Dan’s DVD Discount Dungeon

Question

Dan’s DVD Discount Dungeon is going out of business and selling all of its DVDs. The first customer
bought 1/6
of all the DVDs plus 1; the second buyer again took 1/6
of the remaining DVDs plus 2; the third
customer bought 1/6
of the remaining DVDs plus 3, and so on. When Dan had sold all of the DVDs, he figured out
that surprisingly each customer had purchased the same amount of DVDs. How many DVDs did each customer
buy and what was Dan’s takings if he sold each DVD for $5 each?

Answer 1

I love this question! The pattern is very entertaining, yet not that tough once you catch on to the method.

Instead of numbering customers, I will avoid confusion by calling them A, B, C and so on. We will begin with customer A (shocking, but true!), and use "X" for the total of all DVDs.

A: bought X/6 + 1  DVDs, which means the remainder is X - (x/6 + 1) = 5X/6 - 1

B: bought 1/6 of the previous remainder, plus 2, or:

(1/6) * (5X/6 - 1) +2

= 5X/36 - 1/6 + 2

= 5X/36 + 11/6

We now know enough to solve the question, or at least the first stage:

A = B    in terms of purchases, not remainders, so we get

X/6 + 1 = 5X/36 + 11/6            ...and if we multiply the entire equation by the LCD of 36 to clear all fractions...

6X + 36 = 5X + 66

-5X          -5X

X    + 36 =        66

       - 36          -36

      X      =        30    DVDs total. You can go back and check manually, as I did, to be certain that the pattern always shows a purchase of 6 DVDs each:

A: (1/6) *30 + 1 = 5 + 1 = 6   , with a remainder of 24. Plug that into B and get 6 again, then C, D, and finally E.

So: we know that

1) There were 30 DVDs total

2) Each customer bought 6

3) 30 * $5 = $150 "takings".