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 $:\text{D}$