Question

The average weight of $10$ men in a boat is increased by $1.8$ kg, when one of the crew members whose weighed $53$ kg is replaced by a new man. Find the weight of the new man?

• A. $75$ kg
• B. $71$ kg
• C. $68$ kg
• D. $80$ kg

Answer 1

The correct option is B $71$ kg

Let the average weight of $10$ men be $\overline{x}$

Let the $10$ men weights be $a_1,a_2,...,a_{10}$

Therefore, $\overline{x}=\frac{a_1+a_2+...+a_{10}}{10}$

$⟹a_1+a_2+...+a_{10}=10\overline{x}$ -------(1)

Given that one of members weight is $53$

Let $a_1=53$ ------(2)

substituting (2) in (1) we get

$53+a_2+...+a_{10}=10\overline{x}$

$⟹a_2+...+a_{10}=10\overline{x}-53$ -------(3)

$53$ weight man was replaced by the new man. Let the new man weight be $a_{11}$

Given that average is increased by $1.8kg$ after replacement.

Therefore, $\overline{x}+1.8=\frac{a_2+...+a_{10}+a_{11}}{10}$

$⟹a_2+...+a_{10}+a_{11}=10\overline{x}+18$ -----(4)

substituting (3) in (4) we get

$10\overline{x}-53+a_{11}=10\overline{x}+18$

$⟹a_{11}=18+53=71$

Therefore, the weight of the new man is $71kg$