighed 80 kilograms, so the total weight of the 3 instructors was:
\(80 + 112 = 192\) kilograms
Therefore, the total weight of the 4 junior climbers was:
\(385 - 192 = 193\) kilograms
Since \(193 < 212\), the average weight of the junior climbers was less than 53 kilograms.
Sufficient.
(2) The three lightest people in the group were junior climbers, each weighing 47 kilograms.
Let \(x\) be the weight of the fourth junior climber, and let \(y\) and \(z\) be the weights of the other two instructors.
Then:
\(47 + 47 + 47 + x + y + z + 80 = 385\)
\(x + y + z = 164\)
If the average weight of the 4 junior climbers were not less than 53 kilograms, then their total weight would be at least 212 kilograms:
\(47 + 47 + 47 + x \geq 212\)
\(x \geq 71\)
Then:
\(y + z \leq 164 - 71 = 93\)
But since the three 47-kilogram junior climbers were the lightest people, each of the other two instructors must weigh at least 47 kilograms. So:
\(y + z \geq 47 + 47 = 94\)
This is impossible.
Therefore, the average weight of the junior climbers must be less than 53 kilograms.
Sufficient.
Answer: D
At a climbing gym, 4 junior climbers and 3 instructors were weighed for harness fitting. If the average (arithmetic mean) weight of all 7 people was 55 kilograms and the heaviest instructor weighed 80 kilograms, was the average (arithmetic mean) weight of the 4 junior climbers less than 53 kilograms?
The total weight of all 7 people was:
\(7 * 55 = 385\) kilograms
The question asks whether the total weight of the 4 junior climbers was less than:
\(4 * 53 = 212\) kilograms
(1) The average weight of the other two instructors was 56 kilograms.
So the total weight of those two instructors was:
\(2 * 56 = 112\) kilograms
The heaviest instructor weighed 80 kilograms, so the total weight of the 3 instructors was:
\(80 + 112 = 192\) kilograms
Therefore, the total weight of the 4 junior climbers was:
\(385 - 192 = 193\) kilograms
Since \(193 < 212\), the average weight of the junior climbers was less than 53 kilograms.
Sufficient.
(2) The three lightest people in the group were junior climbers, each weighing 47 kilograms.
Let \(x\) be the weight of the fourth junior climber, and let \(y\) and \(z\) be the weights of the other two instructors.
Then:
\(47 + 47 + 47 + x + y + z + 80 = 385\)
\(x + y + z = 164\)
If the average weight of the 4 junior climbers were not less than 53 kilograms, then their total weight would be at least 212 kilograms:
\(47 + 47 + 47 + x \geq 212\)
\(x \geq 71\)
Then:
\(y + z \leq 164 - 71 = 93\)
But since the three 47-kilogram junior climbers were the lightest people, each of the other two instructors must weigh at least 47 kilograms. So:
\(y + z \geq 47 + 47 = 94\)
This is impossible.
Therefore, the average weight of the junior climbers must be less than 53 kilograms.
Sufficient.
Answer: D
