# Time taken to complete a piece of work

A can complete a piece of work in 80 days. He worked work for 10 days , after that B completed the remaining work in 42 days. If A and B work together how many days will it take them to complete the entire piece of work?

a)30
b)25
c)40
d)none of these

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A completed $\frac{10}{80} = \frac 18$ of the work. Then B completed $\frac 78$ of the work in 42 days. So that means that he'll complete the whole work for:

$$\frac 78 x = 42$$ $$\frac 78x \frac87 = 42 \cdot \frac 87$$ $$x = 48$$

We know that A will do the whole work for 80 days so for 1 day he'll finish $\frac {1}{80}$ of the work, while B will complete $\frac {1}{48}$

Now we have:

$$\frac {x}{80} + \frac {x}{48} = 1$$

Now you can continue on your own. Solve the equation for x to obtain the number of days.

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$A$ worked for 10 days, and hence completed $\frac 18$ of the project. $\frac 78$ of the project was then completed by $B$ in $42$ days. B then could complete the entire project in $x$ days, where $$\dfrac 78 x = 42 \iff x = \dfrac 87\cdot 42 = 56\;\text{days}$$

Let $y$ be the number of days to complete the project if both work together. Then we need only solve for:

$$\dfrac y{80} + \dfrac y{48} = 1$$ $$\dfrac{3y + 5y}{240} = 1 \iff 8y = 240 \iff y = 30$$ So the correct answer is $$(a)\quad 30$$

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