Question

Tinu takes 9 days more than his father to do a certain piece of work .Together they can do the work in 6 days .How many days will Tinu take to do that work alone?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{Tinu}\text{'}\mathrm{s}\mathrm{father}\mathrm{complete}\mathrm{the}\mathrm{work}\mathrm{alone}\mathrm{in}x\mathrm{days}. \\ \mathrm{Then}\mathrm{Tinu}\text{will}\mathrm{take}\left(x+9\right)\mathrm{days}\mathrm{to}\mathrm{complete}\mathrm{the}\mathrm{same}\mathrm{work}. \\ \mathrm{Thus},\text{in}\mathrm{one}\mathrm{day},\mathrm{work}\text{done by}\mathrm{Tinu}\text{'}\mathrm{s}\mathrm{father}\mathrm{is}\frac{1}{x}\mathrm{and}\text{in}\mathrm{one}\mathrm{day}, \\ \mathrm{work}\text{done by}\mathrm{Tinu}\mathrm{is}\frac{1}{x+9} \\ \mathrm{By}\mathrm{the}\mathrm{given}\mathrm{condition},\mathrm{we}\text{get}: \\ \frac{1}{x}+\frac{1}{x+9}=\frac{1}{6} \\ \Rightarrow \frac{x+9+x}{x\left(x+9\right)}=\frac{1}{6} \\ \Rightarrow \frac{2x+9}{x\left(x+9\right)}=\frac{1}{6} \\ \Rightarrow x\left(x+9\right)=6\left(2x+9\right) \\ \Rightarrow x^2+9x=12x+54 \\ \Rightarrow x^2-3x-54=0 \\ \mathrm{On}\mathrm{splitting}\mathrm{the}\mathrm{middle}\mathrm{term}-3x\mathrm{as}6x-9x,\mathrm{we}\mathrm{get}: \\ {x}^2+6x-9x-54=0 \\ \Rightarrow x\left(x+6\right)-9\left(x+6\right)=0 \\ \Rightarrow \left(x+6\right)\left(x-9\right)=0 \\ \Rightarrow x+6=0\mathrm{or}x-9=0 \\ \Rightarrow x=-6\mathrm{or}x=9 \end{gathered}$$

Since x is the number of days, which cannot be negative, therefore, x = 9
Thus, Tinu takes x+9 = 9+9 = 18 days to complete the work .