Friday Puzzler -- Wine business

Hints

Alex Y

Well, not really hints -- more keeping you from going off in the wrong direction, and maybe encouraging the tack you are taking?

1) No trickery involved. If you think anything I wrote could have a double meaning, that is unintentional--please ask and I will clarify.

2) The mathematics involved is nothing beyond plain middle school or high school algebra.

3) While the mathematics is basic, the problem is quite difficult IMO.

Lewis Carroll claims to have thought up these problems and solutions on evening walks. While I accept that he was probably several orders of magnitude smarter than I am, I still have to take that claim with a grain of salt!

Re: Friday Puzzler -- Wine business

Larry Barrett

Too hot for some of the week and too many chores for the rest of the week to think about this.

So this morning I came up with the following:

Each of the 5 contributed 24 bottles of wine for a total of 120.

Cost per bottle is $7.50, selling price is cost + 10% = $8.25

Number disposed of on day 1 = 43; salesman drank 1 and sold 42, kept profit on 1 bottle. Profit turned in to treasurer = $24.

Number sold of on day 2 = 33. Profit turned in to treasurer = $24.

Number disposed of on day 3 = 44; salesman drank 1, sold 43, kept profit on 1 bottle. Profit turned in to treasurer = $24.

Approach

Alex Y

This problem starts out with several variables. However, relationships are given that allow the problem to be reduced to a single variable in a quadratic equation.

Close, but no

Alex Y

Sounds like you are on the right track, but may have missed or misinterpreted some of the puzzle.

1) The treasurer found after day 2 that the inventory was worth 330 retail value. I think your solution has it worth 330 at cost.

2) In your solution, the treasurer has received $72 of profit. That is only $0.60 per bottle that they started out with. Should be $0.75

I haven't verified it, but would bet that fixing those two things will lead to the right answer.

Road map

Alex Y

This one seems not to be getting much traction. Without solving it, here are the steps I took to get to the solution (only after seeing the general direction that Lewis Carroll took in his solution). Quit reading when you think you have enough of a hint, and find your own solution.

Note: Almost every step I took had alternatives that will get to the same answer, so if my step doesn't seem logical to you, go a different route.

1) Set variables x, y, and z for the number of bottles disposed of (sold or drunk by the salesman) on days 1, 2, and 3. Set v as the cost of each bottle, so selling price is 1.1v and margin is .1v

2) Determine the dollar amount of profit each day in terms of x and v, y and v, and z and v.

3) Since the profit is the same each day, set these equal to each other and eliminate extraneous variables to allow x, y, and z to be expressed in terms of only one variable. (Lewis Carroll chose to express y and z in terms of x, but I chose to express in terms of z because of the next step.)

4) The treasurer tells us the value (at retail) of the z bottles remaining after two days, and that lets us express v in terms of z. So now each fact we know about the wine can be expressed in terms of a single variable.

5) The treasurer says that the profit on all the bottles invested in the business was $0.75 per bottle. We also know the profit each day, and can set the dollar profit in total equal to the sum of the daily profits (or three times any day's profits since we know that they are the same).

6) Solve the resulting quadratic equation, and see that only one of the two solutions is possible with five investors putting in the same number of bottles each.

Like I said, middle school or high school algebra, but MUCH harder than any problem I saw back in the day.

Re: Road map helps, but error(s) somewhere

Larry Barrett

My first answer (in error) started out the same way but wandered off the path somewhere. Trying again, but I must be making a basic error somewhere.

These are the equations I come up with, using your notation:

Retail value at the end of day 2: z(1.1v) = $330, so zv=$300

Same profit each day: v[(x-1)1.1 -x] = v[(y-1)1.1 - (y-1)] =v[(z-2)1.1 - (z-1)],

and from this I think x=z-1. y=z-2

Total Profit (TP) = 3(.1z - 1.2)(300/z)

Profit per bottle: TP/(x+y+z) = $.75,

which becomes (z-12)40 = z^2 - z or z^2 - 41z + 480 = 0.

I don't see a solution to this, so must be an error somewhere.

Re: Road map helps, but error(s) somewhere

Alex Y

Same profit each day: v[(x-1)1.1 -x] = v[(y-1)1.1 - (y-1)] =v[(z-2)1.1 - (z-1)],

and from this I think x=z-1. y=z-2

Your formulas for the three days profits are correct. Maybe tried to skip too many steps in getting to relationship between x, y, and z?

v[(x-1)1.1 -x]

=v(1.1x-1.1-x)

=.1v(x-11)

v[(y-1)1.1 - (y-1)]

=v(1.1y-1.1-y+1)

=.1v(y-1)

v[(z-2)1.1 - (z-1)]

=v(1.1z-2.2-z+1)

=.1v(z-12)

x=z-1

y=z-11

I think you are golden otherwise.

Re: Road map helps, but error(s) somewhere

Larry Barrett

I think that fixed it:

x=23, y=13, z=24 x+y+z=60

v=$12.50, 1.1v=$13.75

Correct!