The mean of $6$ numbers is $42$ . If one number is excluded, the mean of remaining numbers is $45$ . Find the included number.

26

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25

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28

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27

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Solution

The correct option is D $27$
Given that mean of $6$ numbers is $42$
Let the numbers be $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$

Therefore the mean $= \frac{a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6}}{6} = 42$

$⟹ a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = 42 \times 6 = 252$ ————(1)

Let the excluded number be $a_{6}$

Given that mean of the remaining numbers is $45$

Therefore the mean $= \frac{a_{1} + a_{2} + a_{3} + a_{4} + a_{5}}{5} = 45$

$⟹ a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = 45 \times 5 = 225$ ————(2)

Substituting (2) in (1) we get

$225 + a_{6} = 252 ⟹ a_{6} = 252 - 225 = 27$

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