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A and B can do a piece of work in 12 days; B and C in 15 days; C and A in 20 days. How much time will A alone take to finish the work? - Mathematics

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ConceptTime and Work

Question

A and B can do a piece of work in 12 days; B and C in 15 days; C and A in 20 days. How much time will A alone take to finish the work?

Solution

\[\text{ Time taken by }  \left( A + B \right) \text{ to do the work = 12 days } \]
\[\text{ Time taken by }  \left( B + C \right) \text{ to do the work = 15 days } \]
\[ \text{ Time taken by }  \left( A + C \right) \text{ to do the work = 20 days } \]
\[ \text{ Now, } \]
\[\text{ Work done by } \left( A + B \right) = \frac{1}{12}\]
\[ \text{ Work done by } \left( B + C \right) = \frac{1}{15}\]
\[ \text{ Work done by } \left( A + C \right) = \frac{1}{20}\]
\[ \therefore \text{ Work done together  } = \left( A + B \right) + \left( B + C \right) + \left( A + C \right)\]
\[ = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}\]
\[ = \frac{5 + 4 + 3}{60} = \frac{12}{60}\]
\[ = \frac{1}{5}\]
\[ \therefore \text{ Work done together } = 2\left( A + B + C \right) = \frac{1}{5}\]
\[ \therefore \text{ Work done by }  \left( A + B + C \right) = \frac{1}{10}\]
\[ \therefore \text{ Work done by A alone }  = \left( A + B + C \right) - \left( B + C \right)\]
\[ = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}\]
\[\text{ Thus, A alone can do the work in 30 days }  .\]

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Solution A and B can do a piece of work in 12 days; B and C in 15 days; C and A in 20 days. How much time will A alone take to finish the work? Concept: Time and Work.
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