Below is a text of a problem:

A new bakery shop sells $3$ sizes of cakes: big, medium, and small. Normally each cake costs a positive integer number of dollars. The ratio of the prices of two sizes of cake, in lowest terms, is the ratio of two consecutive integers, and only four consecutive positive integers are needed to list the numerators and denominators of the three (lowest terms) ratios of different pairs of cakes. On Sundays, however, the prices are discounted by $1$ dollar each.

A customer came into the bakery one Sunday and asked the total cost of one cake of each size. He placed that amount, plus $\$1$, on the counter, and asked whether he could buy a collection of cakes for that sum. This was not possible, but if the customer had put any larger sum of money on the counter, it would have been possible to buy a collection of cakes at that price.

What are the costs of the three sizes of cakes on a weekday?

I cannot understand the question (2nd paragraph). Customer asked for a total amount, added $1 extra - how can it be so, that is not enough to buy? He's giving more money then he was told, by definition it is enough to buy. Even if he was told full amount (without discount), it's Sunday, so full set should be 3 dollars cheaper, than on weekdays. Maybe I have a problem with my English? P.S. I don't ask for a solution, I'm asking for clarification of the problem's text.

  • 4
    $\begingroup$ I think the implicit assumption here is that the bakery cannot give change. $\endgroup$
    – Klaus
    Feb 7, 2020 at 14:20
  • 1
    $\begingroup$ I understand that Amount+1 is not a linear combination of the three prices, whereas Amount+2 and above is. That means, among others, that the smallest price must be 2. $\endgroup$
    – user65203
    Feb 7, 2020 at 14:20
  • $\begingroup$ When the problem says "for that sum", read it as "costing exactly as much as the money on the counter." That is, he wants to spend all that money on cakes and not have any money left over. $\endgroup$
    – David K
    Feb 7, 2020 at 15:29
  • $\begingroup$ Big spender!${}$ $\endgroup$
    – mjw
    Feb 7, 2020 at 15:30


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