The average time taken to deliver orders booked on a particular day is computed as follows. Let the number of orders delivered the next day be x and the number of orders delivered the day after be y. Then the average time to deliver order is $\frac{x+2y}{x+y}$. On which of the following days was the average time taken to deliver orders booked the least?

The correct option is 14th.

Step-by-step Explanation:

An order booked on day N can be delivered either on the next day (day N + 1) or the day after (day N + 2). Let the number of orders delivered on the next day be x and those delivered on the day after be y. Any order that is not delivered by the end of day N + 2 is declared lost. Therefore, the increase in cumulative lost orders on day N + 2 represents the lost orders that were originally booked on day N.

From the given problem data, we can compute the daily bookings and track the delivery days for the orders booked on the 13th, 14th, 15th, and 16th:

1. For orders booked on the 13th:
- Total orders booked on the 13th = 219 (cumulative on 13th) - 188 (cumulative on 12th) = 31 orders.
- Delivered the next day (on the 14th): 21 orders. Thus, $x=21$.
- Delivered the day after (on the 15th): 8 orders. Thus, $y=8$.
- Applying the average time formula:

$\text{Average Time}=\frac{x+2y}{x+y}=\frac{21+2\left(8\right)}{21+8}=\frac{21+16}{29}=\frac{37}{29}\approx 1.276\text{ days}$

2. For orders booked on the 14th:
- Total orders booked on the 14th = 249 (cumulative on 14th) - 219 (cumulative on 13th) = 30 orders.
- Delivered the next day (on the 15th): 15 orders. Thus, $x=15$.
- Delivered the day after (on the 16th): 3 orders. Thus, $y=3$.
- Applying the average time formula:

$\text{Average Time}=\frac{x+2y}{x+y}=\frac{15+2\left(3\right)}{15+3}=\frac{15+6}{18}=\frac{21}{18}=\frac{7}{6}\approx 1.167\text{ days}$

3. For orders booked on the 15th:
- Total orders booked on the 15th = 277 (cumulative on 15th) - 249 (cumulative on 14th) = 28 orders.
- Delivered the next day (on the 16th): 8 orders. Thus, $x=8$.
- Delivered the day after (on the 17th): 8 orders. Thus, $y=8$.
- Applying the average time formula:

$\text{Average Time}=\frac{x+2y}{x+y}=\frac{8+2\left(8\right)}{8+8}=\frac{24}{16}=1.5\text{ days}$

4. For orders booked on the 16th:
- Total orders booked on the 16th = 302 (cumulative on 16th) - 277 (cumulative on 15th) = 25 orders.
- Delivered the next day (on the 17th): 13 orders. Thus, $x=13$.
- Delivered the day after (on the 18th): 10 orders. Thus, $y=10$.
- Applying the average time formula:

$\text{Average Time}=\frac{x+2y}{x+y}=\frac{13+2\left(10\right)}{13+10}=\frac{33}{23}\approx 1.435\text{ days}$

Comparing the average delivery times computed above:
- 13th: ~1.276 days
- 14th: ~1.167 days
- 15th: 1.500 days
- 16th: ~1.435 days

The average time taken to deliver orders booked was the least on the 14th with approximately 1.167 days.