Question

a motorboat takes 4 hours to travel 192 kilometers going upstream. the return trip takes 3 hours going downstream. what is the rate of the boat in still water and what is the rate of the current? rate of the boat in still water: \\(\square \frac{\text{km}}{\text{h}}\\) rate of the current: \\(\square \frac{\text{km}}{\text{h}}\\)

Explanation:

Step1: Find upstream and downstream rates

Upstream rate: Distance ÷ Time = \( \frac{192}{4} = 48 \) km/h.
Downstream rate: \( \frac{192}{3} = 64 \) km/h.

Step2: Define variables and set equations

Let \( b \) = boat rate in still water, \( c \) = current rate.
Upstream: \( b - c = 48 \)
Downstream: \( b + c = 64 \)

Step3: Solve the system of equations

Add the two equations:
\( (b - c) + (b + c) = 48 + 64 \)
\( 2b = 112 \)
\( b = \frac{112}{2} = 56 \)

Substitute \( b = 56 \) into \( b + c = 64 \):
\( 56 + c = 64 \)
\( c = 64 - 56 = 8 \)

Answer:

Rate of the boat in still water: \( \boldsymbol{56} \frac{\text{km}}{\text{h}} \)
Rate of the current: \( \boldsymbol{8} \frac{\text{km}}{\text{h}} \)