The sum of fractions of work done by each person to complete the work is = 1
Since A and B together finish the work together in 16 days, \(\frac{16}{A}\) + \(\frac{16}{B}\) = 1 or \(\frac{1}{A}\) + \(\frac{1}{B}\) = \(\frac{1}{16}\) - Equation (1)
Since B and C finish the work together in 24 days, \(\frac{24}{B}\) + \(\frac{24}{C}\) = 1 or \(\frac{1}{B}\) + \(\frac{1}{C}\) = \(\frac{1}{24}\) - Equation (2)
Also \(\frac{4}{A}\) + \(\frac{7}{B}\) + \(\frac{23}{C}\) = 1 - Equation (3)
Since we need to find the number of days in which C finishes the work, we need to find A and B in terms of C.
From Equation (2) \(\frac{1}{B}\) = \(\frac{1}{24}\) - \(\frac{1}{C}\) - Equation (4)
Subtracting Equation (1) from (2)
\(\frac{1}{A}\) + \(\frac{1}{B}\) - [\(\frac{1}{B}\) + \(\frac{1}{C}\)] = \(\frac{1}{16}\) - \(\frac{1}{24}\)
\(\frac{1}{A}\) - \(\frac{1}{C}\) = \(\frac{1}{16}\) - \(\frac{1}{24}\) = \(\frac{1}{48}\)
\(\frac{1}{A}\) = \(\frac{1}{48}\) + \(\frac{1}{C}\) - (Equation 5)
Putting 4 and 5 in Equation (3)
4 * (\(\frac{1}{48}\) + \(\frac{1}{C}\)) + 7 * (\(\frac{1}{24}\) - \(\frac{1}{C}\)) + \(\frac{23}{C}\) = 1
\(\frac{4}{48}\) + \(\frac{4}{C}\) + \(\frac{7}{24}\) - \(\frac{7}{C}\) + \(\frac{23}{C}\) = 1
\(\frac{20}{C}\) = 1 - \(\frac{4}{48}\) - \(\frac{7}{24}\) = \(\frac{5}{8}\)
Therefore C = 32 days
Option A
Arun Kumar
_________________