2005 AMC 12B Problems

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Problem 1

A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?

$\mathrm{(A)}\ 100 \qquad \mathrm{(B)}\ 200 \qquad \mathrm{(C)}\ 300 \qquad \mathrm{(D)}\ 400 \qquad \mathrm{(E)}\ 500$

Solution

Problem 2

The equations $2x+7=3$ and $bx-10=-2$ have the same solution for $x$ . What is the value of $b$ ?

$\mathrm{(A)}\ -8 \qquad \mathrm{(B)}\ -4 \qquad \mathrm{(C)}\ -2 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 8$

Solution

Problem 3

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

$\mathrm{(A)}\ \frac14x^2 \qquad \mathrm{(B)}\ \frac25x^2 \qquad \mathrm{(C)}\ \frac12x^2 \qquad \mathrm{(D)}\ x^2 \qquad \mathrm{(E)}\ \frac32x^2$

Solution

Problem 4

The average (mean) of $20$ numbers is $30, and the average of$ 30 $other numbers is$ 20 $. What is the average of$ \mathrm{(A)}\ \frac14x^2 \qquad \mathrm{(B)}\ \frac25x^2 \qquad \mathrm{(C)}\ \frac12x^2 \qquad \mathrm{(D)}\ x^2 \qquad \mathrm{(E)}\ \frac32x^2 $