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

Among the following days, the largest fraction of orders booked on which day was lost?

The correct option is 15th.

To determine the day on which the largest fraction of booked orders was lost, we need to calculate both the number of orders booked on each day and the number of those booked orders that were eventually lost.

1. Understanding the delivery and loss timeline:
An order booked on Day D can be delivered on Day D+1 (next day) or Day D+2 (the day after). If it is not delivered by the end of Day D+2, it is declared lost at the end of Day D+2. Thus, the number of lost orders booked on Day D is equal to the increase in cumulative lost orders on Day D+2:
${\text{Lost}}_D=L_{D+2}-L_{D+1}$
where $L_t$ represents the cumulative orders lost up to and including Day t.

2. Calculating orders booked on each day:
The number of orders booked on Day D (for days after the 13th) is the difference in cumulative bookings between Day D and Day D-1:
${\text{Booked}}_D={\text{Cumulative Booked}}_D-{\text{Cumulative Booked}}_{D-1}$

Let us calculate the bookings for the options provided:

• 14th: 249 − 219 = 30 orders booked.
• 15th: 277 − 249 = 28 orders booked.
• 16th: 302 − 277 = 25 orders booked.

For the 13th, we calculate the bookings using the delivery components:
The total orders delivered on Day D is the sum of orders booked on D-1 delivered in 1 day and orders booked on D-2 delivered in 2 days:
${\text{Delivered}}_D={\text{Delivered}}_{D-1,1}+{\text{Delivered}}_{D-2,2}$
From the given data:

• For the 13th: $11={\text{Delivered}}_{12,1}+4\Rightarrow {\text{Delivered}}_{12,1}=7$
• For the 14th: $27={\text{Delivered}}_{13,1}+6\Rightarrow {\text{Delivered}}_{13,1}=21$

Orders booked on the 13th are either delivered on the 14th (1 day), delivered on the 15th (2 days), or lost on the 15th. Therefore:
${\text{Booked}}_{13}={\text{Delivered}}_{13,1}+{\text{Delivered}}_{13,2}+{\text{Lost}}_{13}$
We are given that 8 orders booked on the 13th took 2 days to deliver. The lost orders from the 13th (which are declared lost on the 15th) are:
$L_{15}-L_{14}=94-92=2$
Thus, the total bookings on the 13th are:
$21+8+2=31$

3. Calculating the lost fraction for each day:

• 13th:
  • Booked: 31
  • Lost (on the 15th): 94 − 92 = 2
  • Fraction lost: $\frac{2}{31}\approx 0.0645\text{ (or 6.45\%)}$
• 14th:
  • Booked: 30
  • Lost (on the 16th): 106 − 94 = 12
  • Fraction lost: $\frac{12}{30}=0.40\text{ (or 40\%)}$
• 15th:
  • Booked: 28
  • Lost (on the 17th): 118 − 106 = 12
  • Fraction lost: $\frac{12}{28}\approx 0.4286\text{ (or 42.86\%)}$
• 16th:
  • Booked: 25
  • Lost (on the 18th): 120 − 118 = 2
  • Fraction lost: $\frac{2}{25}=0.08\text{ (or 8\%)}$

Comparing these values, the largest fraction of orders lost is on the 15th with approximately 42.86%.