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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. $53$
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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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