Question

A and B can complete work together in 5 days. If A works at twice his speed and B at half of his speed this works can be finished in 4 days. How many days would it take for A alone to complete the job ?

• A. $10.0$
• B. $12.0$
• C. $15.0$
• D. $18.0$

Answer 1

Answer: (A)

$10.0$

Let $A$'s one day work be $\frac{1}{x}$ and $B$'s one day work $\frac{1}{y}$.

It is given that $A$ and $B$ can complete work together in $5$ days, therefore, we have:

$A+B=\frac{1}{x}+\frac{1}{y}=\frac{1}{5}\Rightarrow \frac{y}{xy}+\frac{x}{xy}=\frac{1}{5}\Rightarrow \frac{y+x}{xy}=\frac{1}{5}\Rightarrow 5(x+y)=xy.....\left(1\right)$

$A$ worked at twice his speed means $\frac{2}{x}$

$B$ worked at half of his speed means $\frac{1}{2y}$

Together they complete work in $4$ days, therefore, we have:

$\frac{2}{x}+\frac{1}{2y}=\frac{1}{4}\Rightarrow \frac{4y}{2xy}+\frac{x}{2xy}=\frac{1}{4}\Rightarrow \frac{4y+x}{2xy}=\frac{1}{4}\Rightarrow 4(x+4y)=2xy\Rightarrow 2(x+4y)=xy\Rightarrow 2x+8y=xy.....\left(2\right)$

Substitute equation 2 in 1:

$5x+5y=2x+8y\Rightarrow 5x-2x=8y-5y\Rightarrow 3x=3y\Rightarrow x=y$

$x=y$ means that number of days worked by $A$ alone is equal to $B$ alone.

Now, substitute the value in equation 1:

$\frac{1}{x}+\frac{1}{x}=\frac{1}{5}\Rightarrow \frac{1+1}{x}=\frac{1}{5}\Rightarrow \frac{2}{x}=\frac{1}{5}\Rightarrow x=5\times 2\Rightarrow x=10$

Hence, it would take $10$ days for $A$ alone to complete the job.