MP Board · Class 10 · Mathematics · Quadratic EquationsTwo water taps together can fill a tank in $9\frac{3}{8}$ 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.

Step-by-Step Solution:

1. Given Data:

• Total time taken by both taps together to fill the tank = $9\frac{3}{8} \text{ hours} = \frac{75}{8} \text{ hours}$.

2. Assume Variables:

• Let the time taken by the smaller tap alone to fill the tank = $x \text{ hours}$.
• Then, the time taken by the larger tap alone to fill the tank = $(x - 10) \text{ hours}$.

3. Determine Work Rates per Hour:

• Part of the tank filled by the smaller tap in 1 hour = $\frac{1}{x}$
• Part of the tank filled by the larger tap in 1 hour = $\frac{1}{x - 10}$
• Part of the tank filled by both taps together in 1 hour = $\frac{1}{\frac{75}{8}} = \frac{8}{75}$

4. Formulate the Equation:

• The sum of the work done by both taps in 1 hour equals their combined work rate per hour: $$\frac{1}{x} + \frac{1}{x - 10} = \frac{8}{75}$$

5. Simplify and Solve the Equation:

• Take the LCM on the left-hand side: $$\frac{(x - 10) + x}{x(x - 10)} = \frac{8}{75}$$ $$\frac{2x - 10}{x^2 - 10x} = \frac{8}{75}$$
• Cross-multiply to clear the fractions: $$75(2x - 10) = 8(x^2 - 10x)$$ $$150x - 750 = 8x^2 - 80x$$
• Rearranging all terms to one side to form a standard quadratic equation: $$8x^2 - 80x - 150x + 750 = 0$$ $$8x^2 - 230x + 750 = 0$$
• Divide the entire equation by 2 to simplify: $$4x^2 - 115x + 375 = 0$$

6. Factorization by Splitting the Middle Term:

• We need two numbers whose product is $4 \times 375 = 1500$ and whose sum is $-115$.
• These numbers are $-100$ and $-15$ (since $(-100) \times (-15) = 1500$ and $-100 + (-15) = -115$).
• Splitting the middle term: $$4x^2 - 100x - 15x + 375 = 0$$ $$4x(x - 25) - 15(x - 25) = 0$$ $$(4x - 15)(x - 25) = 0$$

7. Find the Final Values:

• Either $4x - 15 = 0 \implies x = \frac{15}{4} = 3.75$
• Or $x - 25 = 0 \implies x = 25$

8. Test the Validity of Roots:

• If $x = 3.75$, then the time taken by the larger tap would be $3.75 - 10 = -6.25$ hours, which is negative and impossible.
• Therefore, we discard $x = 3.75$.
• Hence, $x = 25$.

Conclusion:

• Time taken by the smaller tap alone = $25 \text{ hours}$.
• Time taken by the larger tap alone = $25 - 10 =$ $15 \text{ hours}$.