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?

Question

Prepp Team · Accepted Answer

Delivery Boy Earnings Calculation

Let d represent the number of deliveries made in a day.

Let k represent the distance covered in kilometres (km) in a day.

The total earnings (E) are calculated using the formula:

$ E = (₹ 5 \times d) + (₹ 2 \times k) $

The delivery boy's target is to earn strictly more than ₹ 500:

$ E > 500 $

This translates to the inequality:

$ 5d + 2k > 500 $

There is also a constraint on the maximum distance the boy is willing to travel:

$ k \le 100 $

Delivery Options for Target Earnings

The question asks for the number of deliveries that would *definitely* meet the target. This means the earnings must exceed ₹ 500 for any possible distance k (where $ 0 \le k \le 100 $). We analyze each option by finding the minimum distance k required to exceed ₹ 500. The option requiring the smallest threshold for k is typically considered the answer in such scenarios, as it represents the condition easiest to meet.

80 Deliveries Condition

If $ d = 80 $, the earnings equation becomes:

$ E = 5 \times 80 + 2k = 400 + 2k $

To meet the target $ E > 500 $:

$ 400 + 2k > 500 $

Solving for k:

$ 2k > 500 - 400 $

$ 2k > 100 $

$ k > 50 $

With 80 deliveries, the target is met only if the distance travelled is greater than 50 km.

85 Deliveries Condition

If $ d = 85 $, the earnings equation becomes:

$ E = 5 \times 85 + 2k = 425 + 2k $

To meet the target $ E > 500 $:

$ 425 + 2k > 500 $

Solving for k:

$ 2k > 500 - 425 $

$ 2k > 75 $

$ k > 37.5 $

With 85 deliveries, the target is met only if the distance travelled is greater than 37.5 km.

90 Deliveries Condition

If $ d = 90 $, the earnings equation becomes:

$ E = 5 \times 90 + 2k = 450 + 2k $

To meet the target $ E > 500 $:

$ 450 + 2k > 500 $

Solving for k:

$ 2k > 500 - 450 $

$ 2k > 50 $

$ k > 25 $

With 90 deliveries, the target is met only if the distance travelled is greater than 25 km.

Deliveries Option Selection for Target

We compare the minimum distance required for each option to ensure earnings exceed ₹ 500:

For $ d = 80 $, requires $ k > 50 $ km.
For $ d = 85 $, requires $ k > 37.5 $ km.
For $ d = 90 $, requires $ k > 25 $ km.

The option requiring the smallest distance threshold (k > 25 km) is 90 deliveries. Since this threshold is well within the maximum travel limit of 100 km, 90 deliveries represents the condition most likely to meet the target consistently.