登录
3 某商场销售甲、乙两种衬衫.若售出3件甲衬衫和2件乙衬衫,总销售额为380元;若售出2件甲衬衫和4件乙衬衫,总销售额为440元.
（1）甲、乙两种衬衫的单价分别是多少元？
（2）商场搞促销,顾客A购买2件甲衬衫和3件乙衬衫,共花费312元;顾客B购买3件甲衬衫和2件乙衬衫,共花费328元.甲、乙两种衬衫分别打几折？
（1）甲、乙两种衬衫的单价分别是多少元？
（2）商场搞促销,顾客A购买2件甲衬衫和3件乙衬衫,共花费312元;顾客B购买3件甲衬衫和2件乙衬衫,共花费328元.甲、乙两种衬衫分别打几折？
答案
(1) 设甲衬衫的单价为$x$元,乙衬衫的单价为$y$元.
列方程组
$\begin{cases} 3x + 2y = 380, \\ 2x + 4y = 440, \end{cases}$
解得$x = 80$,$y = 70$.
(2) 设甲衬衫折扣为$x$,乙衬衫折扣为$y$.
列方程组
$\begin{cases} 2 × 80x + 3 × 70y = 312, \\ 3 × 80x + 2 × 70y = 328, \end{cases}$
解得$x = 0.9$,$y = 0.8$.
甲衬衫打九折,乙衬衫打八折.
列方程组
$\begin{cases} 3x + 2y = 380, \\ 2x + 4y = 440, \end{cases}$
解得$x = 80$,$y = 70$.
(2) 设甲衬衫折扣为$x$,乙衬衫折扣为$y$.
列方程组
$\begin{cases} 2 × 80x + 3 × 70y = 312, \\ 3 × 80x + 2 × 70y = 328, \end{cases}$
解得$x = 0.9$,$y = 0.8$.
甲衬衫打九折,乙衬衫打八折.
解析
【分析】
（1）本题有甲、乙衬衫单价两个未知量，题干给出了两种销售组合的总销售额，刚好对应两个等量关系，因此可以设两个未知数列二元一次方程组求解。两个等量关系分别为：3件甲衬衫的销售额+2件乙衬衫的销售额=380元；2件甲衬衫的销售额+4件乙衬衫的销售额=440元。
（2）第一问已求出两种衬衫的原价，折扣后商品的售价=原价×折扣系数，结合两位顾客的实际消费总额可得到两个新的等量关系，再次列二元一次方程组求解出折扣系数，再转换为对应折扣即可。
【解析】
（1）设甲衬衫的单价为$x$元，乙衬衫的单价为$y$元，根据题意列方程组：
$\begin{cases} 3x + 2y = 380 \quad \mathrm{①} \\ 2x + 4y = 440 \quad \mathrm{②} \end{cases}$
将①式两边同时乘2得：$6x + 4y = 760 \quad \mathrm{③}$
用③式减去②式消去$y$，可得：$4x=320$，解得$x=80$。
将$x=80$代入①式，得$3×80 + 2y = 380$，解得$y=70$。
（2）设甲衬衫的折扣系数为$x$，乙衬衫的折扣系数为$y$，根据题意列方程组：
$\begin{cases} 2 × 80x + 3 × 70y = 312 \quad \mathrm{④} \\ 3 × 80x + 2 × 70y = 328 \quad \mathrm{⑤} \end{cases}$
化简得$\begin{cases} 160x + 210y = 312 \\ 240x + 140y = 328 \end{cases}$
用消元法求解可得：$x=0.9$，$y=0.8$，即甲衬衫打九折，乙衬衫打八折。
【答案】
(1) 甲衬衫单价为80元，乙衬衫单价为70元；
(2) 甲衬衫打九折，乙衬衫打八折。
【知识点】
二元一次方程组的应用；二元一次方程组的解法；打折销售问题
【点评】
本题结合日常购物场景考查方程知识的实际应用，解题核心是准确提取题干中的等量关系建立方程组，整体计算难度不高，能有效锻炼用数学知识解决实际问题的能力。
【难度系数】
0.8
（1）本题有甲、乙衬衫单价两个未知量，题干给出了两种销售组合的总销售额，刚好对应两个等量关系，因此可以设两个未知数列二元一次方程组求解。两个等量关系分别为：3件甲衬衫的销售额+2件乙衬衫的销售额=380元；2件甲衬衫的销售额+4件乙衬衫的销售额=440元。
（2）第一问已求出两种衬衫的原价，折扣后商品的售价=原价×折扣系数，结合两位顾客的实际消费总额可得到两个新的等量关系，再次列二元一次方程组求解出折扣系数，再转换为对应折扣即可。
【解析】
（1）设甲衬衫的单价为$x$元，乙衬衫的单价为$y$元，根据题意列方程组：
$\begin{cases} 3x + 2y = 380 \quad \mathrm{①} \\ 2x + 4y = 440 \quad \mathrm{②} \end{cases}$
将①式两边同时乘2得：$6x + 4y = 760 \quad \mathrm{③}$
用③式减去②式消去$y$，可得：$4x=320$，解得$x=80$。
将$x=80$代入①式，得$3×80 + 2y = 380$，解得$y=70$。
（2）设甲衬衫的折扣系数为$x$，乙衬衫的折扣系数为$y$，根据题意列方程组：
$\begin{cases} 2 × 80x + 3 × 70y = 312 \quad \mathrm{④} \\ 3 × 80x + 2 × 70y = 328 \quad \mathrm{⑤} \end{cases}$
化简得$\begin{cases} 160x + 210y = 312 \\ 240x + 140y = 328 \end{cases}$
用消元法求解可得：$x=0.9$，$y=0.8$，即甲衬衫打九折，乙衬衫打八折。
【答案】
(1) 甲衬衫单价为80元，乙衬衫单价为70元；
(2) 甲衬衫打九折，乙衬衫打八折。
【知识点】
二元一次方程组的应用；二元一次方程组的解法；打折销售问题
【点评】
本题结合日常购物场景考查方程知识的实际应用，解题核心是准确提取题干中的等量关系建立方程组，整体计算难度不高，能有效锻炼用数学知识解决实际问题的能力。
【难度系数】
0.8
Choose the best answer.（根据以下内容，选择最恰当的答案）
How We Deal with Problems
In our daily life, we often face various problems. However, the key to a happy and successful life is how we deal with these problems. Let's take a look at a story about Tom.
Tom is a middle school student in Grade 8. He lives in a small town. Every day, he rides his bike to school, which takes him about 20 minutes. But recently, he has been facing a problem. The road to his school has been under construction for a few weeks. The workers are repairing the road because it was damaged by the heavy trucks last month. As a result, the road has become very rough and full of holes. Every time Tom rides on it, he has to be very careful. Sometimes, he even gets his bike tires punctured（刺破）.
Tom thought about this problem. He knew he couldn't just give up and stop going to school. So, he started to look for solutions. First, he considered taking a different route. He asked his parents and some of his neighbors if there was another way to get to school. They told him there was a longer but smoother road. However, this new route would take him about 40 minutes, which was much longer than his usual time.
Then, Tom had another idea. He decided to ask his father to drive him to school. But his father was very busy with his work and couldn't drive him every day. Also, Tom didn't want to bother his father too much.
Finally, Tom came up with a great solution. He asked his parents to buy him a new bike with thicker tires. Thicker tires would be less likely to get punctured on the rough road. His parents agreed, and they bought him a new bike. Now, Tom can ride to school safely and quickly again.
Tom's story shows us that when we face problems, we should think actively and look for different solutions. We should never give up easily. There is always a way to solve our problems if we try hard enough.
（
A. Because it is too long.
B. Because it is under construction and very rough.
C. Because there are too many cars on the road.
D. Because he lost his way on the road.
（
A. About 10 minutes.
B. About 20 minutes.
C. About 30 minutes.
D. About 40 minutes.
（
A. Asking his father to drive him to school.
B. Buying a new bike.
C. Taking a different route.
D. Stopping going to school.
（
A. Because his father didn't want to.
B. Because his father had no car.
C. Because his father was too busy with work.
D. Because Tom didn't like to go by car.
（
A. Taking the longer route.
B. Asking his father to drive him every day.
C. Buying a new bike with thicker tires.
D. Waiting for the road construction to finish.
How We Deal with Problems
In our daily life, we often face various problems. However, the key to a happy and successful life is how we deal with these problems. Let's take a look at a story about Tom.
Tom is a middle school student in Grade 8. He lives in a small town. Every day, he rides his bike to school, which takes him about 20 minutes. But recently, he has been facing a problem. The road to his school has been under construction for a few weeks. The workers are repairing the road because it was damaged by the heavy trucks last month. As a result, the road has become very rough and full of holes. Every time Tom rides on it, he has to be very careful. Sometimes, he even gets his bike tires punctured（刺破）.
Tom thought about this problem. He knew he couldn't just give up and stop going to school. So, he started to look for solutions. First, he considered taking a different route. He asked his parents and some of his neighbors if there was another way to get to school. They told him there was a longer but smoother road. However, this new route would take him about 40 minutes, which was much longer than his usual time.
Then, Tom had another idea. He decided to ask his father to drive him to school. But his father was very busy with his work and couldn't drive him every day. Also, Tom didn't want to bother his father too much.
Finally, Tom came up with a great solution. He asked his parents to buy him a new bike with thicker tires. Thicker tires would be less likely to get punctured on the rough road. His parents agreed, and they bought him a new bike. Now, Tom can ride to school safely and quickly again.
Tom's story shows us that when we face problems, we should think actively and look for different solutions. We should never give up easily. There is always a way to solve our problems if we try hard enough.
（
B
）(1) Why has the road to Tom's school become a problem for him?A. Because it is too long.
B. Because it is under construction and very rough.
C. Because there are too many cars on the road.
D. Because he lost his way on the road.
（
B
）(2) How long did it usually take Tom to ride to school?A. About 10 minutes.
B. About 20 minutes.
C. About 30 minutes.
D. About 40 minutes.
（
C
）(3) What was Tom's first solution to the problem?A. Asking his father to drive him to school.
B. Buying a new bike.
C. Taking a different route.
D. Stopping going to school.
（
C
）(4) Why couldn't Tom's father drive him to school every day?A. Because his father didn't want to.
B. Because his father had no car.
C. Because his father was too busy with work.
D. Because Tom didn't like to go by car.
（
C
）(5) What was Tom's final solution to the problem?A. Taking the longer route.
B. Asking his father to drive him every day.
C. Buying a new bike with thicker tires.
D. Waiting for the road construction to finish.
答案
(1) B (2) B (3) C (4) C (5) C
解析
【分析】
这是一篇记叙文类阅读理解，解题时可遵循以下思路：首先通读全文，了解文章讲述的Tom解决上学路问题的核心内容；再逐题抓取题干关键词，回到原文定位对应信息句，将原文内容和选项逐一比对，排除错误选项后锁定正确答案，所有选项判断都要找到原文依据，避免主观臆断。
【解析】
(1) 定位原文第二段内容：“The road to his school has been under construction for a few weeks... As a result, the road has become very rough and full of holes. Every time Tom rides on it, he has to be very careful. Sometimes, he even gets his bike tires punctured”，可知道路正在施工、路面坑洼不平是困扰Tom的原因，对应选项B。
(2) 定位原文第二段“Every day, he rides his bike to school, which takes him about 20 minutes”，可知Tom平时骑车上学通常耗时约20分钟，对应选项B。
(3) 定位原文第三段“First, he considered taking a different route”，可知Tom想到的第一个解决方案是换一条路线上学，对应选项C。
(4) 定位原文第四段“But his father was very busy with his work and couldn't drive him every day”，可知父亲没法每天送他上学的原因是工作太忙，对应选项C。
(5) 定位原文第五段“Finally, Tom came up with a great solution. He asked his parents to buy him a new bike with thicker tires”，可知Tom最终的解决方案是让父母给他买了一辆轮胎更厚的新自行车，对应选项C。
【答案】
(1) B (2) B (3) C (4) C (5) C
【知识点】
细节信息定位，文意理解
【点评】
本文通过Tom解决上学出行难题的小故事，引导学生明白遇到问题要积极思考、不轻言放弃的道理。题目均为细节考查题，难度较低，只要认真回读原文查找对应信息，就能快速选出正确答案，主要考查学生的信息检索和基础文本理解能力。
【难度系数】
0.8
这是一篇记叙文类阅读理解，解题时可遵循以下思路：首先通读全文，了解文章讲述的Tom解决上学路问题的核心内容；再逐题抓取题干关键词，回到原文定位对应信息句，将原文内容和选项逐一比对，排除错误选项后锁定正确答案，所有选项判断都要找到原文依据，避免主观臆断。
【解析】
(1) 定位原文第二段内容：“The road to his school has been under construction for a few weeks... As a result, the road has become very rough and full of holes. Every time Tom rides on it, he has to be very careful. Sometimes, he even gets his bike tires punctured”，可知道路正在施工、路面坑洼不平是困扰Tom的原因，对应选项B。
(2) 定位原文第二段“Every day, he rides his bike to school, which takes him about 20 minutes”，可知Tom平时骑车上学通常耗时约20分钟，对应选项B。
(3) 定位原文第三段“First, he considered taking a different route”，可知Tom想到的第一个解决方案是换一条路线上学，对应选项C。
(4) 定位原文第四段“But his father was very busy with his work and couldn't drive him every day”，可知父亲没法每天送他上学的原因是工作太忙，对应选项C。
(5) 定位原文第五段“Finally, Tom came up with a great solution. He asked his parents to buy him a new bike with thicker tires”，可知Tom最终的解决方案是让父母给他买了一辆轮胎更厚的新自行车，对应选项C。
【答案】
(1) B (2) B (3) C (4) C (5) C
【知识点】
细节信息定位，文意理解
【点评】
本文通过Tom解决上学出行难题的小故事，引导学生明白遇到问题要积极思考、不轻言放弃的道理。题目均为细节考查题，难度较低，只要认真回读原文查找对应信息，就能快速选出正确答案，主要考查学生的信息检索和基础文本理解能力。
【难度系数】
0.8
