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A class consists of $20$ boys and $30$ girls. In the mid-semester examination, the average score of the girls was $5$ higher than that of the boys. In the final exam, however, the average score of the girls dropped by $3$ while the average score of the entire class increased by $2$. The increase in the average score of the boys is

1. $9.5$
2. $10$
3. $4.5$
4. $6$

Let, in the mid-semester examination, the total score of girls and boys be $g$ and $b$ respectively.

So, $\frac{g}{30} = \frac{b}{20} + 5 \quad \longrightarrow (1)$

Average score of the class $= \frac{g+b}{50}$

Let, in the final exam, the total score of girls and boys be ${g}’$ and ${b}’$ respectively.

Now, $\frac{{g}’}{30} = \frac{g}{30} – 3 \Rightarrow \boxed{ {g}’ = g-90}$

And, $\frac{{g}’+{b}'}{50} = \frac{g+b}{50} + 2$

$\Rightarrow \frac{{g}’+{b}'}{50} = \frac{g+b+100}{50}$

$\Rightarrow {g}’+{b}' = g+b+100$

$\Rightarrow g-90+{b}' = g+b+100$

$\Rightarrow \boxed { {b}' = b+190 }$

So, the new average of boys $= \frac{{b}'}{20} = \frac{b+190}{20} = \frac{b}{20} + \frac{190}{20} = \frac{b}{20} + 9.5$

$\therefore$ The increase in the average score of the boys is $9.5.$

Correct Answer $: \text{A}$
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