AMC8 2009

AMC8 2009 · Q11

AMC8 2009 · Q11. It mainly tests Money / coins, GCD & LCM.

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil?

Amaco 中学的书店出售铅笔，每支铅笔的价格是一个整数美分。有些七年级学生每人买了一支铅笔，总共支付了 1.43 美元。30 名六年级学生中的一些人也每人买了一支铅笔，他们总共支付了 1.95 美元。有多少名六年级学生比七年级学生多买了铅笔？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The number of sixth graders who bought a pencil is 195 divided by the cost of a pencil. Similarly the number of seventh graders who bought a pencil is 143 divided by the cost of a pencil. That means both 195 and 143 are multiples of the price of the pencil. Factor $195 = 1\cdot3\cdot5\cdot13$ and $143 = 1\cdot11\cdot13$. The only common divisors are 1 and 13. If a pencil cost 1 cent, then 195 sixth graders bought a pencil. However, there are only 30 sixth graders, so a pencil must cost 13 cents. Using that fact, $\frac{195}{13}-\frac{143}{13}=15-11=4$ more sixth graders than seventh graders bought pencils.

答案（D）：购买铅笔的六年级学生人数等于$195$除以一支铅笔的价格。同样，购买铅笔的七年级学生人数等于$143$除以一支铅笔的价格。这意味着$195$和$143$都是铅笔价格的倍数。分解因数：$195 = 1\cdot3\cdot5\cdot13$，$143 = 1\cdot11\cdot13$。它们唯一的公因数是$1$和$13$。如果一支铅笔价格是$1$美分，那么会有$195$名六年级学生买了铅笔。然而六年级只有$30$名学生，所以铅笔价格必须是$13$美分。利用这一点，$\frac{195}{13}-\frac{143}{13}=15-11=4$，即买铅笔的六年级学生比七年级学生多$4$人。

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