Let’s break this problem down step-by-step.

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Step 1: Define variables

• Let $v_f$ be the speed of the faster boat in still water (this is what we need to find).
• Let $v_s$ be the speed of the slower boat in still water.
• Let $c$ be the speed of the current.

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Step 2: Downstream case

• Downstream speed for a boat = Boat speed + Current speed.

Thus:

• Speed of the faster boat downstream = $v_f + c$.
• Speed of the slower boat downstream = $v_s + c$.

The problem states:

1. The faster boat is 1.5 times as fast as the slower boat downstream: $v_f + c = 1.5 \times (v_s + c)$

2. The faster boat is ahead by 8 km for every hour: This means the difference in their downstream speeds is 8 km/h: $(v_f + c) - (v_s + c) = 8$ Simplifying:

   \tag{1}$$

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Step 3: Upstream case

• Upstream speed for a boat = Boat speed - Current speed.

Thus:

• Speed of the faster boat upstream = $v_f - c$.
• Speed of the slower boat upstream = $v_s - c$.

The problem states:

1. The faster boat covers the upstream distance (B to A) in half the time as the slower boat: $\frac{v_f - c}{v_s - c} = 2$

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Step 4: Solve the system of equations

Use Equation (1):

$v_f = v_s + 8$

Substituting $v_f = v_s + 8$ into the ratio equation:

$\frac{(v_s + 8) - c}{v_s - c} = 2$ Simplify the numerator: $\frac{v_s + 8 - c}{v_s - c} = 2$ Cross-multiply: $v_s + 8 - c = 2(v_s - c)$ Distribute on the right-hand side: $v_s + 8 - c = 2v_s - 2c$ Rearrange terms: $8 = 2v_s - v_s - 2c + c$ $8 = v_s - c$ Thus:

\tag{2}$$ --- ### Step 5: Substitute into Equation (1) From Equation (2), $$v_s = c + 8$$. Substituting into Equation (1): $$v_f = (c + 8) + 8 = c + 16$$ --- ### Step 6: Determine the speed in still water We need to determine $$v_f$$ (the speed of the faster boat in still water). Notice that $$c$$ (the current speed) cancels out across different steps. If