Question

$A$ takes $m$ times as long to do a piece of work as $B$ and $C$ together; $B$ takes $n$ times as long as $C$ and $A$ together; and $C$ takes $x$ times as long as $A$ and $B$ together. Then $x,$ in terms of $m$ and $n,$ is:

$\dfrac{2mn}{m+n}$

$\dfrac1{2(m+n)}$

$\dfrac1{m+n-mn}$

$\dfrac{1-mn}{m+n+2mn}$

$\dfrac{m+n+2}{mn-1}$

Answer: E

Concepts:rate system of equations algebraic manipulation

Difficulty rating: 2170

Small Hint:

Let the work rates of $A,$ $B,$ and $C$ be $a,$ $b,$ and $c$

Big Hint:

Translate the first two conditions as $b+c=ma$ and $c+a=nb$

Solution:

Let the work rates be $a,$ $b,$ and $c.$ The time statements give $\begin{gathered} b+c=ma,\\ c+a=nb,\\ a+b=xc. \end{gathered}$ From the first two, $\frac ba=\frac{m+1}{n+1}$ and $\frac ca=\frac{mn-1}{n+1}.$ Hence $x=\frac{a+b}{c} =\frac{m+n+2}{mn-1}.$

Therefore, the correct answer is E.

1965 AMC 12 Problem 38

Answer: E

Difficulty rating: 2170

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years