Question

The average weight of $8$ person’s increase by $2.5{\text{kg}}$ when a new person comes in the place of one of them weighing $65{\text{kg}}$. What might be the weight of the new person?
A. $76{\text{kg}}$
B. $76.5{\text{kg}}$
C. $85{\text{kg}}$
D. \[{\text{data inadequate}}\]
E. none of these.

Answer 1

Hint: Let us take the individual weights of $8$ persons as ${w_1},{w_2},{w_3},...........{w_8}$ is given by $\dfrac{{{w_1} + {w_2} + {w_3} + ...........{w_8}}}{8}$. Then according to the given data form the equation.

Complete step-by-step answer:
Here we are given in the question that the average weight of $8$ person’s increase by $2.5{\text{kg}}$ when a new person comes in the place of one of them weighing$65{\text{kg}}$ and we need to find what might be the weight of the new person.
So let the weight of the $8$ persons be ${w_1},{w_2},{w_3},...........{w_8}$
Let $W$ be the weight of the new person which comes at place of one of the person whose weight is $65{\text{kg}}$
Let us suppose that $W$ comes in the place of ${w_5}$
So we can say that ${w_5} = 65{\text{kg}}$
Let us assume that initial mean of all the persons be $A$
So we can say that
$A = \dfrac{{{w_1} + {w_2} + {w_3} + ...........{w_8}}}{8}$
We know that ${w_5} = 65{\text{kg}}$
$\Rightarrow$ $A = \dfrac{{65 + {w_1} + {w_2} + {w_3} + {w_4} + {w_6} + {w_7} + {w_8}}}{8}$
$\Rightarrow$ ${w_1} + {w_2} + {w_3} + {w_4} + {w_6} + {w_7} + {w_8} = 8A - 65$$ - - - - (1)$
Now for the 2nd case we can say that the mean is increased by $2.5$
So we can write that
$\Rightarrow$ $A + 2.5 = \dfrac{{{w_1} + {w_2} + {w_3} + {w_4} + W + {w_6} + {w_7} + {w_8}}}{8}$
$\Rightarrow$ ${w_1} + {w_2} + {w_3} + {w_4} + {w_6} + {w_7} + {w_8} + W = 8(A + 2.5)$$ - - - - - (2)$
Subtracting the equation (1) and (2) we get that
$
   - W = 8A - 65 - 8A - 20 = - 85 \\
\Rightarrow W = 85{\text{kg}} \\
 $
Hence we get that the weight of the new person is $85{\text{kg}}$.

Note: If we are replacing any number by the other number which is greater than the initial number then we can also say that new mean will also increase than the initial mean or if the number is lesser than the initial one then we will get the mean also lesser than the initial mean.