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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

1. $49$
2. $48$
3. $51$
4. $53$

Given that, Ramesh score $= 82 \cdot 5$

And, the average score of the $21$ students other than Gautam is $62.$

$\frac{\text{Ramesh + 20 others (not include Gautam)}} {21} = 62$

$\Rightarrow 20$ others students score $= 62 \times 21 – 82 \cdot 5 = 1302 – 82 \cdot 5 = 1219 \cdot 5$

The average score of all the $22$ students is one more than the average score of the $21$ students other than Ramesh.

$\frac{ \text {Ramesh + Gautam +20 others}} {22} = \frac{ \text {Gautam + 20 others (Ramesh not include)}}{21} + 1$

$\Rightarrow \frac{ 82\cdot5 + \text{Gautam} + 1219\cdot5 }{22} = \frac{ \text{Gautam} + 1219\cdot5 }{21} + 1$

$\Rightarrow \frac{1302 + \text{Gautam}}{22} = \frac{\text{Gautam} + 1219\cdot5 + 21}{21}$

$\Rightarrow \frac{1302 + \text{Gautam}}{22} = \frac{\text{Gautam} + 1240\cdot5}{21}$

$\Rightarrow 1302 \times 21 + 21 \; \text{Gautam} = 22 \; \text{Gautam} + 1240 \cdot5 \times 22$

$\Rightarrow 27342 + 21 \; \text{Gautam} = 22 \; \text{Gautam} + 27291$

$\Rightarrow 27342 – 27291 = 22 \; \text{Gautam} – 21 \; \text{Gautam}$

$\Rightarrow \boxed{\text{Gautam} = 51}$

$\therefore$ The score of Gautam $= 51.$

Correct Answer $: \text{C}$
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