SOLUTION: hi to all of you who help. can you please help me? A person traveling 4 hr by plane and 25 hr by ship covers 1580 miles. if the speed of the plane had been one-half of the actual

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Question 161627: hi to all of you who help. can you please help me? A person traveling 4 hr by plane and 25 hr by ship covers 1580 miles. if the speed of the plane had been one-half of the actual speed and the speed of the ship had been one-fourth greater, the person would have traveled only 1315 miles in the same length of time.find the speeds of the plane and ship.thanks for the help.
Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
let x = rate of the plane.
let y = rate of the ship.
first equation is:
4*x + 25*y = 1580.
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since the plane traveled 4 hours at half rate of speed, and the ship traveled 25 hours at 1.25 rate of speed, the second equation becomes
2*x + 31.25*y = 1315.
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this is because 4 * x/2 = 2*x, and 25 * (5/4*y) = 125/4 * y = 31.25 * y.
the distance traveled under these new rates of speed is 1315.
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we have 2 equations with 2 unknowns.
4*x + 25*y = 1580
2*x + 31.25*y = 1315.
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multiply the bottom equation by 2 so we can eliminate the x.
equations become
4*x + 25*y = 1580
4*x + 62.5*y = 2630
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subtract top equation from bottom equation to get
37.5*y = 1050
y = 28
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if y = 28, we can solve for x in either equation.
use 4*x + 25*y = 1580
this becomes
4*x + 700 = 1580
4*x = 880
x = 220
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x = 220 = rate of plane
y = 28 = rate of ship
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to prove, take 1/2 the rate of the plane and 1.25 times the rate of the ship to solve the second equation.
1/2 * 220 = 110.
5/4 * 28 = 35
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4*110 + 25*35 = 1315 which proves the values of x and y are good in both equations.
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answer is:
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plane travels at 220 miles per hour for 4 hours.
ship travels at 28 miles per hour for 25 hours.
total distance traveled is 1580 miles.
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plane travels at 110 miles per hour for 4 hours.
ship travels at 35 miles per hour for 25 hours.
total distance traveled is 1315 miles.
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110 is half of 220.
35 is 5/4 * 28.
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x = 220
y = 28