Kunal bought $65$ books for ₹ $1050$ from one shop and $50$ books for ₹ $1020$ from another. What is the average price he paid per book?

### Concept & Formula When finding the average cost across multiple bulk purchases, you cannot simply average the total prices. You must calculate the combined total cost of all items and divide it by the combined total number of items. This is a classic Weighted Average scenario. $$ \text{Average Price} = \frac{\text{Total Cost of All Items}}{\text{Total Number of Items}} $$ ### Step-by-Step Solution * **Given:** Purchase 1: $65$ books for a total of ₹ $1050$. Purchase 2: $50$ books for a total of ₹ $1020$. * **Calculation:** Step 1: Find the grand total cost. $\text{Total Cost} = 1050 + 1020 = 2070$ Step 2: Find the grand total number of books. $\text{Total Books} = 65 + 50 = 115$ Step 3: Apply the average formula. $\text{Average Price} = \frac{2070}{115}$ Step 4: Simplify the fraction. Both end in $0$ or $5$, so divide by $5$. $\frac{2070 \div 5}{115 \div 5} = \frac{414}{23}$ Now, divide $414$ by $23$. Since $23 \times 10 = 230$ and $23 \times 20 = 460$, the answer is slightly less than $20$. $23 \times 8 = 184$. $230 + 184 = 414$. Thus, $23 \times 18 = 414$. $\text{Average Price} = 18$ ### Exam Strategy & Shortcut **Unit Digit Estimation:** You need to solve $\frac{2070}{115}$. Look at the fraction $\frac{414}{23}$. You are looking for a number which, when multiplied by $23$, gives $414$. Check the unit digit: The multiplier must have a unit digit that, when multiplied by $3$, ends in $4$. $3 \times 8 = 24$ (ends in $4$). Looking at the options, $18$ ends in $8$. Multiplying $23 \times 18$ confirms it exactly. This avoids tedious long division. ### Common Pitfall The most dangerous pitfall is finding the average of the two costs ($\frac{1050 + 1020}{2}$) or the average of the individual book prices from each shop without weighting them by quantity. Always pool the total money and total items first. ### Final Answer **Therefore, the correct answer is ₹ 18.**