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Anu, Vinu and Manu can complete a work alone in $15 \; \text{days}, 12 \; \text{days}$ and $20 \; \text{days},$ respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is

1. $6$
2. $5$
3. $8$
4. $7$

We can draw a table for better understanding.

$$\begin{array} {llll} & \text{Anu} & \text{Vinu} & \text{Manu} \\\hline \text{Time :} & 15\;\text{days} & 12 \;\text{days} & 20 \;\text{days} \\ \text{Total work :} & \text{LCM(15,12,20)} & = & 60\;\text{units} \\ \text{Efficiency :} & 4\;\text{units/day} & 5\;\text{units/day} & 3 \;\text{units/day} \end{array}$$

Now,

• First day: Vinu  + Anu $= 5+4 =9\;\text{units}$
• Second day: Vinu  + Manu $= 5+3 =8\;\text{units}$
• Third day: Vinu  + Anu $= 5+4 =9\;\text{units}$
• Fourth day: Vinu  + Manu $= 5+3 =8\;\text{units}$
• Fifth day: Vinu  + Anu $= 5+4 =9\;\text{units}$
• Sixth day: Vinu  + Manu $= 5+3 =8\;\text{units}$
• Seven day: Vinu  + Anu $= 5+4 =9\;\text{units}$

Total work $= 60\; \text{units}$

$\text{(Or)}$

In odd days, Vinu + Anu $=5+4 = 9\;\frac{\text{units}}{\text{day}}$

In even days, Vinu + Manu $=5+3 = 8\;\frac{\text{units}}{\text{day}}$

• $2$ days $\longrightarrow 17\;\text{units}$
• $6$ days $\longrightarrow 51\;\text{units}$

In the Seventh day (Vinu + Anu) $= 5+4 = 9\;\frac{\text{units}}{\text{day}}$

• $7$ days $\longrightarrow 60\;\text{units}$

$\therefore$ The number of days needed to complete the work is $7$ days.

Correct Answer $:\text{D}$

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