$\text{XYZ}$ organization got into the business of delivering groceries to home at the beginning of the last month. They have a two-day delivery promise. However, their deliveries are unreliable. An order booked on a particular day may be delivered the next day or the day after. If the order is not delivered at the end of two days, then the order is declared as lost at the end of the second day. $\text{XYZ}$ then does not deliver the order, but informs the customer, marks the order as lost, returns the payment and pays a penalty for non-delivery.
The following table provides details about the operations of $\text{XYZ}$ for a week of the last month. The first column gives the date, the second gives the cumulative number of orders that were booked up to and including that day. The third column represents the number of orders delivered on that day. The last column gives the cumulative number of orders that were lost up to and including that day.
It is known that the numbers of orders that were booked on the $11 \text{th}, 12 \text{th},$ and $13 \text{th}$ of the last month that took two days to deliver were $4,6,$ and $8$ respectively.
| Day |
Cumulative orders booked |
Orders delivered on day |
Cumulative orders lost |
| $13\text{th}$ |
$219$ |
$11$ |
$91$ |
| $14\text{th}$ |
$249$ |
$27$ |
$92$ |
| $15\text{th}$ |
$277$ |
$23$ |
$94$ |
| $16\text{th}$ |
$302$ |
$11$ |
$106$ |
| $17\text{th}$ |
$327$ |
$21$ |
$118$ |
| $18\text{th}$ |
$332$ |
$13$ |
$120$ |
| $19\text{th}$ |
$337$ |
$14$ |
$129$ |
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 $(x+2y)/(x+y).$ On which of the following days was the average time taken to deliver orders booked the least $?$
- $ 14 \text{th} $
- $ 16 \text{th} $
- $ 15 \text{th} $
- $ 13 \text{th} $