Question

For the same amount of work, A takes 6 hours less than B. If together they complete the work in 13 hours 20 minutes; find how much time will B alone take to complete the work.
A) 20 hrs
B) 30 hrs
C) 10 hrs
D) None of the above

Answer 1

Hint: We consider the time taken by A alone to complete the work to be $x$ hrs. By using the given condition, we get a quadratic equation in $x$ . Hence, we get the required answer by finding the roots of the obtained quadratic equation by using $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .

Complete step-by-step answer:
For the same amount of work, A takes 6 hours less than B and together they complete the work in 13 hours 20 minutes.
Let us consider the time taken by A alone to complete the work to be $x$ hrs.
So, the time taken by B alone to complete the work is $x+6$ hrs.
We were given that the time taken by A and B together to complete the work is 13 hours 20 minutes.
So, we get,
$\begin{align}
  & \dfrac{1}{x}+\dfrac{1}{x+6}=\dfrac{1}{13\text{ }hours\text{ }20\text{ }minutes} \\
 & \\
 & \Rightarrow \dfrac{1}{x}+\dfrac{1}{x+6}=\dfrac{1}{\dfrac{40}{3}} \\
 & \\
 & \Rightarrow \dfrac{1}{x}+\dfrac{1}{x+6}=\dfrac{3}{40} \\
 & \\
 & \Rightarrow \dfrac{x+6+x}{x\left( x+6 \right)}=\dfrac{3}{40} \\
 & \\
 & \Rightarrow \dfrac{2x+6}{{{x}^{2}}+6x}=\dfrac{3}{40} \\
 & \\
 & \Rightarrow 40\left( 2x+6 \right)=3\left( {{x}^{2}}+6x \right) \\
 & \\
 & \Rightarrow 3{{x}^{2}}+18x-80x-240=0 \\
 & \\
 & \Rightarrow 3{{x}^{2}}-62x-240=0 \\
\end{align}$
Let us consider the formula for the roots of the quadratic equation, that is, the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
By applying the above formula to the obtained quadratic equation, we get,
$\begin{align}
  & x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
 & \\
 & \Rightarrow x=\dfrac{62\pm \sqrt{{{62}^{2}}-4\left( 3 \right)\left( -240 \right)}}{2\left( 3 \right)} \\
 & \\
 & \Rightarrow x=\dfrac{62\pm \sqrt{3844+2800}}{6} \\
 & \\
 & \Rightarrow x=\dfrac{62\pm \sqrt{6724}}{6} \\
 & \\
 & \Rightarrow x=\dfrac{62\pm 82}{6} \\
\end{align}$
$\Rightarrow x=\dfrac{62+82}{6}$ and $\Rightarrow x=\dfrac{62-82}{6}$
$\Rightarrow x=24$ and $x=-\dfrac{20}{6}$
As $x$ is the time taken by A to complete the work, it cannot be negative.
So, we get,
$x=24$
As the time taken by B to complete the work is $\left( x+6 \right)hrs$ , we get,
The time taken by B alone to complete the work is 24+6 = 30 hours.

Hence, the answer is Option B.

Note: The possibilities for making mistakes in this type of problems are,
One may make a mistake by considering \[\dfrac{1}{x}+\dfrac{1}{x+6}=13\text{ }hours\text{ }20\text{ }minutes\] instead of taking \[\dfrac{1}{x}+\dfrac{1}{x+6}=\dfrac{1}{13\text{ }hours\text{ }20\text{ }minutes}\] .
Time and work are directly proportional to each other. If the work increases then the time taken to finish the work increases. If the work decreases then the time taken to finish the work decreases.