Question

A & B working together can complete a task in 8 days, B & C can do it in 16 days. A worked at it for 5 days & B worked at it for 1 5 days & B worked at it for 6 days C alone finished the remaining work in 16 days. How many days would C take to complete the same work alone?

Answer 1

Hint:
We convert the work done into unit work then solve the equations and we will find the work done by C working alone
Formula used: $\dfrac{1}{\text{A}}+\dfrac{1}{\text{B}}+\dfrac{1}{\text{C}}=\dfrac{1}{\text{tag}}$
i.e. work done by A in one day$+$ work done by B in one day$+$ work done by A, B, C working together in one day.

Complete step by step solution:
 Let us assume that total work=x
A & B can complete the work in 8 days
     $\therefore $ \[\left( \text{A}+\text{B} \right)\times 8=\text{x}\]
         $\left( \text{A}+\text{B} \right)=\dfrac{\text{x}}{8}$ …… (2)
B & C can complete the work in 16 days
       $\begin{align}
  & \left( \text{B}+\text{C} \right)\times 16=\text{x} \\
 & \left( \text{B}+\text{C} \right)=\dfrac{\text{x}}{16}......(2) \\
\end{align}$
If A works for 5 days then work done by A$=5\times \text{A}$
                                  =5A
 If B works for 6 days then work done by B$=6\times \text{B}$
                              =6B
If C works for 16 days then work done by C =\[16\times \text{C}\]
                              =16C
Then according to condition
     5A$+$6B$+$16C=x
$\left( 5\text{A}+5\text{B} \right)+\left( \text{B}+\text{C} \right)+15\text{ C=x}$
From (1) & (2) we get
       $\left( 5\text{A}+5\text{B} \right)+\left( \text{B}+\text{C} \right)+15\text{ C=x}$
     $\begin{align}
  & 5\left( \dfrac{\text{x}}{8} \right)+\dfrac{\text{x}}{16}+15\text{ C=x} \\
 & \dfrac{5\text{x}}{8}+\dfrac{\text{x}}{16}+15\text{ C=x} \\
 & \text{15 C=x}-\dfrac{\text{5x}}{8}-\dfrac{\text{x}}{16} \\
\end{align}$
$\begin{align}
  & 15\text{ C}=\dfrac{16\text{x}-10\text{x}-\text{x}}{16} \\
 & 15\text{ C}=\dfrac{5\text{x}}{16} \\
 & \text{C}=\dfrac{5\text{x}}{16\times 15} \\
 & \text{ }=\dfrac{\text{x}}{48} \\
\end{align}$

Therefore C alone can complete the work in 48 days.

Note:
Additional information: We can assume the total work as 1. Then solve the question in the similar manner.
Note: We also convert the work done in one day to work so it’s always easy to solve the problem at work.