A student obtained $60$, $75$ and $85$ marks respectively in three monthly examinations in Physics and $95$ marks in the final examination. The three monthly examinations are of equal weightage whereas the final examination is weighted twice as much as a monthly examination. His average marks in Physics are

Concept & Formula This problem requires calculating a weighted average. When different components of a total have different levels of importance, we assign them "weights". The weighted average is the sum of the products of each value and its corresponding weight, divided by the total sum of the weights. $$ \text{Weighted Average} = \frac{w_1x_1 + w_2x_2 + \dots + w_nx_n}{w_1 + w_2 + \dots + w_n} $$ Step-by-Step Solution * Given marks for monthly exams: $60$, $75$, and $85$. * Given mark for the final exam: $95$. * Let the weight of each monthly exam be $1$. * The final exam is weighted twice as much, so its weight is $2$. * Sum of all weights = $1 + 1 + 1 + 2 = 5$. * Calculate the weighted sum of marks: * Monthly 1: $60 \times 1 = 60$ * Monthly 2: $75 \times 1 = 75$ * Monthly 3: $85 \times 1 = 85$ * Final Exam: $95 \times 2 = 190$ * Total Weighted Marks = $60 + 75 + 85 + 190 = 410$. * Weighted Average = $410 / 5 = 82$. Exam Strategy & Shortcut To calculate $410 / 5$ rapidly in your head, use the "double and divide by 10" trick. Multiply the number by $2$ and then move the decimal point one place to the left. $410 \times 2 = 820$. $820 / 10 = 82$. This avoids traditional long division and saves precious seconds. Common Pitfall The most common mistake is ignoring the weights entirely and calculating a simple arithmetic mean: $(60 + 75 + 85 + 95) / 4 = 315 / 4 = 78.75$. Notice that the examiner deliberately included $78.75$ as option (a) to trap students who forget to apply the weightage. Final Answer **Therefore, the correct answer is 82.**