A shopkeeper employs a delivery boy and gives him a motorcycle for home delivery. For every delivery, the boy is given ₹5. At the end of the day, he also gets ₹2 for every kilometre of the distance covered in the day. The boy wants to earn more than ₹500 a day, but does not want to travel more than 100 km. Which of the following numbers of deliveries would definitely meet his target?

The correct option is D - The question cannot be answered due to insufficient data.

Solution

Let the number of deliveries be 'n' and the total distance covered be 'd' km.
The total daily earning (E) is given by the formula:
$E=(5\times n)+(2\times d)$

According to the conditions provided:
1. The boy wants $E>500$.
2. The boy does not want to travel more than 100 km, so d ≤ 100.

To find the number of deliveries (n) that definitely meets the target, we must consider the "worst-case scenario" for the distance. Since the distance (d) contributes to his earnings, the minimum possible earning for a fixed number of deliveries occurs when the distance traveled is at its minimum.

The problem states the maximum distance (100 km), but it does not provide a minimum distance required per delivery or a minimum total distance for the day. If the boy makes deliveries to neighbors very close to the shop, 'd' could be close to 0 km.

Why this is correct

To ensure the target of ₹500 is definitely met regardless of the distance covered, we must assume the distance (d) is 0 km (the lowest possible value).
$5n+2(0)>500$
$5n>500$
$n>100$

None of the numerical options (80, 85, or 90) are greater than 100. If the boy makes 90 deliveries but only travels 10 km in total, his earning would be $(5\times 90)+(2\times 10)=450+20$ = ₹470, which is less than ₹500. Because the distance 'd' is a variable that can range anywhere from 0 to 100 km, we cannot determine a delivery count below 101 that guarantees the target is met without knowing the actual distance covered.

Why the other options are incorrect

• Option (a) - 80: If the boy travels the maximum distance of 100 km, his earning is $(5\times 80)+(2\times 100)$ = ₹600. However, if he travels only 10 km, his earning is ₹420. Thus, 80 deliveries do not definitely meet the target.
• Option (b) - 85: Similar to option (a), if the distance covered is small (e.g., 20 km), the earning is $(5\times 85)+(2\times 20)=425+40$ = ₹465, which fails the > ₹500 condition.
• Option (c) - 90: Even with 90 deliveries, if the distance covered is only 10 km, the earning is ₹470. Since the distance is not fixed or given a minimum value, this number does not guarantee the target.

Key Concept

In "definitely" or "must be true" questions involving multiple variables, one must test the boundary conditions (minimums and maximums) to see if the condition holds true in all possible scenarios.