Question : $P_1$ and $P_2$ can do a piece of work together in 14 days, $P_2$ and $P_3$ can do the same work together in 21 days, while $P_3$ and $P_1$ can do it together in 42 days. How much work can all the 3 together do in 12 days?
Option 1: $\frac{5}{6}$
Option 2: $\frac{5}{7}$
Option 3: $\frac{6}{7}$
Option 4: $\frac{3}{7}$
Correct Answer: $\frac{6}{7}$
Solution :
$P_1$ and $P_2$ can do a piece of work together in = 14 days
$P_2$ and $P_3$ can do the same work together in = 21 days
$P_3$ and $P_1$ can do it together in = 42 days
Total work = Time × Efficiency
Total work = LCM(14, 21, 42) = 42 units
Efficiency of $P_1$ and $P_2$ = $\frac{42}{14}$ = 3 units/day
Efficiency of $P_2$ and $P_3$ = $\frac{42}{21}$ = 2 units/day
Efficiency of $P_3$ and $P_1$ = $\frac{42}{42}$ = 1 units/day
Efficiency of $P_1$, $P_2$ and $P_3$ = $\frac{(3 + 2 + 1)}{2}$ = 3 units/day
Work done by all 3 in 12 days = 3 × 12 = 36 units
Part of work done in 12 days = $\frac{36}{42}$ = $\frac{6}{7}$ unit
Hence, the correct answer is $\frac{6}{7}$ unit.
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