Question

Two pipes running together can fill a tank in $11{\displaystyle \frac{1}{9}}$ minutes. If one pipe takes 5 minutes more than the other to fill the tank, find the product of the time in which each pipe would fill the tank.

Answer 1

Hint: Here we go through by assuming the time taken by one pipe be ‘x’ then we automatically calculate the time taken by the second pipe according to the question, then by applying a unitary method we will get our answer.

Complete step-by-step answer:
Here in the question it is given that two pipes running together can fill a tank in $11{\displaystyle \frac{1}{9}}$ minutes.

Now Let the time taken by the faster pipe to fill the tank be x minutes.

Therefore, time taken by slower pipe to fill the tank =(x+5) minutes

Since the faster pipe takes x minutes to fill the tank.

Portion of tank filled by the faster pipe in 1 minute$={\displaystyle \frac{1}{x}}$.

Portion of tank filled by the faster pipe in 1 minute$={\displaystyle \frac{1}{x+5}}$

Portion of tank filled by the two pipes in 1 minute$={\displaystyle \frac{1}{\left(11{\displaystyle \frac{1}{9}}\right)}}={\displaystyle \frac{9}{100}}$

Now according to the question-

$$\begin{gathered} \Rightarrow {\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{x+5}}={\displaystyle \frac{9}{100}} \\ \Rightarrow {\displaystyle \frac{x+5+x}{x(x+5)}}={\displaystyle \frac{9}{100}} \\ \Rightarrow 9(x^2+5x)=200x+500 \\ \Rightarrow 9x^2-155x-500=0 \end{gathered}$$

Now we apply quadratic formula to get the value of x. i.e. $x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$.

$$\begin{gathered} \Rightarrow x={\displaystyle \frac{155\pm \sqrt{{(155)}^2-4\times 9\times (-500)}}{2\times 9}} \\ \Rightarrow x={\displaystyle \frac{155\pm 205}{18}} \\ \therefore x=20,-{\displaystyle \frac{50}{18}} \end{gathered}$$

And we know that time cannot be negative therefore the value of x is 20.

Hence the faster pipe to fill the tank will be 20 minutes and slower pipe to fill the tank =(x+5)

=20+5=25minutes.

Therefore, the product of the time in which each pipe would fill the tank $=20\times 25=500$.

Note: Whenever we face such type of question the key concept for solving the question is whenever in the question two working things are given then always assume one of its working time as variable then according to question we will find the working time of other one then by applying the unitary method we will find their work for unit time because we know that we have add their work in unit time as we learn in chapter of unitary method. By these steps we will get our answer.