If Samantha and Logan can do their paper route together in 45 minutes, how many hours would it take each of them to do it alone, given that Samantha is three times as fast as Logan?
Logans rate = 1/x
Samantha's rate = 3/x
combined rate = 1/x + 3/x = 4/x
1/(4/x) = 45
x/4 = 45
x = 180
time for Logan alone = 1/(1/180) = 180 minutes
time for Samantha alone = 1/(3/18) = 60 minutes
To solve this problem, we can first find the time it takes Samantha and Logan to complete the paper route together. Let's denote Logan's speed as "L" (in routes per hour) and Samantha's speed as "3L" since Samantha is three times as fast as Logan.
Their combined speed is the sum of their individual speeds:
Combined speed = Logan's speed + Samantha's speed
Since the time taken to complete the paper route is inversely proportional to the speed, we can write:
Time taken = 1 / Combined speed
Given that it takes them 45 minutes (or 0.75 hours) to complete the paper route together, we can substitute the values:
0.75 hours = 1 / (L + 3L)
Simplifying the equation, we have:
0.75 = 1 / 4L
To find the value of L, we can solve for L:
L = 1 / (0.75 x 4)
L = 1.333
Now we have Logan's speed, L, which is 1.333 routes per hour. To find the time it takes each of them to do the paper route alone, we divide the distance (1 route) by their individual speeds:
Time taken by Logan = 1 route / 1.333 routes per hour
Time taken by Logan = 0.75 hours
Time taken by Samantha = 1 route / (3 x 1.333) routes per hour
Time taken by Samantha = 0.25 hours
Therefore, it would take Logan 0.75 hours (or 45 minutes) to do the paper route alone, while Samantha would take 0.25 hours (or 15 minutes) to do it alone.