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CAT 2021 Slot 2 — QA Question 10

Change in AverageEasy

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 

Answer & solution

Answer: 10

Solution

Easy

Let nn be the matches already played and gg the goals so far. Each scenario gives "total goals over total matches = stated average." Form two linear equations and solve.

1

Scenario 1 (1 goal in next 10). Overall average 0.150.15 over n+10n+10 matches.

g+1=0.15 (n+10)⇒ g=0.15 n+1.5−1=0.15 n+0.5…(1)\begin{aligned} &g+1=0.15\,(n+10)\\ &\Rightarrow\ g=0.15\,n+1.5-1=0.15\,n+0.5 \quad\dots(1) \end{aligned}
2

Scenario 2 (2 goals in next 10). Overall average 0.20.2 over n+10n+10 matches.

g+2=0.2 (n+10)⇒ g=0.2 n+2−2=0.2 n…(2)\begin{aligned} &g+2=0.2\,(n+10)\\ &\Rightarrow\ g=0.2\,n+2-2=0.2\,n \quad\dots(2) \end{aligned}
3

Solve (1) and (2). Set the two expressions for gg equal.

0.2 n=0.15 n+0.5⇒ 0.05 n=0.5⇒ n=10\begin{aligned} &0.2\,n=0.15\,n+0.5\\ &\Rightarrow\ 0.05\,n=0.5\\ &\Rightarrow\ n=10 \end{aligned}
Matches already played=10\text{Matches already played}=10
CAT 2021 Slot 2 QA Q10: In a football tournament, a player has played a certain number of matches and 10 more matches are to be played — Solution | TheCATExam