Question 176860: A person traveling 4 hours by plane and 25 by ship covers 1580. 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 lengh of time. Find the speed of the plane and the ship.
Found 2 solutions by gonzo, MathTherapy:
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! let x = rate of plane
let y = rate of ship
rate times time = distance, so you have
4*x + 25*y = 1580
half the rate of x is x/2
1.25 * the rate of y is 1.25y
second equation is:
4*x/2 + 25*1.25y = 1315
take the first equation and solve for x or y in terms of the other.
i used x in terms of y.
i got 4*x = 1580 - 25*y
this means x = (1580-25*y)/4
i substituted that value for x in the second equation to get:
4 * (1580-25*y)/4/2 + 25*1.25*y = 1315
this became:
(1580-25*y)/2 + 31.25*y = 1315
multiply both sides of this equation by 2 to get:
1580 - 25*y + 62.5*y = 2630
combine like terms to get:
1580 + 37.5*y = 2630
subtract 1580 from both sides of this equation to get:
37.5*y = 2630 - 1580 = 1050
divide both sides of this equation by 37.5 to get:
y = 28
substitute in the first equation of:
4*x + 25*y = 1580
to get:
4*x + 25*28 = 1580
which becomes:
4*x + 700 = 1580
subtract 700 from both sides to get:
4*x = 880
divide both sides by 4 to get:
x = 220
your answer should be:
speed of plane is 220 miles per hour.
speed of boat is 28 miles per hour.
but you need to verify this is true.
substitute these values in the second equation of:
4*x/2 + 25*1.25*y = 1315
to get:
4*220/2 + 25*1.25*28 = 1315
simplify each term to get:
440 + 875 = 1315
combine like terms to get:
1315 = 1315
which is true so the values for x and y are good and your answer is:
speed of the plane is 220 miles per hour and the speed of the ship is 28 miles per hour.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! Use simultaneous equations to solve, doing the following:
Let speed of plane be p
Let speed of ship be s
Therefore, 4p + 25s = 1,580 --------- (i) x 1
And, .5(4p) + 1.25(25s) = 1,315, or 2p + 31.25s = 1,315 --------- (ii) x -2
4p + 25s = 1,580 --------- (iii)
-4p - 62.5s = -2,630 --------- (iv)
- 37.5s = - 1,050
s = 28
Substituting 28 for s in eq. (i), we get: 4p + 700 = 1,580
4p = 880
p = 220
Speed of plane is 220 mph, and speed of ship is 28 mph.
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