# 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?

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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 \[\dfrac{{xy}}{{x + y}}\] 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 = \dfrac{x}{2}$

A takes thrice as much as C. Therefore C takes one third of what time A takes.

$ \Rightarrow C = \dfrac{x}{3}$

When all of them work together, they can finish work in 2 days.

$

\Rightarrow \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{2} \\

\Rightarrow \dfrac{1}{x} + \dfrac{2}{x} + \dfrac{3}{x} = \dfrac{1}{2} \\

\Rightarrow \dfrac{6}{x} = \dfrac{1}{2} \\

$

Now, Cross multiply

$ \Rightarrow x = 12$

Now, we calculate how much time taken by B to complete work.

$

B = \dfrac{x}{2} = \dfrac{{12}}{2} \\

\Rightarrow B = 6 \\

$

B takes 6 days to complete work.

Now, we calculate how much time taken by C to complete work.

$

C = \dfrac{x}{3} = \dfrac{{12}}{3} \\

\Rightarrow C = 4 \\

$

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

$

\Rightarrow \dfrac{1}{B} + \dfrac{1}{C} \\

\Rightarrow \dfrac{1}{6} + \dfrac{1}{4} \\

$

Take LCM

$

\Rightarrow \dfrac{{2 + 3}}{{12}} \\

\Rightarrow \dfrac{5}{{12}} \\

$

So, If B and C work together they will take $\dfrac{{12}}{5}$ 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.

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 = \dfrac{x}{2}$

A takes thrice as much as C. Therefore C takes one third of what time A takes.

$ \Rightarrow C = \dfrac{x}{3}$

When all of them work together, they can finish work in 2 days.

$

\Rightarrow \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{2} \\

\Rightarrow \dfrac{1}{x} + \dfrac{2}{x} + \dfrac{3}{x} = \dfrac{1}{2} \\

\Rightarrow \dfrac{6}{x} = \dfrac{1}{2} \\

$

Now, Cross multiply

$ \Rightarrow x = 12$

Now, we calculate how much time taken by B to complete work.

$

B = \dfrac{x}{2} = \dfrac{{12}}{2} \\

\Rightarrow B = 6 \\

$

B takes 6 days to complete work.

Now, we calculate how much time taken by C to complete work.

$

C = \dfrac{x}{3} = \dfrac{{12}}{3} \\

\Rightarrow C = 4 \\

$

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

$

\Rightarrow \dfrac{1}{B} + \dfrac{1}{C} \\

\Rightarrow \dfrac{1}{6} + \dfrac{1}{4} \\

$

Take LCM

$

\Rightarrow \dfrac{{2 + 3}}{{12}} \\

\Rightarrow \dfrac{5}{{12}} \\

$

So, If B and C work together they will take $\dfrac{{12}}{5}$ 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.

Last updated date: 23rd Sep 2023

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