Ram, Ravi and Reena can complete a piece of work in 16, 20 and 24 days respectively. They started the work together but Ravi left 5 days before the completion of the work. In how many days did the other two together complete the work?

Solution:

Work rates:

Ram = $ \frac{1}{16} $
Ravi = $ \frac{1}{20} $
Reena = $ \frac{1}{24} $

Step 1: Let total time = $x$ days

Ravi works for $(x - 5)$ days,
Ram & Reena work for full $x$ days

Step 2: Total work = 1

$ x\left(\frac{1}{16} + \frac{1}{24}\right) + (x - 5)\left(\frac{1}{20}\right) = 1 $

Step 3: Simplify

$ \frac{1}{16} + \frac{1}{24} = \frac{5}{48} $

So,

$ x \cdot \frac{5}{48} + (x - 5)\cdot \frac{1}{20} = 1 $

Multiply by 240:

$ 25x + 12(x - 5) = 240 $

$ 25x + 12x - 60 = 240 $

$ 37x = 300 $

$ x = \frac{300}{37} = 8\frac{4}{37} $

Step 4: Time taken by Ram & Reena together

Total time = $8\frac{4}{37}$ days