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    Topics || Problems

    When A and B both work, they can paint a certain house in 8 days. Also, they could paint this house if A worked 12 days and B worked 6 days. How long would it take each to paint the house alone? Solution:

    Rate of A: \(\frac{1}{A}\)

    Rate of B: \(\frac{1}{B}\)

    \(\frac{1}{A} + \frac{1}{B} = \frac{1}{8}\)

    \(\frac{1}{A} = - \frac{1}{B} + \frac{1}{8}\)

    \(\frac{12}{A} + \frac{6}{B} = 1\)

    \(12(-\frac{1}{B} + \frac{1}{8})+ \frac{6}{B} = 1\)

    \( \frac{6}{B} = \frac{3}{2}-1\)

    \(B = 12\)

    \(\frac{1}{A} + \frac{1}{12} = \frac{1}{8}\)

    \(\frac{1}{A} = \frac{1}{24}\)

    \(A = 24\)

    A would paint the house in 24 days alone and B in 12 days alone.