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Question
A and B can do a piece of work in 18 days; B and C in 24 days and A and C in 36 days. In what time can they do it, all working together?
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Solution
\[\text{ Time taken by } \left( A + B \right) \text{ to do the work = 18 days } \]
\[\text{ Time taken by } \left( B + C \right) \text{ to do the work = 24 days } \]
\[\text{ Time taken by } \left( A + C \right) \text{ to do the work = 36 days } \]
\[\text{ Now } , \]
\[\text{ Work done by } \left( A + B \right) = \frac{1}{18}\]
\[ \text{ Work done by } \left( B + C \right) = \frac{1}{24}\]
\[ \text{ Work done by } \left( A + C \right) = \frac{1}{36}\]
\[ \therefore \text{ Work done together } = \left( A + B \right) + \left( B + C \right) + \left( A + C \right)\]
\[ = \frac{1}{18} + \frac{1}{24} + \frac{1}{36}\]
\[ = \frac{4 + 3 + 2}{72} = \frac{9}{72}\]
\[ = \frac{1}{8}\]
\[ \therefore \text{ Work done together } = 2\left( A + B + C \right) = \frac{1}{8}\]
\[ \therefore \text{ Work done by } \left( A + B + C \right) = \frac{1}{16}\]
\[ \text{ Thus, together they can finish the work in 16 days } .\]
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