The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest
number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean
becomes 2 more than the middle number, while the difference between the largest and the smallest
numbers becomes 64.Then, the largest number in the original set of three numbers is
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Explanatory Answer
Let the numbers be a (smallest), b (middle), c (largest).
Given: (a + b + c)/3 = 28 ⇒ a + b + c = 84. ...(1)
After changes:
smallest → a + 7
middle → b
largest → c − 10
New mean:
[(a + 7) + b + (c − 10)] / 3 = (a + b + c − 3)/3.
But a + b + c = 84, so new mean = (84 − 3)/3 = 81/3 = 27.
Given: new mean = b + 2 ⇒ 27 = b + 2 ⇒ b = 25. ...(2)
Given difference after changes:
(c − 10) − (a + 7) = 64
⇒ c − a − 17 = 64
⇒ c − a = 81. ...(3)
From (1) and (2): a + c = 84 − b = 84 − 25 = 59. ...(4)
Solve (3) and (4):
Add: (c − a) + (a + c) = 81 + 59 ⇒ 2c = 140 ⇒ c = 70.
Answer:
The largest number originally is 70.
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