Question

A candidate obtained the following percentage of marks in an examination English 60, Math 90, Physics 75, chemistry 66. If weights 2, 4, 3, 3 are allotted to these subjects respectively, then find the weighted mean.

Answer 1

Hint: First we will note the formula to find the weighted mean of $n$ numbers which are ${{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{n}}$ and their weights are given by ${{w}_{1}},{{w}_{2}},{{w}_{3}}...{{w}_{n}}$. So, the weighted mean is given by the formula $\dfrac{{{a}_{1}}{{w}_{1}}+{{a}_{2}}{{w}_{2}}+{{a}_{3}}{{w}_{3}}...\,+{{a}_{n}}{{w}_{n}}}{{{w}_{1}}+{{w}_{2}}+{{w}_{3}}...+{{w}_{n}}}$. Then we will replace the variable given in the formula by the values given in the question to get our result.

Complete step-by-step solution:
The weighted mean of $n$ numbers which are ${{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{n}}$ and their weights are given by ${{w}_{1}},{{w}_{2}},{{w}_{3}}...{{w}_{n}}$. So, the weighted mean is given by the formula
$\dfrac{{{a}_{1}}{{w}_{1}}+{{a}_{2}}{{w}_{2}}+{{a}_{3}}{{w}_{3}}...+{{a}_{n}}{{w}_{n}}}{{{w}_{1}}+{{w}_{2}}+{{w}_{3}}...+{{w}_{n}}}$ …. (i)
Where ${{w}_{1}},{{w}_{2}},{{w}_{3}}...{{w}_{n}}$ are the weights of the corresponding values ${{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{n}}$.
Now according to the question the value of $n$ is 4. Similarly, the corresponding values and their weights are given by ${{a}_{1}}=60,{{a}_{2}}=90,{{a}_{3}}=75,{{a}_{4}}=66$ and their corresponding weights are given by ${{w}_{1}}=2,{{w}_{2}}=4,{{w}_{3}}=3,{{w}_{4}}=3$.
Now replacing the above values in the corresponding equation (i) we get
$=\dfrac{2\times 60+4\times 90+3\times 75+3\times 66}{2+4+3+3}$
$=\dfrac{120+360+225+198}{12}$
$=\dfrac{903}{12}$
$=75.25$
So, the weighted mean of the marks along with their corresponding weights is 75.25.

Note: While calculating weighted mean we must keep in mind the difference between the arithmetic mean and the weighted mean, while the former is just the normal average that we calculate in day to day life while the latter is calculated by multiplying the corresponding weights with the values and then dividing with the total weight.