Arithmetic

Time, Speed & Distance

Hard

Given below are two statements based on the following:

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours.

Statement I: Speed of the boat in still water is 12 km per hour.

Statement II: Speed of the stream is 4 km per hour.

In the light of the above statements, choose the correct answer from the options given below:

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is false but Statement II is true

Statement I is true but Statement II is false

✅ Correct Option: 3

Let speed in still water = $x$ km/hr

Stream speed = $y$ km/hr

From first journey (30 km upstream + 28 km downstream in 7 hours):

$\frac{30}{x-y} + \frac{28}{x+y} = 7$ ...(1)

From second journey (21 km upstream + 21 km downstream in 5 hours):

$\frac{21}{x-y} + \frac{21}{x+y} = 5$ ...(2)

For Statement II ( $y = 4$ ):

From equation (2):

$\frac{21}{x-4} + \frac{21}{x+4} = 5$

$\frac{42x}{(x-4)(x+4)} = 5$

$42x = 5(x^2-16)$

$5x^2 - 42x - 80 = 0$

Solving quadratic:

$a = 5$ , $b = -42$ , $c = -80$

$x = \frac{-b ± \sqrt{b^2-4ac}}{2a}$

$x = \frac{42 ± \sqrt{1764+1600}}{10}$

$x = \frac{42 ± \sqrt{3364}}{10}$

$x = \frac{42 ± 58}{10}$

$x = 10$ (ignore the negative alternative)

Verifying $x = 10$ and $y = 4$ in equation (1):

$\frac{30}{6} + \frac{28}{14} = 5 + 2 = 7$ ✓

Therefore Statement II ( $y = 4$ ) is true and $x = 10$ (not 12)

So Statement I ( $x = 12$ ) is false.

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