Stephen drove at a constant speed from Town X to Town Y at 9 A.M. yesterday. Half an hour later, Cole drove from Town X to Town Y at a constant speed that was 30km/h faster than Stephen's. By 9:30 A.M., Stephen traveled 40 km. Cole caught up with Stephen at Town Y, arriving at the same time as Stephen. At what speed was Stephen driving? What was the distance between the 2 towns?
Let's solve this step-by-step:
Step 1: Convert the given information into variables.
Let's assume Stephen's speed as S km/h and the distance between Town X and Town Y as D km.
Step 2: Calculate the time taken by Stephen.
Stephen traveled 40 km in half an hour, which is 0.5 hours. So we can use the formula:
Speed = Distance / Time
S = 40 km / 0.5 hours
Step 3: Calculate Cole's speed.
Cole drove 30 km/h faster than Stephen, so his speed can be represented as (S + 30) km/h.
Step 4: Time taken by Cole.
Both Stephen and Cole arrived at Town Y at the same time. Since Cole started half an hour later, Cole's total time traveled would be 0.5 hours less than Stephen's time. So, we can represent Cole's time as (0.5 hours + Stephen's time) or (0.5 + 0.5) hours.
Step 5: Use the formula Distance = Speed * Time to calculate the distance between Town X and Town Y.
For Stephen:
D = S * 0.5
For Cole:
D = (S + 30) * (0.5 + 0.5)
Step 6: Equate the distances traveled by Stephen and Cole.
Since both Stephen and Cole traveled the same distance, we can equate the expressions from Step 5:
S * 0.5 = (S + 30) * 1
Step 7: Solve the equation.
Distribute on the right side of the equation:
S * 0.5 = S + 30
Multiply through by 2 to remove the fraction:
S = 2S + 60
Subtract 2S from both sides:
0 = S + 60
Subtract 60 from both sides:
-60 = S
Therefore, Stephen was driving at a speed of -60 km/h. However, since the speed cannot be negative, there must be an error in the data or calculations.
So, we cannot determine the speed at which Stephen was driving or the distance between the two towns with the given information.
To solve this problem, let's break it down step by step.
Step 1: Understanding the problem
We have two drivers, Stephen and Cole, who are driving from Town X to Town Y. Stephen starts at 9 A.M. and Cole starts half an hour later, driving 30 km/h faster than Stephen. By 9:30 A.M., Stephen has traveled 40 km, and Cole catches up to him at Town Y.
Step 2: Defining the variables
Let's assign variables to the unknowns in the problem. Let's call Stephen's speed "S" km/h and Cole's speed "C" km/h. The distance between Town X and Town Y will be represented by "D" km.
Step 3: Setting up the equations
We know that Stephen traveled 40 km in half an hour, so we can set up the equation:
Distance = Speed × Time
40 km = S km/h × 0.5 h
We know that Cole caught up to Stephen at Town Y, so they both took the same amount of time to reach their destination. Since Stephen drove for 30 minutes and Cole drove for the same time plus an additional half-hour, we can express this as:
Time = 0.5 h + 0.5 h = 1 h
Since Cole's speed was 30 km/h faster than Stephen's, we can express this as:
C km/h = S km/h + 30 km/h
Step 4: Solving the equations
We have two equations with two variables, so we can solve them simultaneously. Let's substitute the value of "S" from the second equation into the first equation:
40 km = (S km/h) × 0.5 h
40 km = (S + 30 km/h) × 0.5 h
Now, let's solve for "S" by multiplying both sides of the equation by 2:
80 km = S km/h + 30 km/h
Subtracting 30 km/h from both sides, we get:
80 km - 30 km = S km/h
50 km = S km/h
So Stephen was driving at a speed of 50 km/h.
Step 5: Finding the distance between the towns
Now that we know Stephen's speed, we can find the distance between the towns by using the equation:
Distance = Speed × Time
Since Stephen drove for 1 hour until Cole caught up with him, we can substitute the values into the equation:
D km = 50 km/h × 1 h
D km = 50 km
Therefore, the distance between Town X and Town Y is 50 km.