Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is

Correct Option: 3 (51)

Let the total score of all 22 students be $S$, the score of Ramesh be $R$, and the score of Gautam be $G$.
We are given:
$R=82.5$

The average score of the 21 students other than Gautam is 62. Therefore, the sum of scores of these 21 students is:
$S-G=21\times 62=1302$
Thus, the total score of all 22 students is:
$S=1302+G$

The average score of the 21 students other than Ramesh is:
$\frac{S-R}{21}=\frac{S-82.5}{21}$

We are given that the average score of all 22 students is one more than the average score of the 21 students other than Ramesh:
$\frac{S}{22}=\frac{S-82.5}{21}+1$

Let us solve this equation for $S$:
$\frac{S}{22}=\frac{S-82.5+21}{21}$
$\frac{S}{22}=\frac{S-61.5}{21}$
$21S=22(S-61.5)$
$21S=22S-1353$
$S=1353$

Now we substitute $S$ back to find Gautam's score $G$:
$G=S-1302$
$G=1353-1302=51$

Hence, the score of Gautam is 51.