A can do a piece of work in 120 days and B can do it in 150 days. They work together for 20 days. Then B leaves and A continues the work alone. 12 days after C joins A and the work is completed in 48 days more. In lion many days can C do it if he works alone? (Work & Time)

প্রশ্নে বলা হচ্ছে যে, A একটি কাজ 120 দিনে এবং B কাজটি 150 দিনে করতে পারে। A এবং B একত্রে কাজটি  20 দিন করে B কাজটি ছেড়ে চলে গেল এবং A কাজটি চালিয়ে গেল। 12 দিন পরে C এসে A এর সাথে যোগ দিলে কাজটি শেষ করতে আরো 48 দিন সময় লাগল । C একা কাজটি কত দিনে করতে পারবে?

  $$\begin{gathered} A \& B can do in 1 day = \left(\frac{1}{120} + \frac{1}{150}\right) = \left(\frac{5 + 4}{600}\right) = \frac{9}{600} = \frac{3}{200} Part of the work \\ \therefore A \& B can do in 20 days = \left(\frac{3}{200} \times 20\right) = \frac{3}{10} Part of the work \\ Remaining work = 1 - \frac{3}{10} = \frac{10-3}{10} =- \frac{7}{10} Part of the work \end{gathered}$$

$$\begin{gathered} In 12 days, A can complete = \frac{1}{120} \times 12 = \frac{1}{10} Part of the work \\ \therefore Remaining work =\left(\frac{7}{10} - \frac{1}{10}\right) = \frac{7-1}{10} = \frac{6}{10} = \frac{3}{5} Part of the work \end{gathered}$$

Let, C can do the work is x days

So, A & C can do in 1 day $= \left(\frac{1}{20} + \frac{1}{x}\right) = \frac{x+ 120}{120x} Part of the work$

$$\begin{gathered} \therefore A \& C can do in 48 days = \frac{48 \left(x+120\right)}{120x} Part of the work \\ According to question, \frac{48 \left(x+120\right)}{120x} = \frac{3}{5} \\ \Rightarrow 5\times 48 (x + 120) = 360x \\ \Rightarrow 240x + 28,800 = 360x \\ \Rightarrow 360x-240x = 28,800 \\ \Rightarrow 120x=28,800 \\ \therefore x = \frac{28800}{120} = 240 \\ \therefore C can do the work 240 days. \end{gathered}$$