SOLUTION: Please help me solve this word problem: An airplane can travel a distance of 6,000 km in 6 hours with the wind. The return trip against the wind takes 7.5 hours. The rate of the a

Question 202504: Please help me solve this word problem:
An airplane can travel a distance of 6,000 km in 6 hours with the wind. The return trip against the wind takes 7.5 hours. The rate of the airplane in still air is
a. 1,000 km/hr
b. 994 km/hr
c. 900 km/hr
d. 800 km/hr
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
plane travels 6000 km in 6 hours with the wind.
plane travels 6000 km in 7.5 hours against the wind.
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let p = speed of plane in still air.
let w = speed of wind
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total speed = plane speed plus or minus wind speed.
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let (p+w) = speed of plane with the wind.
let (p-w) = speed of plane against the wind.
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rate * time = distance
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with the wind:
6 * (p+w) = 6000
against the wind:
7.5 * (p-w) = 6000
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since they both = 6000, they equal each other, so:
6*(p+w) = 7.5*(p-w)
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this becomes:
6*p + 6*w = 7.5*P - 7.5*w
subtracting 6*p from both sides of this equation and adding 7.5*w to both sides of this equation gets:
1.5*p = 13.5*w
dividing both sides of this equation by 1.5 gets:
p = 9*w
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we can substitute 9*w for p in either equation.
take with the wind.
6*p + 6*w = 6000 becomes:
6*9*w + 6*w = 6000 after substituting 9*w for p.
this becomes:
54*w + 6*w = 6000 which becomes:
60*w = 6000
w = 100
substituting w = 100 in the second equation gets:
7.5*p - 7.5*(100) = 6000
adding 7.5*(100) to both sides of this equation gets:
7.5*p = 6000 + 750 = 6750
dividing both sides of this equation by 7.5 gets:
p = 6750 / 7.5 = 900
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we have:
p = 900
w = 100
with the wind, the equation becomes:
6 * (100+900) = 6 * 1000 = 6000 which checks out ok.
against the wind, the equation becomes:
7.5 * (900 - 100) = 7.5 * (800) = 6000 which checks out ok again.
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speed of the plane in still air is 900 kmph (kilometers per hour).

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