2014 AMC amc10a 真题 答案 详解

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中英双语真题

2014 AMC 10A
Problem 1
What is $10 \cdot (\frac{1}{2} + \frac{1}{5} + \frac{1}{10})^{-1}$?
$10 \cdot (\frac{1}{2} + \frac{1}{5} + \frac{1}{10})^{-1}$ 的值是多少？
(A) 3 (B) 8 (C) $\frac{25}{2}$ (D) $\frac{170}{3}$ (E) 170

Problem 2
$\frac{1}{3}$
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
$\frac{1}{4}$
Roy 的猫每天早上吃一罐猫粮的 $\frac{1}{3}$，每天晚上吃一罐猫粮的 $\frac{1}{4}$。某个周一的早上，在 Roy 开始喂猫之前，他打开了一个盒子，里面有 6 罐猫粮。猫会在周几吃完盒子里所有的猫粮？
(A) Tuesday (B) Wednesday (C) Thursday (D) Friday (E) Saturday

Problem 3
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $2.50 each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $0.75 for her to make. In dollars, what is her profit for the day?
Bridget 为她的面包店烤了 48 片面包，她早上以每片 2.5 美元的价格卖出去了一半，下午的时候，她卖出了剩下的三分之二，而因为面包不新鲜了，所以她以半价出售。傍晚时，她以每片 1 美元的价格把剩下的都卖了出去，已知每片面包的成本是 0.75 美元，问她这天的利润是多少美元？
(A) 24 (B) 36 (C) 44 (D) 48 (E) 52

英文真题

2014 AMC 10A Problems

Problem 1
What is $10 \cdot (\frac{1}{2} + \frac{1}{5} + \frac{1}{10})^{-1}$?
(A) 3 (B) 8 (C) $\frac{25}{2}$ (D) $\frac{170}{3}$ (E) 170

Problem 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
(A) Tuesday (B) Wednesday (C) Thursday (D) Friday (E) Saturday

Problem 3
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $2.50 each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $0.75 for her to make. In dollars, what is her profit for the day?
(A) 24 (B) 36 (C) 44 (D) 48 (E) 52

Problem 4
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 5
On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest scored 100 points. What is the difference between the mean and median score of the students' scores on this quiz?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 6
Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?
(A) $\frac{bde}{ac}$ (B) $\frac{ac}{bde}$ (C) $\frac{abde}{c}$ (D) $\frac{bcde}{a}$ (E) $\frac{abc}{de}$

Problem 7
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
(I) $x+y < a+b$
(II) $x-y < a-b$
(III) $xy < ab$
(IV) $\frac{x}{y} < \frac{a}{b}$
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Problem 8
Which of the following numbers is a perfect square?

真题文字详解(英文)

Problem 1 What is $10 \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}$?
(A) 3 (B) 8 (C) $\frac{25}{2}$ (D) $\frac{170}{3}$ (E) 170

Solution We have
$10 \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}$
Making the denominators equal gives
$\Rightarrow 10 \cdot \left(\frac{5}{10} + \frac{2}{10} + \frac{1}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{5+2+1}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{8}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{4}{5}\right)^{-1}$
$\Rightarrow 10 \cdot \frac{5}{4}$
$\Rightarrow \frac{50}{4}$
Finally, simplifying gives
$\boxed{\text{(C)}\ \frac{25}{2}}$

Solution 2 We have
$\left(\frac{1}{10}\right)^{-1} \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}$
By Distributive Property,
$\left(\frac{1}{20} + \frac{1}{50} + \frac{1}{100}\right)^{-1}$
Now, we want to find the least common multiple of 20, 50, and 100, so
$lcm(20, 50, 100) = lcm(2^2 \cdot 5, 2 \cdot 5^2, 2^2 \cdot 5^2) = 2^2 \cdot 5^2 = 100$
Converting everything to a denominator of 100,
$\left(\frac{5}{100} + \frac{2}{100} + \frac{1}{100}\right)^{-1} = \left(\frac{8}{100}\right)^{-1} = \frac{100}{8}$
Now, we use Euclidean Algorithm, to find if this fraction is reducible, so
$gcd(100, 8) = gcd(12, 8) = gcd(4, 8) = gcd(4, 4)$
Thus, both the numerator and denominator are divisible by 4, so
$\frac{100}{8} \cdot \frac{4}{4} = \frac{100}{4} \cdot \frac{4}{8} = 25 \cdot \frac{1}{2} = \boxed{\frac{25}{2}}$

• kante314

Problem 2 Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?

真题文字详解(中英双语)

Problem 1 What is $10 \cdot \left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}$?
(A) 3 (B) 8 (C) $\frac{25}{2}$ (D) $\frac{170}{3}$ (E) 170

Solution We have
$10 \cdot \left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}$
Making the denominators equal gives
$\Rightarrow 10 \cdot \left(\frac{5}{10}+\frac{2}{10}+\frac{1}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{5+2+1}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{8}{10}\right)^{-1}$
$\Rightarrow 10 \cdot \left(\frac{4}{5}\right)^{-1}$
$\Rightarrow 10 \cdot \frac{5}{4}$
$\Rightarrow \frac{50}{4}$
Finally, simplifying gives
(C) $\frac{25}{2}$

Solution 2 We have
$\left(\frac{1}{10}\right)^{-1} \cdot \left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}$
By Distributive Property,
$\left(\frac{1}{20}+\frac{1}{50}+\frac{1}{100}\right)^{-1}$
Now, we want to find the least common multiple of 20, 50, and 100, so
lcm(20, 50, 100) = lcm($2^2 \cdot 5$, $2 \cdot 5^2$, $2^2 \cdot 5^2$) = $2^2 \cdot 5^2 = 100$
Converting everything to a denominator of 100,
$\left(\frac{5}{100}+\frac{2}{100}+\frac{1}{100}\right)^{-1}=\left(\frac{8}{100}\right)^{-1}=\frac{100}{8}$
Now, we use Euclidean Algorithm, to find if this fraction is reducible, so
gcd(100, 8) = gcd(12, 8) = gcd(4, 8) = gcd(4, 4)
Thus, both the numerator and denominator are divisible by 4, so
we have the distributive property, now, we want to find the least common multiple of 20, 50, and 100, so that all fractions can be converted to a denominator of 100, now, we use the Euclidean algorithm to find whether this fraction is reducible, so therefore, the numerator and denominator can both be divided by 4,
$\frac{100}{8} \cdot \frac{4}{4} = \frac{100}{8} \cdot \frac{4}{4} = 25 \cdot \frac{1}{2} = \boxed{\frac{25}{2}}$