A boat covers 150 km downstream and 40 km upstream in 4 hours while it covers 70 km downstream and 24 km upstream in 2 hours. Determine the velocity of the current?

When a boat travels downstream (with the current), its effective speed is $(b + c)$ where $b$ is the boat's speed in still water and $c$ is the current's speed.

When traveling upstream (against the current), its effective speed is $(b - c)$.

Using Time $= \dfrac{\text{Distance}}{\text{Speed}}$

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For the first journey: 150 km downstream and 40 km upstream in 4 hours

$\dfrac{150}{b+c} + \dfrac{40}{b-c} = 4$ ... equation (1)

For the second journey: 70 km downstream and 24 km upstream in 2 hours

$\dfrac{70}{b+c} + \dfrac{24}{b-c} = 2$ ... equation (2)

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Let $x = \dfrac{1}{b+c}$ and $y = \dfrac{1}{b-c}$

The equations become:

$150x + 40y = 4$ ... (1)

$70x + 24y = 2$ ... (2)

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Multiplying equation (1) by 3:

$450x + 120y = 12$

Multiplying equation (2) by 5:

$350x + 120y = 10$

Subtracting:

$100x = 2$

$x = \dfrac{1}{50}$

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Since $x = \dfrac{1}{b+c}$:

$b + c = 50$ km/hr

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Substituting $x = \dfrac{1}{50}$ into equation (1):

$150 \cdot \dfrac{1}{50} + 40y = 4$

$3 + 40y = 4$

$40y = 1$

$y = \dfrac{1}{40}$

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Since $y = \dfrac{1}{b-c}$:

$b - c = 40$ km/hr

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From the two equations:

$b + c = 50$

$b - c = 40$

Subtracting the second from the first:

$2c = 10$

$c = 5$ km/hr

Therefore, the velocity of the current is $5$ km/hr.