Question

Suppose A takes twice as much time as B and thrice as much time as C to complete a work. If all of them work together, they can finish the work in 2 days. How much time B and C working together to finish it?

Answer 1

Hint- If one person does a work in x days and another person does it in y days then together they can finish that work in $${\displaystyle \frac{xy}{x+y}}$$ days.
Let’s take work done by A be x
A takes twice as much as B Therefore B takes half of what time A takes.
$\Rightarrow B={\displaystyle \frac{x}{2}}$
A takes thrice as much as C. Therefore C takes one third of what time A takes.
$\Rightarrow C={\displaystyle \frac{x}{3}}$
When all of them work together, they can finish work in 2 days.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{1}{A}}+{\displaystyle \frac{1}{B}}+{\displaystyle \frac{1}{C}}={\displaystyle \frac{1}{2}} \\ \Rightarrow {\displaystyle \frac{1}{x}}+{\displaystyle \frac{2}{x}}+{\displaystyle \frac{3}{x}}={\displaystyle \frac{1}{2}} \\ \Rightarrow {\displaystyle \frac{6}{x}}={\displaystyle \frac{1}{2}} \end{gathered}$$
Now, Cross multiply
$\Rightarrow x=12$
 Now, we calculate how much time taken by B to complete work.
$$\begin{gathered} B={\displaystyle \frac{x}{2}}={\displaystyle \frac{12}{2}} \\ \Rightarrow B=6 \end{gathered}$$
B takes 6 days to complete work.
Now, we calculate how much time taken by C to complete work.
$$\begin{gathered} C={\displaystyle \frac{x}{3}}={\displaystyle \frac{12}{3}} \\ \Rightarrow C=4 \end{gathered}$$
C takes 4 days to complete work.
Now, we calculate how much time taken by B and C to work together.
 B does work in 6 days and C does it in 4 days.
 If B and C can work together
$$\begin{gathered} \Rightarrow {\displaystyle \frac{1}{B}}+{\displaystyle \frac{1}{C}} \\ \Rightarrow {\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{4}} \end{gathered}$$
Take LCM
$$\begin{gathered} \Rightarrow {\displaystyle \frac{2+3}{12}} \\ \Rightarrow {\displaystyle \frac{5}{12}} \end{gathered}$$
So, If B and C work together they will take ${\displaystyle \frac{12}{5}}$ days.

Note- Whenever we face such types of problems we use some important points. Like we calculate how much time taken by a single person to complete their work then we calculate how much time taken by persons when they work together.