It takes 4.5 hours for a ship traveling downriver to get from part A to port B. The return journey takes 6.3 hours. The river flows at 40 meters per minute. What is the distance between the two ports?
The Correct Answer and Explanation is:
Given:
- Time downstream (Port A to Port B): 4.5 hours
- Time upstream (Port B to Port A): 6.3 hours
- Speed of river current: 40 meters per minute
Let the speed of the ship in still water be xxx meters per minute.
When the ship is going downstream, the current aids the ship, so its effective speed is x+40x + 40x+40 meters per minute.
When the ship is going upstream, the current opposes the ship, so its effective speed is x−40x – 40x−40 meters per minute.
Let the distance between the two ports be ddd meters.
Step 1: Convert time to minutes
- Downstream time: 4.5 hours=4.5×60=2704.5 \text{ hours} = 4.5 \times 60 = 2704.5 hours=4.5×60=270 minutes
- Upstream time: 6.3 hours=6.3×60=3786.3 \text{ hours} = 6.3 \times 60 = 3786.3 hours=6.3×60=378 minutes
Step 2: Write equations
Using the formula:
Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}Distance=Speed×Time
We get two equations:
- d=(x+40)(270)d = (x + 40)(270)d=(x+40)(270)
- d=(x−40)(378)d = (x – 40)(378)d=(x−40)(378)
Since both equations equal ddd, we can set them equal to each other:(x+40)(270)=(x−40)(378)(x + 40)(270) = (x – 40)(378)(x+40)(270)=(x−40)(378)
Step 3: Solve the equation
Expand both sides:270x+10800=378x−15120270x + 10800 = 378x – 15120270x+10800=378x−15120
Bring like terms together:10800+15120=378x−270x⇒25920=108x⇒x=25920108=24010800 + 15120 = 378x – 270x \Rightarrow 25920 = 108x \Rightarrow x = \frac{25920}{108} = 24010800+15120=378x−270x⇒25920=108x⇒x=10825920=240
So, the ship’s speed in still water is 240 meters per minute.
Step 4: Find the distance
Now substitute back into one of the equations:d=(x+40)(270)=(240+40)(270)=280×270=75,600 metersd = (x + 40)(270) = (240 + 40)(270) = 280 \times 270 = 75,600 \text{ meters}d=(x+40)(270)=(240+40)(270)=280×270=75,600 meters
✅ Final Answer: The distance between the two ports is 75,600 meters.
Explanation
This problem involves relative motion and how a flowing river affects travel time depending on direction. When a ship travels on a river, the flow of the water either aids or opposes its motion. The ship’s effective speed downstream is faster because the current helps push it along, while upstream the current slows it down.
To determine the distance between the two ports, we first convert the travel times into minutes. This is necessary because the river current speed is given in meters per minute. Then we apply the basic distance formula, Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}Distance=Speed×Time, separately for downstream and upstream.
In both cases, the actual distance between the ports is the same. The only difference is how long it takes the ship to cover the distance based on whether the current helps or hinders it. This gives us two expressions for the same distance, each involving the ship’s unknown still-water speed.
By setting these two expressions equal to each other, we eliminate the unknown distance and solve for the ship’s speed in still water. Once we find that, we plug it back into either equation to get the actual distance.
This type of question helps reinforce understanding of how currents influence motion in opposite directions and how algebra can be used to find unknowns when multiple conditions apply to a single variable. The correct answer is 75,600 meters, which reflects how far apart the two ports are along the river.
