Question : Rekha alone can complete a task in 16 days and Bina alone can complete the same task in 12 days. Starting with Rekha, they work on alternate days. The task will be completed in:
Option 1: $12 \frac{3}{4}\ \text{days}$
Option 2: $12\ \text{days}$
Option 3: $13\ \text{days}$
Option 4: $13 \frac{3}{4}\ \text{days}$
Correct Answer: $13 \frac{3}{4}\ \text{days}$
Solution :
Given: Rekha alone can complete the task in 16 days.
Bina alone can complete the same task in 12 days.
Let the total work be the LCM of (16, 12) = 48 units.
So, Rekha's efficiency is $\frac{48}{16} = 3$ units per day,
and Bina's efficiency is $\frac{48}{12} = 4$ units per day.
They work on alternate days, starting with Rekha.
So, two days of work = [(Efficiency of Rekha) × 1 day) + (Efficiency of Bina) × 1 day)]
= (3 + 4) = 7 units
So, 12 alternate days of work = 42
So, work left = 48 – 42 = 6 units
Now, Rekha has started again and does 3 units of work on the 13
th
day.
So, their 13 alternate days of work are 45 units.
Work left now is 3 units.
Now, Bina starts and does 4 units of work in 1 day.
So, Bina does 1 unit of work in $\frac{1}{4}$ days.
So, 3 units of work are done in $\frac{1}{4} × 3 =\frac{3}{4} $ days.
So, the total time taken is 13 days + $\frac{3}{4}$ days = $13 \frac{3}{4}$ days
Hence, the correct answer is $13 \frac{3}{4}\ \text{days}$.
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