# Ex.4.3 Q9 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

Two water taps together can fill a tank in \begin{align} 9\frac{3}{8} \end{align} hours. The tap of larger diameter takes $$10$$ hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank?

Video Solution
Ex 4.3 | Question 9

## Text Solution

What is known?

i) Two water taps together can fill the tank in \begin{align} 9\frac{3}{8} \end{align} hours.

ii) The tap of larger diameter takes $$10$$ hours less than the smaller one to fill the tank separately.

What is the Unknown?

Time taken by smaller tap and larger tap to fill the tank separately.

Reasoning:

Let the tap of smaller diameter fill the tank in $$x$$ hours.

Tap of larger diameter takes $$\left( {x - 10} \right)$$ hours.

In $$x$$ hours, smaller tap fills the tank.

In one hour, part of tank filled by the smaller tap \begin{align} =\frac{1}{x} \end{align}

In $$\left( {x - 10} \right)$$ hours, larger tap fills the tank.

In one hour, part of tank filled by the larger tap\begin{align} = \frac{1}{{\left( {x - 10} \right)}} \end{align}

In $$1$$ hours, the part of the tank filled by the smaller and larger tap together:

\begin{align}\frac{1}{x} + \frac{1}{{x - 10}}\end{align}

\begin{align}\therefore \quad \frac{1}{x} + \frac{1}{{x - 10}} = \frac{1}{{9\frac{3}{8}}} \end{align}

Steps:

\begin{align}\frac{1}{x} + \frac{1}{{x - 10}} = \frac{1}{{\frac{{75}}{8}}}\end{align}

By taking LCM and cross multiplying:

\begin{align}\frac{{x - 10 + x}}{{x(x - 10)}} &= \frac{8}{{75}}\\ \frac{{2x - 10}}{{{x^2} - 10x}} &= \frac{8}{{75}}\\75\left( {2x - 10} \right) &= 8\left( {{x^2} - 10x} \right)\\150x - 750x &= 8{x^2} - 80x\\8{x^2} - 80x - 150x + 750& = 0\\8{x^2} - 230x + 750& = 0\\4{x^2} - 115x + 375& = 0\end{align}

Comparing with $$ax^\text{2}+bx+c=0$$

$a = 4,\;b = - 115,\;c = 375$

\begin{align} b{}^{2}-4ac&={{\left( -115 \right)}^{2}}-4\left( 4 \right)\left( 375 \right) \\ & =13225-6000 \\ & =7225 \\ b{}^{2}-4ac&>0\end{align}

$$\therefore$$ Real roots exist.

\begin{align}{{x}} &= \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\\ {{x}} &= \frac{{115 \pm \sqrt {7225} }}{8}\\{{x}} &= \frac{{115 + 85}}{8} \qquad {{x}} = \frac{{115 - 85}}{8}\\{{x}} &= \frac{{200}}{8} \qquad \;\qquad {{x}} = \frac{{30}}{8}\\{{x}} &= 25 \qquad\;\;\; \qquad {{x}} = 3.75\end{align}

$$x$$ cannot be $$3.75$$ hours because the larger tap takes $$10$$ hours less than $$x$$

Time taken by smaller tap $$x = 25$$ hours

Time taken by larger tap $$(x - 10) =15$$ hours.

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