GMAT
题目
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.
解析
1. 先看两家店的优惠活动：
- 斯克鲁奇商店：花\( c \)美元可以买\( n \)块巧克力（定额）。
- 市场综合商店：花\( c+3 \)美元可以买\( n \)块巧克力（定额）。
2. 巧克力的数量\( n \)必须是整数。因为丽塔用1美元纸币付款，所以\( c \)和\( c+3 \)也必须是整数。
3. 市场综合商店的优惠，每美元能买到的巧克力比斯克鲁奇商店少2块。因此：
\[ \frac{n}{c} - \frac{n}{c+3} = 2 \]
4. 对等式变形：
\[ \frac{n}{c} - \frac{n}{c+3} = 2 = n\left( \frac{3}{c(c+3)} \right) \implies n = \frac{2}{3} \times c(c + 3) \]
因为\( n \)是整数，所以等式右边也必须是整数，这意味着\( c(c + 3) \)能被3整除。而只有当\( c \)能被3整除时，这个条件才成立。
5. 现在列举可能的情况：
- 若\( c = 3 \)，则\( n = \frac{2}{3} \times 3 \times (3 + 3) = 12 \)，该情况成立。
- 若\( c = 6 \)，则\( n = \frac{2}{3} \times 6 \times (6 + 3) = 36 \)，但\( n \)不能超过20，因此不成立。
- \( c \)更大的话，\( n \)会超过20，同样不成立。
6. 此时两家店的优惠活动为：
- 斯克鲁奇商店：花3美元可以买12块巧克力。
- 市场综合商店：花6美元可以买12块巧克力。
7. 这说明丽塔买了12块巧克力，若在市场综合商店购买需要支付6美元。
8. 最终答案为：1-12和2-6。
### 公式标准排版
步骤3的等式：
\[ \frac{n}{c} - \frac{n}{c+3} = 2 \]
步骤4的变形：
\[ \frac{n}{c} - \frac{n}{c+3} = 2 = n\left( \frac{3}{c(c+3)} \right) \implies n = \frac{2}{3} \times c(c + 3) \]
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