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. 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 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 11th, 12th, and 13th 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
13th	219	11	91
14th	249	27	92
15th	277	23	94
16th	302	11	106
17th	327	21	118
18th	332	13	120
19th	337	14	129

On which of the following days was the number of orders booked the highest?

The correct option is 13th.

To find the day on which the highest number of orders were booked, let us first calculate the number of orders booked on each of the relevant days from the given data.

The table provides the Cumulative orders booked up to and including each day. We can calculate the individual orders booked on any day d (where d > 13th) by subtracting the cumulative orders booked up to day d - 1 from the cumulative orders booked up to day d.

Let's find the orders booked on the 13th, 14th, and 15th:
- For the 14th:
Cumulative orders booked up to 14th = 249
Cumulative orders booked up to 13th = 219
Orders booked on the 14th = 249 - 219 = 30

- For the 15th:
Cumulative orders booked up to 15th = 277
Cumulative orders booked up to 14th = 249
Orders booked on the 15th = 277 - 249 = 28

Now, let us determine the number of orders booked on the 12th and 13th.
An order booked on a particular day can be:
1. Delivered the next day (1-day delivery).
2. Delivered the day after (2-day delivery).
3. Lost at the end of the second day (non-delivery).

Therefore, for any day d, the orders booked on day d will eventually be resolved in one of three ways:
Orders booked on day d = (Orders booked on day d delivered on day d + 1) + (Orders booked on day d delivered on day d + 2) + (Orders booked on day d lost at the end of day d + 2).

Let us analyze the deliveries and losses on the 14th:
The orders delivered on the 14th can only come from orders booked on the 12th (taking 2 days to deliver) or the 13th (taking 1 day to deliver).
$\text{Total delivered on 14th}=27$
We are given that the orders booked on the 12th that took 2 days to deliver was 6.
Thus, the orders booked on the 13th that were delivered on the 14th (taking 1 day) = 27 - 6 = 21.

Let us analyze the losses on the 14th:
An order is declared lost at the end of the second day. Thus, cumulative orders lost up to the 14th includes orders booked up to the 12th that were not delivered by the end of the 14th.
The increase in cumulative losses on the 14th is:
$\text{New losses on 14th}=92-91=1$
These lost orders must have been booked on the 12th.

Let us analyze the deliveries and losses on the 15th:
The orders delivered on the 15th can only come from orders booked on the 13th (taking 2 days to deliver) or the 14th (taking 1 day to deliver).
We are given that the orders booked on the 13th that took 2 days to deliver was 8.
Thus, the orders booked on the 14th that were delivered on the 15th (taking 1 day) = 23 - 8 = 15.

The increase in cumulative losses on the 15th is:
$\text{New losses on 15th}=94-92=2$
These lost orders must have been booked on the 13th.

Now we can compute the total orders booked on the 13th:
Orders booked on the 13th = (Orders booked on 13th delivered on 14th) + (Orders booked on 13th delivered on 15th) + (Orders booked on 13th lost on 15th)
$\text{Orders booked on 13th}=21+8+2=31$

Now let us compute the total orders booked on the 12th:
The orders booked on the 12th can be delivered on the 13th (taking 1 day), delivered on the 14th (taking 2 days), or lost on the 14th.
From the table, the total deliveries on the 13th was 11. These must come from orders booked on the 11th (taking 2 days) or the 12th (taking 1 day).
We are given that the orders booked on the 11th that took 2 days to deliver was 4.
Thus, the orders booked on the 12th delivered on the 13th (taking 1 day) = 11 - 4 = 7.
We already know:
- Orders booked on the 12th delivered on the 14th (taking 2 days) = 6
- Orders booked on the 12th lost on the 14th = 1
Therefore:
$\text{Orders booked on 12th}=7+6+1=14$

Comparing the number of orders booked on the given options:
- 12th: 14 orders
- 13th: 31 orders
- 14th: 30 orders
- 15th: 28 orders

The number of orders booked was highest on the 13th (with 31 orders).