Elaine takes 77 seconds to climb up a flight of 50 steps if she runs up 20 steps and walks up the rest of the way. She will take 98.5 seconds to climb the flight of 50 steps if she runs up 10 steps and walks the rest of the way. How long will Elaine take if she runs all the way up the 50 steps? (Assuming that Elaine’s running and walking speeds are constant.)
Question
Elaine takes 77 seconds to climb up a flight of 50 steps if she runs up 20 steps and walks up the rest of the way. She will take 98.5 seconds to climb the flight of 50 steps if she runs up 10 steps and walks the rest of the way. How long will Elaine take if she runs all the way up the 50 steps? (Assuming that Elaine’s running and walking speeds are constant.)
Solution
Let's denote the time Elaine takes to run up one step as R and the time she takes to walk up one step as W.
From the problem, we have two equations based on the two scenarios given:
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20R + 30W = 77 (This is from the scenario where Elaine runs up 20 steps and walks up the rest)
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10R + 40W = 98.5 (This is from the scenario where Elaine runs up 10 steps and walks up the rest)
We can solve this system of equations to find the values of R and W.
First, let's multiply the first equation by 2 and the second equation by 4:
2*(20R + 30W) = 277 4(10R + 40W) = 4*98.5
This gives us:
40R + 60W = 154 40R + 160W = 394
Now, let's subtract the first equation from the second:
100W = 240
Solving for W, we get:
W = 240 / 100 = 2.4 seconds
Substitute W = 2.4 into the first equation:
20R + 30*2.4 = 77 20R + 72 = 77 20R = 77 - 72 20R = 5 R = 5 / 20 = 0.25 seconds
Now that we know R and W, we can find out how long Elaine will take if she runs all the way up the 50 steps:
50R = 50*0.25 = 12.5 seconds
So, Elaine will take 12.5 seconds to run up all 50 steps.
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