1965 AMC 12 Problem 38

Attempt Problem 38 of the 1965 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1965 AMC 12 solutions, or check the answer key.

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38.

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.

Problem 38 in other years