The average of 11 numbers is 44.5. If the average of the first six numbers is 42.5 and that of the last six numbers is 45, then find the middle number.

Correct option is C

Given:

The average of 11 numbers = 44.5.

The average of the first 6 numbers = 42.5.

The average of the last 6 numbers = 45.

Formula Used:

Sum = Average $$\times$$​ Number of terms

Solution:

Sum of all 11 numbers: $$44.5 \times 11 = 489.5$$​​

Sum of the first 6 numbers: $$42.5 \times 6 = 255$$​​

Sum of the last 6 numbers: $$45 \times 6 = 270$$​​

Now, let the middle number be x. Since the first 6 numbers and the last 6 numbers overlap at the middle number, we have:

Sum of the first 6 numbers + Sum of the last 6 numbers - x = Sum of all 11 numbers

255 + 270 - x = 489.5

525 - x = 489.5

x = 525 - 489.5 = 35.5

The middle number is 35.5.

Thus, the correct option is (c) 35.5

Alternate Method:

Middle Number = (6 $$\times$$ 42.5) + (6 $$\times$$ 45) - (11 $$\times$$ 44.5)

Middle Number = 255 + 270 - 489.5

Middle Number = 525 - 489.5 = 35.5
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