Question

Amaco Middle School 的书店出售铅笔，每支铅笔价格为整数美分。一些七年级学生每人买了一支铅笔，总共支付 $1.43$ 美元。 $30$ 名六年级学生中的一些人每人买了一支铅笔，总共支付 $1.95$ 美元。买铅笔的六年级学生比七年级学生多多少人？

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

$1$

$2$

$3$

$4$

$5$

答案：D

知识点：最大公约数质因数分解

难度评级：1290

解答：

买铅笔的七年级学生人数等于 $143$ 除以一支铅笔的价格。同样，买铅笔的六年级学生人数等于 $195$ 除以一支铅笔的价格。

因此铅笔价格同时整除 $143$ 和 $195$ 。质因数分解得和同时整除 $143$ 和 $195$ 的数只有 $1$ 和 $13$ 。 $143 = 11 \cdot 13$ $195 = 3 \cdot 5 \cdot 13.$

如果一支铅笔价格是 $1$ 美分，那么会有 $195$ 名六年级学生买铅笔，这是不可能的。因此一支铅笔价格是 $13$ 美分。

于是 $143 \div 13 = 11$ 名七年级学生买铅笔， $195 \div 13 = 15$ 名六年级学生买铅笔。所以六年级学生比七年级学生多 $4$ 人。

所以正确答案是 D。

The number of seventh graders that bought a pencil is $143$ divided by the price of a pencil. Similarly, the number of sixth graders that bought a pencil is $195$ divided by the price of a pencil.

This means that the price of a pencil divides both $143$ and $195.$ Prime factorizing, we get $143 = 11 \cdot 13$ and $195 = 3 \cdot 5 \cdot 13.$ The only numbers that divide both $143$ and $195$ are $1$ and $13.$

If $1$ cent was the price of the pencil, that means $195$ sixth graders bought pencils, which is not possible. Therefore, the price of a pencil is $13$ cents.

This means that $143 \div 13 = 11$ seventh graders bought a pencil, and $195 \div 13 = 15$ sixth graders bought a pencil. Therefore, $4$ more sixth graders than seventh graders bought pencils.

Thus, D is the correct answer.

2009 AMC 8 第 11 题

答案：D

难度评级：1290

解答：

其他年份的第 11 题