1954 AMC 12 Problem 30

Question

A

and

B

together can do a job in

2

days;

B

and

C

can do it in four days; and

A

and

C

in

2\dfrac25

days. The number of days required for

A

to do the job alone is:

1

3

6

12

2.8

Let

a,b,c

be the fractions of the job completed per day

Add

a+b=\frac{1}{2}

and

a+c=\frac{5}{12}

, then subtract

b+c=\frac{1}{4}

Let

a,b,c

be the individual daily rates. Then

\begin{aligned} a+b&=\frac12,\\ b+c&=\frac14,\\ a+c&=\frac5{12}. \end{aligned}

Thus

\begin{aligned} 2a&=(a+b)+(a+c)\\ &\quad{}-(b+c)\\ &=\frac12+\frac5{12}-\frac14\\ &=\frac23, \end{aligned}

so

a=\frac{1}{3}.

Therefore

A

needs

3

days.

Thus, the correct answer is B.

Problem 30 in other years