In the first $10$ overs of a cricket game, the run rate was only $3.2$. What should be the run rate in the remaining $40$ overs to reach the target of $282$ runs?

Concept & Strategy The total runs scored in any segment of a cricket match is the product of the number of overs and the run rate. To find the required run rate, we must determine the remaining runs needed and divide it by the remaining overs. $$ \text{Total Runs} = \text{Overs} \times \text{Run Rate} $$ Step-by-Step Solution * Given: Target score = $282$ runs. First $10$ overs run rate = $3.2$. Remaining overs = $40$. * Calculate the runs scored in the first $10$ overs: $10 \times 3.2 = 32$ runs. * Determine the remaining runs required to reach the target: $282 - 32 = 250$ runs. * Calculate the required run rate for the remaining $40$ overs: $250 / 40 = 25 / 4 = 6.25$. Exam Strategy & Shortcut Instead of writing out full equations, you can calculate this mentally. Mentally shift the decimal: $10 \times 3.2 = 32$. Subtract $32$ from $282$ in your head to get $250$. The fraction $250/40$ simplifies to $25/4$. Since $24/4 = 6$ and $1/4 = 0.25$, the sum is $6.25$. Common Pitfall A frequent mistake is dividing the total target ($282$) by the total overs ($50$) to get the overall run rate, and then trying to subtract the initial run rate ($3.2$), which mathematically does not yield the correct required run rate. Always work with absolute runs, not averages of averages. Final Answer **Therefore, the correct answer is 6.25.**