$A$ and $B$ separately do a work in $10$ and $15$ days respectively. They worked together for some days and then $A$ completed the remaining work in $5$ days. For how many days had $A$ and $B$ worked together?

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Solution

Given,
$A$ can do a work in = $10$ days
$B$ can do a work in = $15$ days

$A$ ’s, $1$ day's work = $\frac{1}{10}$ ... (1)
$B$ ’s, $1$ day's work = $\frac{1}{15}$

$(A + B)$ ’s $1$ day's work = $\frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}$

Now, let us consider $(A + B)$ worked together for $x$ days.

So, $(A + B)$ ’s, $x$ day work = $\frac{x}{6}$

Remaining work $= 1 - \frac{x}{6}$

It is given that,
$A$ can do the remaining work, $(1 - \frac{x}{6})$ in $= 5$ days
So, $A$ ’s $1$ day's work $= \frac{1 - \frac{x}{6}}{5} = \frac{6 - x}{30}$ ... (2)

From (1) and (2), we get
$\frac{1}{10} = \frac{6 - x}{30} \Rightarrow \frac{30}{10} = 6 - x \Rightarrow 3 = 6 - x \Rightarrow x = 6 - 3 = 3$

$∴$ Both $A$ and $B$ can do the work together in $3$ days.

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