Question

The average weight of 7 men is diminished by 3kg when one of them who weighs 73 kgs is replaced by a new man. The weight of the new man is
[a] 50 kg
[b] 52 kg
[c] 53 kg
[d] 54 kg

Answer 1

Hint: Use the fact that $Average=\dfrac{Sum\text{ }of\text{ }all\ observations}{Number\ of\ all\ observations}$. Hence find the total weight of the persons initially and finally. Use the fact that the change in total weight is equal to the weight of the new person – the weight of the previous person. Hence find the weight of the new person.

Complete step-by-step answer:
Let the weight of the new person be w and the initial average be a
We know that $Average=\dfrac{Sum\text{ }of\text{ }all\ observations}{Number\ of\ all\ observations}$.
Hence, we have $a=\dfrac{total\ weight}{7}\Rightarrow total\ weight=7a$
Since the final average weight is
3 less than what was initially, we have final average weight = a-3.
Hence, we have $a-3=\dfrac{new\ total\ weight}{7}\Rightarrow new\ total\ weight=7a-21$
Hence change in total weight $=7a-21-7a=-21$
We know that the change in total weight is equal to the weight of the new person – the weight of the previous person. Hence find the weight of the new person.
Hence, we have $-21=w-73$
Adding 73 on both sides of the equation, we get
$w=73-21=52$
Hence the weight of the new person is 52kgs.
Hence option [b] is correct.

Note: [1] Alternative solution:
Best method:
A person’s weight reflects as $\dfrac{1}{7}$ per kg over the average
Hence if the average decline by 3, the net decline in total weight should be $7\times 3=21kg$
Hence, we have 73 -w = 21
Adding w on both sides, we get
21+w = 73
Subtracting 21 from both sides, we get
w = 73-21 =52, which is the same as obtained above.
[2] Assume that the weight of the first six persons be S.
Hence, we have $\dfrac{S+73}{7}-\dfrac{S+w}{7}=3$
Multiplying both sides by 7 and solving the equation yields w = 52
Hence the weight of the new person is 52kgs.