At 9:00 on Saturday morning, two bicyclists heading in opposite directions pass each other on a bicycle path.The bicyclist heading north is riding 6 km/hour faster than the bicyclist heading south. At 10:15, they are 42.5 km apart. Find the two bicyclists’ rates.
Let's represent the speed of the bicyclist heading south as x km/hour.
Since the bicyclist heading north is riding 6 km/hour faster, their speed can be represented as (x+6) km/hour.
From 9:00am to 10:15am, which is a time span of 1 hour and 15 minutes, the combined distance the two bicyclists travel is 42.5 km.
To calculate their combined distance, we can use the formula: Distance = speed * time.
For the bicyclist heading south, the distance traveled is x * 1.25 hours (since 1 hour and 15 minutes is equal to 1.25 hours).
For the bicyclist heading north, the distance traveled is (x+6) * 1.25 hours.
Adding these two distances together gives us the equation:
x * 1.25 + (x+6) * 1.25 = 42.5
Expanding the equation, we get:
1.25x + 1.25x + 7.5 = 42.5
Combine like terms:
2.5x + 7.5 = 42.5
Subtract 7.5 from both sides:
2.5x = 35
Divide both sides by 2.5:
x = 14
Therefore, the speed of the bicyclist heading south is 14 km/hour, and the speed of the bicyclist heading north is 14 + 6 = 20 km/hour.
Let's set up equations to solve this problem step by step:
Let's assume the rate of the bicyclist heading south is "x" km/hour.
The rate of the bicyclist heading north is then "x + 6" km/hour (6 km/hour faster than the southbound cyclist).
Now let's calculate the time they have been traveling:
From 9:00 to 10:15 is a time difference of 1 hour and 15 minutes, which is equivalent to 1.25 hours.
Now let's calculate the distance traveled by each cyclist:
The distance traveled by the southbound cyclist is "x * 1.25" km (rate multiplied by time).
The distance traveled by the northbound cyclist is "(x + 6) * 1.25" km.
According to the problem, the total distance between them at 10:15 is 42.5 km.
Therefore, we can set up the equation:
x * 1.25 + (x + 6) * 1.25 = 42.5
Simplifying the equation:
1.25x + 1.25(x + 6) = 42.5
1.25x + 1.25x + 7.5 = 42.5
2.5x + 7.5 = 42.5
2.5x = 35
x = 14
So the rate of the southbound cyclist is 14 km/hour.
Substituting this value back into the equation to find the rate of the northbound cyclist:
x + 6 = 14 + 6 = 20
So the rate of the northbound cyclist is 20 km/hour.