Question:medium

A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B . If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is

Updated On: Jan 13, 2026
  • \(3(\sqrt 5-1)\)
  • \(3(3+\sqrt 5)\)
  • \(3(3-\sqrt5)\)
  • \(12(\sqrt5-2)\)
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The Correct Option is C

Solution and Explanation

Let $b$ represent the speed of the first boat, $s$ represent the speed of the second boat, and $r$ represent the speed of the river. Let $d$ be the distance between points A and B.

According to the problem statement:
$\Rightarrow d = 2(b + r)$ and $d = 3(b - r)$

Solving these equations simultaneously yields:
$\Rightarrow b + r = \frac{d}{2}$ and $b - r = \frac{d}{3}$

Subtracting the second equation from the first:
$\Rightarrow (b + r) - (b - r) = \frac{d}{2} - \frac{d}{3}$
$\Rightarrow 2r = \frac{3d - 2d}{6} = \frac{d}{6} \Rightarrow r = \frac{d}{12}$

Considering the time taken by the second boat:
$\frac{d}{s + r} + \frac{d}{s - r} = 6$

Substituting the value of $r = \frac{d}{12}$:
$\Rightarrow \frac{d}{s + \frac{d}{12}} + \frac{d}{s - \frac{d}{12}} = 6$

To simplify, multiply the numerator and denominator by 12:
Alternatively, multiply the entire equation by the least common multiple (LCM) for simplification:

Adjusting the numerator and denominator:
$\Rightarrow \frac{d(12)}{12s + d} + \frac{d(12)}{12s - d} = 6$

Multiplying both sides by $(12s + d)(12s - d)$:
$\Rightarrow 12d(12s - d) + 12d(12s + d) = 6(144s^2 - d^2)$

Simplification leads to:
$144ds - 12d^2 + 144ds + 12d^2 = 6(144s^2 - d^2)$
$\Rightarrow 288ds = 864s^2 - 6d^2$

Rearranging the terms to one side:
$\Rightarrow 144s^2 - 48ds - d^2 = 0$

Solving this quadratic equation for $s$ using the quadratic formula:
$s = \frac{48d + \sqrt{(48d)^2 + 4(144)(d^2)}}{2 \cdot 144}$
$= d\left(\frac{48 + \sqrt{48^2 + 4 \cdot 144}}{2 \cdot 144}\right)$

Further simplification yields:
$s = d\left(\frac{1}{6} + \frac{\sqrt{5}}{12}\right)$

Now, calculate the required value:
$\frac{d}{s + r} = \frac{d}{\frac{d}{6} + \frac{d\sqrt{5}}{12} + \frac{d}{12}}$

$\Rightarrow \frac{1}{\frac{1}{6} + \frac{\sqrt{5}}{12} + \frac{1}{12}} = \frac{1}{\frac{3 + \sqrt{5}}{12}} = \frac{12}{3 + \sqrt{5}}$

Rationalizing the denominator:
$\Rightarrow \frac{12}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{12(3 - \sqrt{5})}{9 - 5} = \frac{12(3 - \sqrt{5})}{4}$

$\Rightarrow 3(3 - \sqrt{5})$

Thus, the correct option is (C): $\boxed{3(3 - \sqrt{5})}$

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The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
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