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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

Course
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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#### APPEARS IN

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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