Doctors have advised Rita, a chocolate freak, not to consume more than 20 chocolates in one day. When she went to the market to buy her daily quota, she found that if she were to buy chocolates from the market-complex, she would have to pay $3 more for the same number of chocolates than she would have spent had she bought them from her uncle Scrooge’s shop, getting two chocolates fewer per dollar at the market complex than at uncle Scrooge’s shop. She finally decided to get them from uncle Scrooge’s shop paying only in one-dollar bills. Choose for 1 the number of chocolates Rita bought and for 2 the amount she would have paid if she had purchased the chocolates from the market complex. Make only two selections, one in each column.

Let n be the number of chocolates bought. Let c be number of chocolates per dollar at uncle's. Let u be number of dollars paid at uncle's. number of chocolates per dollar * number of dollars paid = number of chocolates bought for uncle: c*u = n for market, c-2 * u+3 = n Try values for n, maybe start with larger values like 18,15,12 etc 12 seems most diverse in the sense that it may contain divisors within a difference of +-3 or +-2 12 = 4*3 = 2*6 is consistent! hence, n = 12 and u+3 = 6 works!

1. Let's look at the deals at both shops: - Scrooge's shop: for c dollars you can buy n chocolates (quota). - Market-complex: for c+3 dollars you can buy n chocolates (quota). 2. The number of chocolates, n, has to be a whole number. Since Rita bought the chocolates using one-dollar bills, then c and c+3 must be whole numbers too. 3. The market-complex has a deal that gives 2 chocolates fewer per dollar compared to Scrooge's shop. So, \frac{n}{c} - \frac{n}{c + 3} = 2 . 4. \frac{n}{c} - \frac{n}{c + 3} = 2 = n(\frac{3}{c(c + 3)}) \rightarrow n = \frac{2}{3} * c(c + 3) . Since n is whole, the right hand side must be whole too and that means c(c + 3) is divisible by 3. This is only possible if c is divisible by 3. 5. Now let's look at some possible cases: - c = 3. n = \frac{2}{3} * 3 * (3 + 3) = 12 . This works. - c = 6. n = \frac{2}{3} * 6 * (6 + 3) = 36 . n must be no more than 20. - Larger c will also not work since n will then be larger than 20. 6. The deals will now look like: - Scrooge's shop: for 3 dollars you can buy 12 chocolates. - Market-complex: for 6 dollars you can buy 12 chocolates. 7. That means Rita bought 12 chocolates and would've payed 6 dollars at the market-complex. 8. Our answer will be: 1 - 12 and 2 - 6 .