There’s a lot of work in preparing a birthday dinner. Even after the

Question

There’s a lot of work in preparing a birthday dinner. Even after the turkey is in the oven, there’s still the potatoes and gravy, yams, salad, and cranberries, not to mention setting the table.

Three friends, Asit, Arnold, and Afzal, work together to get all of these chores done. The time it takes them to do the work together is six hours less than Asit would have taken working alone, one hour less than Arnold would have taken alone, and half the time Afzal would have taken working alone.

How long did it take them to do these chores working together?

(A) 20 minutes 
(B) 30 minutes 
(C) 40 minutes 
(D) 50 minutes
(E) 60 minutes

rajatchopra1994 · Answer

Explanation:
Method 1:
Let t is the total hours to do the chores if all friends work together.
Also x, y and z are the time taken by Asit, Arnold and Afzal respectively.
t = x - 6 = y - 1 = z/2
t = 1/x + 1/y + 1/z = xyz / (xy + yz + zx) = x - 6
we know y = x - 5 and z = 2(x - 6)
convert all in terms of x and solve for x.
we get x = 20/3 and t = (20/3) - 6 = 2/3 hours = 40 minutes.

Method 2:

Above solution is very calculation intensive, So by checkin options
y = t + 1
z = 2t
1/t = 1/(t + 6) + 1/(t - 1) + 1/2t
Substitute options and t = 2/3 hours (40 minutes) satisfies.

IMO-C

saynchalk · Answer

Deriving the LCM after substituting the options given looks very calculation intensive as well. Am I missing something? Is there a quick trick for LCMs, or a better way to solve the equation after that?

curiousPope · Answer

Hi  Bunuel ,
can you please help with the solution.

Bunuel · Answer

Solving  \frac{1}{t} = \frac{1}{(t + 6)} + \frac{1}{(t + 1)} + \frac{1}{2t}  would be easier:
 
 \frac{1}{t} = \frac{1}{(t + 6)} + \frac{1}{(t + 1)} + \frac{1}{2t}

\frac{1}{t} - \frac{1}{2t}= \frac{1}{(t + 6)} + \frac{1}{(t + 1)}

\frac{1}{2t}= \frac{(t + 1)+(t + 6)}{(t + 6)(t + 1)}

\frac{1}{2t}= \frac{2t + 7}{(t + 6)(t + 1)}

(t + 6)(t + 1)=2t(2t + 7)

3t^2 + 7 t - 6 = 0

t =\frac{ 2}{3}  or  t = -3 .
 
Answer: C.

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