Question

Two water taps can together fill a tank in \[9\dfrac{3}{8}\] hours. The tap of larger diameter takes 10 hours less than the tap of smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Answer 1

Hint:This is a pipe and cisterns related problem. In this we will consider the time for each tap separately but interrelated. Like the tap with smaller diameter fills the tank in x hours and the larger diameter tap fills it in $x-10$ hours. Now together they can fill the tank in \[9\dfrac{3}{8}\] hours. So there we get an equation. On solving which we will get the value of $x$ that is nothing but the time taken by the taps separately.

Complete step by step answer:
Here the two taps are used to fill the tank. Let the tap with a smaller diameter fill the tank in $x$ hours. Then from the given data that The tap of larger diameter takes 10 hours less than the tap of smaller one to fill the tank we can say it takes x-10 hours. Now together they fill the whole tank in \[9\dfrac{3}{8}\] hours that is in \[\dfrac{{75}}{8}\] hours.Now we will consider the work done by the taps in 1 hour. So tap with smaller diameter fills \[\dfrac{1}{x}\] part in one hour. And that with a larger diameter fills \[\dfrac{1}{{x - 10}}\] part in one hour.

Now in one hour they together can fill \[\dfrac{8}{{75}}\] part of the tank. So the equation can be written as,
\[\dfrac{1}{x} + \dfrac{1}{{x - 10}} = \dfrac{8}{{75}}\]
Now we need to solve this equation. For that we will take LCM on LHS,
\[ \Rightarrow \dfrac{{\left( {x - 10} \right) + x}}{{\left( {x - 10} \right)x}} = \dfrac{8}{{75}}\]
Now solving the numerator we get,
\[ \Rightarrow \dfrac{{2x - 10}}{{\left( {x - 10} \right)x}} = \dfrac{8}{{75}}\]
On cross multiplying we get,
\[ \Rightarrow 75\left( {2x - 10} \right) = 8x\left( {x - 10} \right)\]

On multiplying the brackets,
\[ \Rightarrow 150x - 750 = 8{x^2} - 80x\]
Now rearranging the terms we get,
\[ \Rightarrow 8{x^2} - 80x - 150x + 750 = 0\]
Now if we observe this is turning towards quadratic equation,
\[ \Rightarrow 8{x^2} - 230x + 750 = 0\]
Dividing both sides by 2 we get,
\[ \Rightarrow 4{x^2} - 115x + 375 = 0\]
Now we will split the middle terms as,
\[ \Rightarrow 4{x^2} - 100x - 15x + 375 = 0\]
Taking 4x common from first two terms as 15 common from lats two terms,
\[ \Rightarrow 4x\left( {x - 25} \right) - 15\left( {x - 25} \right) = 0\]

Now separating the brackets,
\[ \Rightarrow \left( {x - 25} \right)\left( {4x - 15} \right) = 0\]
Equating them separately to zero,
\[4x - 15 = 0\] or \[x - 25 = 0\]
Thus value of x can be,
\[x = \dfrac{{15}}{4}\] or \[x = 25\]
Now we will check for the correct value using the time bound of tap with larger diameter,
For \[x = \dfrac{{15}}{4}\]:
\[x - 10 = \dfrac{{15}}{4} - 10
\Rightarrow x = \dfrac{{15 - 40}}{{10}}
\therefore x = \dfrac{{ - 25}}{{10}}\]
But time is never negative. So \[x = 25\] is the correct choice.

Thus a tap with smaller diameter can fill the tank in 25 long hours whereas a tap with larger diameter can fill it in 10 hours less that is in 15 hours.

Note:Here the thing that is to be noted is an obvious fact that, tap with larger diameter can perform the same task in less time so we will definitely subtract 10 hours from the time taken by tap with smaller diameter. Also note that the quadratic equation has two roots but we need a time constraint that is never negative. so we chose 25 as the correct answer. Also note that we can solve that equation by using a quadratic formula.