Question 1171556: From Precious Mirror of the Four Elements ( 1303 A.D. ) . I take along a bottle with some wine in it for an excursion in the spring. On reaching a tavern, I double the bottle’s content and drink 1 9/10 dou in the tavern. After passing 4 taverns, the bottle is empty. How much wine was there at the beginning? Show your solutions.
Found 2 solutions by ankor@dixie-net.com, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! I take along a bottle with some wine in it for an excursion in the spring.
On reaching a tavern, I double the bottle’s content and drink 1 9/10 dou in the tavern.
After passing 4 taverns, the bottle is empty.
How much wine was there at the beginning? Show your solutions.
:
let x = the amt of wine originally
Assuming he drinks 1.9 d at each tavern
2x - 1.9 - 4(1.9) = 0
2x - 1.9 - 7.6 = 0
2x - 9.5 = 0
2x = 9.5
x = 9.5/2
x = 4.75 d originally
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
I take along a bottle with some wine in it for an excursion in the spring.
On reaching a tavern, I double the bottle’s content and drink 1 9/10 dou in the tavern.
After passing 4 taverns, the bottle is empty. How much wine was there at the beginning?
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My interpretation of this problem is different from that of @ankor@dixie-net.com.
THEREFORE, my solution and my answer are different.
We have a sequence of numbers  ,  ,  ,  ,  .
is the amount of wine in the bottle originally.
 is the amount of wine in the bottle at the Exit from the k-th tavern
(same as at the entering to the (k+1)-th tavern), k = 1,2,3,4.
From the problem's condition, we have this recursive formula
 =  , (1)
which is the formal mathematical translation of the problem's description.
We are given that = 0, and they want we find .
Surely, the problem should be solved in a backward manner.
For it, I express from (1) in this way
=  (2)
Now I will go step by step BACKWARD from k= 3 with = 0.
In order for to facilitate my calculations, I use MS Excel in my computer with implemented formula (2) in it.
It provides all calculations in seconds, giving me THIS Table below
k
--------------------------
4 0
3 0.95
2 1.425
1 1.6625
0 1.78125
So, the ANSWER is: the initial amount of wine in the bottle was 1.78125 units of the volume.
SOLVED.
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