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 6 hr less than Asit would have taken working alone, 1 hr 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 min
• B. 30 min
• C. 40 min
• D. 50 min

Hint:

Let the time together be \(T\) hours; then the individual times are \(T+6\), \(T+1\) and \(2T\). Add the three rates and set the sum equal to \(1/T\).

Answer 1

The Correct Option is C

Step 1: Let the time together be \(T\) hours.
Asit alone takes \(T+6\), Arnold alone takes \(T+1\), Afzal alone takes \(2T\).

Step 2: Add the work rates.
\[\frac{1}{T+6}+\frac{1}{T+1}+\frac{1}{2T}=\frac{1}{T}\]Subtract \(\frac{1}{2T}\) from both sides:
\[\frac{1}{T+6}+\frac{1}{T+1}=\frac{1}{2T}\]Step 3: Solve.
\[\frac{2T+7}{(T+6)(T+1)}=\frac{1}{2T}\Rightarrow 2T(2T+7)=T^2+7T+6\Rightarrow 3T^2+7T-6=0\]\[(3T-2)(T+3)=0\Rightarrow T=\tfrac23\ \text{hr}\ (\text{rejecting } T=-3)\]Step 4: Convert.
\(T=\frac23\) hr \(=40\) minutes.

Final Answer:
The correct option is (3): 40 min.