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In a football tournament, a player has played a certain number of matches and $10$ more matches are to be played. If he scores a total of one goal over the next $10$ matches, his overall average will be $0.15$ goals per match. On the other hand, if he scores a total of two goals over the next $10$ matches, his overall average will be $0.2$ goals per match. The number of matches he has played is

Let the total matches played by him be $x,$ and his goals be $n.$

Now, $\frac{n+1}{x+10} = 0.15$

$\Rightarrow n+1 = (0.15)(x+10) \quad \longrightarrow (1)$

And, $\frac{n+2}{x+10} = 0.2$

$\Rightarrow n+2 = (0.2)(x+10) \quad \longrightarrow (2)$

Substract equation $(1),$ from equation $(2).$

$n+2 – (n+1) = 0.2 \times (x+10) – 0.15 \times (x+10)$

$\Rightarrow n+2-n-1 = (x+10) (0.2 – 0.15)$

$\Rightarrow 1 = (x+10) (0.05)$

$\Rightarrow x+10 = \frac{1}{0.05}$

$\Rightarrow x+10 = 20$

$\Rightarrow \boxed{x = 10}$

$\therefore$ The number of matches played by him is $10.$

Correct Answer $: 10$
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