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.

- 13th
- 15th
- 16th
- 14th

- 14th
- 15th
- 12th
- 13th

- 15th
- 14th
- 16th
- 13th

- 14th
- 13th
- 15th
- 16th

From the given table we can easily find the order that took two days to deliver.

For any day ordered book = cumulative order booked on that day – cumulative order booked on previous day

Ordered that took two days to deliver = order booked on any day – Orders delivered on next day – order lost

Now as the order lost was declared on 2nd day so we can say that the order lost for nth day booking will be equal to the number of ordered lost on (n+2)th day

So we can make the following table:

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 23 – 8 = 15 | |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | ||

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | ||

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

From the table we can say ordered booked on 13th = 2+8+21 = 31

For any day ordered book = cumulative order booked on that day – cumulative order booked on previous day Ordered that took two days to deliver = order booked on any day – Orders delivered on next day – order lost Now as the order lost was declared on 2nd day so we can say that the order lost for nth day booking will be equal to the number of ordered lost on (n+2)th day

So we can make the following table:

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 23 – 8 = 15 | |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | ||

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | ||

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

From the table we can say ordered booked on 13th = 2+8+21 = 31

Fraction of order lost over order booked on that day :

On 13th = 2/31

On 14th = 12/30

On 15th = 12/28

On 16th = 2/25

Clearly largest fraction of order lost is 12/28 on 13th day.

For any day ordered book = cumulative order booked on that day – cumulative order booked on previous day

Ordered that took two days to deliver = order booked on any day – Orders delivered on next day – order lost

Now as the order lost was declared on 2nd day so we can say that the order lost for nth day booking will be equal to the number of ordered lost on (n+2)th day

So we can make the following table:

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 23 – 8 = 15 | |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | ||

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | ||

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

From the table we can say ordered booked on 13th = 2+8+21 = 31

Thus we can among the given days , maximum number of order were booked on 13th.

For any day ordered book = cumulative order booked on that day – cumulative order booked on previous day

Ordered that took two days to deliver = order booked on any day – Orders delivered on next day – order lost

Now as the order lost was declared on 2nd day so we can say that the order lost for nth day booking will be equal to the number of ordered lost on (n+2)th day

So we can make the following table:

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 23 – 8 = 15 | |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | ||

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | ||

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

From the table we can say ordered booked on 13th = 2+8+21 = 31

Now for 14th -: order delivered on 2nd day = 30 – 15 – 12 = 3

For 15th :Order delivered on next day = order delivered on 16th – order delivered on 2nd day for 14th = 11– 3 =8

Now for 15th -: order delivered on 2nd day = 28 – 8 -12 = 8

For 16th :Order delivered on next day = order delivered on 17th – order delivered on 2nd day for 15th = 21– 8 =12

Now for 16th -: order delivered on 2nd day = 25 – 12 – 2 = 9

So The table will now look like :-

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 3 | 23 – 8 = 15 |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | 8 | 8 |

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | 12 | 11 |

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

So delivery ratio on 13th =21/8= 2.625

delivery ratio on 14th = 15/3 = 5

delivery ratio on 15th = 8/8 =1

delivery ratio on 16th = 11/12

Thus we can see delivery ratio was highest on 14th.

For any day ordered book = cumulative order booked on that day – cumulative order booked on previous day

Ordered that took two days to deliver = order booked on any day – Orders delivered on next day – order lost

Now as the order lost was declared on 2nd day so we can say that the order lost for nth day booking will be equal to the number of ordered lost on (n+2)th day

So we can make the following table:

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 23 – 8 = 15 | |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | ||

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | ||

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

From the table we can say ordered booked on 13th = 2+8+21 = 31

Now for 14th -: order delivered on 2nd day = 30 – 15 – 12 = 3

For 15th :Order delivered on next day = order delivered on 16th – order delivered on 2nd day for 14th = 11– 3 =8

Now for 15th -: order delivered on 2nd day = 28 – 8 -12 = 8

For 16th :Order delivered on next day = order delivered on 17th – order delivered on 2nd day for 15th = 21– 8 =12

Now for 16th -: order delivered on 2nd day = 25 – 12 – 2 = 9

So The table will now look like :-

Day | Cumulative orders booked | Order on that day | Orders delivered on day | Cumulative orders lost | Order lost on that day | Order lost on of that day booking | Order delivered on 2^{nd} day |
Order delivered on the next day (1^{st} day) of booking |

11^{th} |
4 | |||||||

12^{th} |
6 | 11 – 4 = 7 | ||||||

13th | 219 | 11 | 91 | 2 | 8 | 27- 6 =21 | ||

14th | 249 | 249 – 219 = 30 | 27 | 92 | 92 – 91 = 1 | 12 | 3 | 23 – 8 = 15 |

15th | 277 | 277 – 249 = 28 | 23 | 94 | 94 – 92 =2 | 12 | 8 | 8 |

16th | 302 | 302 – 277 = 25 | 11 | 106 | 106 – 94 = 12 | 2 | 12 | 11 |

17th | 327 | 327 – 302 = 25 | 21 | 118 | 118 – 106 = 12 | 9 | ||

18th | 332 | 332 – 327 = 5 | 13 | 120 | 120 – 118 = 2 | |||

19th | 337 | 337 – 332 = 5 | 14 | 129 | 129 – 120 = 9 |

Average time taken to deliver on 13th = (21+2*8)/29 = 37/29

Average time taken to deliver on 14th = (15+2*3)/(15+3) = 21/18 < 37/29

Average time taken to deliver on 15th = (8+2*8)/(8+8) = 3/2

Average time taken to deliver on 16th = (11+2*12)/(11+12) = 35/23 >3/2

Thus we can see the average time taken to deliver a order was least on 14th.

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