2015 AMC 12A Problems

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Problem 1

What is the value of $(2^0-1+5^2-0)^{-1}\times5?$

$\textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}}\ \frac{5}{24}\qquad\textbf{(E)}\ 25$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 2

Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?

$\textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72$

Solution

Problem 3

Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?

$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95$

Solution

Problem 4

The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?

$\textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}}\ 2 \qquad\textbf{(E)}\ \frac52$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 5

Amelia needs to estimate the quantity $\frac{a}{b} - c$, where $a, b,$ and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\frac{a}{b} - c$?

$\textbf{(A)}\ \text{She rounds all three numbers up.}\\ \qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{down.}\\ \qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{down.} \\ \qquad\textbf{(D)}}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{down.}\\ \qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{down.}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 6

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 7

$\textbf{(A)}\ \qquad\textbf{(B)}\ \qquad\textbf{(C)}\ \qquad\textbf{(D)}}\ \qquad\textbf{(E)}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 8

$\textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 9

$\textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 10

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 11

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 12

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 13

$\textbf{(A)}\ \qquad\textbf{(B)}\ \qquad\textbf{(C)}\ \qquad\textbf{(D)}}\ \qquad\textbf{(E)}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 14

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}}\ 24\qquad\textbf{(E)}\ 36$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 15

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 16

$\textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 17

$\textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 18

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 19

$\textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 20

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 21

$\textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 22

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 23

$\textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 24

$\textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

Problem 25

$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

Solution

See also

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