Two cyclists leave towns apart at the same time and travel toward each other. One cyclist travels slower than the other. If they meet in hours, what is the rate of each cyclist?
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To find the rates of the two cyclists, we need to use the formula:
Distance = Rate × Time
Let's say the slower cyclist's rate is S (in miles per hour) and the faster cyclist's rate is F (in miles per hour).
We know that they travel for 2 hours and meet each other, so the total distance covered by both cyclists is the sum of their individual distances:
Distance of the slower cyclist = S × 2
Distance of the faster cyclist = F × 2
Since they are traveling towards each other, the sum of their distances must equal the total distance between the two towns. Let's call this distance D:
S × 2 + F × 2 = D
Now, since we have two unknowns (S and F), we need another equation to solve for them. This equation can be found using the relationship between their rates. We know that the slower cyclist is, well, slower than the faster cyclist. This means that the slower cyclist's rate is less than the faster cyclist's rate:
S < F
Now we have a system of equations:
S × 2 + F × 2 = D
S < F
Without any specific values for D, we can't solve this system of equations exactly. However, we can provide an example solution.
Let's say that the distance between the two towns D is 30 miles. We can then rewrite the first equation as:
2S + 2F = 30
With this additional equation, we can solve for S and F simultaneously. For example, let's assume that the slower cyclist's rate (S) is 10 mph. Substituting this value into the equation:
2(10) + 2F = 30
20 + 2F = 30
2F = 10
F = 5
Therefore, the slower cyclist travels at a rate of 10 mph, and the faster cyclist travels at a rate of 5 mph.
Please let me know if you have any further questions or if there's anything else I can assist you with!