In a one-day cricket match the captain of one of the teams scored $30$ runs more than the average runs scored by the remaining six batsmen of that team who batted in the match. If the total runs scored by all the batsmen of that team were $310$, how many runs did the captain score?

Concept & Logic The total score of a team is the sum of the runs scored by the captain and the runs scored by the remaining batsmen. By setting a variable for the average of the remaining batsmen, we can form a linear equation based on the total team score. $$ \text{Total Runs} = \text{Captain's Runs} + \text{Remaining Batsmen's Runs} $$ Step-by-Step Solution * Let the average runs scored by the remaining $6$ batsmen be $x$. * The total runs scored by these $6$ batsmen = $6x$. * The captain scored $30$ runs more than their average, so the captain's score = $x + 30$. * The total runs scored by all $7$ batsmen (captain + $6$ others) is given as $310$. * Set up the equation: $(x + 30) + 6x = 310$. * Simplify and solve for $x$: $7x + 30 = 310 \implies 7x = 280 \implies x = 40$. * The captain's score is $x + 30 = 40 + 30 = 70$. Exam Strategy & Shortcut Subtract the captain's "extra" $30$ runs from the team total immediately: $310 - 30 = 280$. This $280$ represents the runs if the captain had scored exactly the average of the other $6$ batsmen. Divide this adjusted total by the total number of batsmen ($7$): $280 / 7 = 40$. This is the average $x$. Add back the $30$ to find the captain's score: $40 + 30 = 70$. Common Pitfall A common error is confusing the total number of players. Students might divide by $11$ (a full cricket team) instead of the $7$ batsmen who actually batted as stated by the constraints of the problem. Always read the specific headcount given. Final Answer **Therefore, the correct answer is 70.**