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

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Let's break down the information clearly:

22 students total (including Ramesh and Gautam)

Ramesh's score: 82.5

Average of 21 students (excluding Gautam): 62

Key condition: Average of all 22 students = Average of 21 students (excluding Ramesh) + 1

We need to find Gautam's score.

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Since the average of 21 students (excluding Gautam) is 62:

Total score of these 21 students = 21 × 62 = 1302

This includes Ramesh's score of 82.5

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Here's the crucial insight: We need to use the condition about the averages.

Let's call the average of 21 students (excluding Ramesh) = $A$

From the given condition:

Average of all 22 students = $A + 1$

Now let's express both sides using totals:

Left side: Average of all 22 students = $\frac{\text{Total of all 22 students}}{22}$

Right side: $A + 1 = \frac{\text{Total of 21 students excluding Ramesh}}{21} + 1$

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The total score of 21 students (excluding Ramesh) = Total of all 22 students - Ramesh's score

Let Total of all 22 students = $T$

So: $A = \frac{T - 82.5}{21}$

From our condition: $\frac{T}{22} = A + 1$

Substituting $A$: $\frac{T}{22} = \frac{T - 82.5}{21} + 1$

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$\frac{T}{22} = \frac{T - 82.5}{21} + 1$

Multiply everything by $22 \times 21 = 462$ to clear fractions:

$21T = 22(T - 82.5) + 22 \times 21$

$21T = 22T - 22 \times 82.5 + 462$

$21T = 22T - 1815 + 462$

$21T = 22T - 1353$

$T = 1353$

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From earlier we know:

Total score of 21 students (excluding Gautam) = 1302

Total score of all 22 students = 1353

Therefore: Gautam's score = 1353 - 1302 = 51