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Difference between revisions of "2014 AMC 12A Problems"

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==Problem 10==
 
==Problem 10==
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Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length <math>1</math>.  The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle.  What is the length of one of the two congruent sides of one of the isosceles triangles?
 +
 +
<math>\textbf{(A) }\dfrac{\sqrt3}4\qquad
 +
\textbf{(B) }\dfrac{\sqrt3}3\qquad
 +
\textbf{(C) }\dfrac23\qquad
 +
\textbf{(D) }\dfrac{\sqrt2}2\qquad
 +
\textbf{(E) }\dfrac{\sqrt3}2</math>
  
 
==Problem 11==
 
==Problem 11==

Revision as of 11:43, 7 February 2014

Problem 1

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$ (Error compiling LaTeX. Unknown error_msg)

Problem 2

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $24.50$. How much would $8$ adult tickets and $6$ child tickets cost?

$\textbf{(A) }35\qquad \textbf{(B) }38.50\qquad \textbf{(C) }40\qquad \textbf{(D) }42\qquad \textbf{(E) }42.50$

Problem 3

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?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$ (Error compiling LaTeX. Unknown error_msg)

Problem 4

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?

$\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$ (Error compiling LaTeX. Unknown error_msg)

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?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$ (Error compiling LaTeX. Unknown error_msg)

Problem 6

The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?

$\textbf{(A) }44\qquad \textbf{(B) }55\qquad \textbf{(C) }77\qquad \textbf{(D) }99\qquad \textbf{(E) }110$

Problem 7

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?

$\textbf{(A) }1\qquad \textbf{(B) }\sqrt[7]3\qquad \textbf{(C) }\sqrt[8]3\qquad \textbf{(D) }\sqrt[9]3\qquad \textbf{(E) }\sqrt[10]3\qquad$

Problem 8

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: $10\%$ off the listed price if the listed price is at least $50$

Coupon 2: $20$ dollars off the listed price if the listed price is at least $100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $100$

For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?

$\textbf{(A) }179.95\qquad \textbf{(B) }199.95\qquad \textbf{(C) }219.95\qquad \textbf{(D) }239.95\qquad \textbf{(E) }259.95\qquad$

Problem 9

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?

$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$ (Error compiling LaTeX. Unknown error_msg)

Problem 10

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

$\textbf{(A) }\dfrac{\sqrt3}4\qquad \textbf{(B) }\dfrac{\sqrt3}3\qquad \textbf{(C) }\dfrac23\qquad \textbf{(D) }\dfrac{\sqrt2}2\qquad \textbf{(E) }\dfrac{\sqrt3}2$

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

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