SOLUTION: A bus averages 6 miles per hour faster than a trolley. If the bus travels 90 miles in the same time it takes the trolley to travel 75 miles, then what is the speed of each?

Question 1118466: A bus averages 6 miles per hour faster than a trolley. If the bus travels 90 miles in the same time it takes the trolley to travel 75 miles, then what is the speed of each?
Found 4 solutions by josgarithmetic, addingup, Theo, greenestamps:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
bus, r+6
trolley, r

Solve this for r, and evaluate r+6.
Answer by addingup(3677)   (Show Source): You can put this solution on YOUR website!
trolley: x
bus: x+6
75/x = 90/(x + 6)
75*(x + 6) = 90*x
75x + 450 = 90x
75x - 90x = -450
-15x = -450
x = 30
The trolley travels at 30 m.p.h. and the bus at 30 + 6 = 36 mph

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
rate * time= distance.

let the rate of the trolley be equal to x.

the rate of the bus is then x + 6.

the bus travels 90 miles in the same time it takes the trolley to travel 75 miles.

what is the speed of each?

let T equal the time for each.

for the bus, the equation becomes (x + 6) * T = 90

for the trolley, the equation becomes x * T = 75

for the bus, the equation becomes x * T + 6 * T = 90

since x * T = 75 from the trolley equation, replace x * T in the bus equation with 75 to get 75 + 6 * T = 90

subtract 75 from both sides of the equation to get 6 * T = 90 - 75.

this becomes 6 * T = 15.

solve for T to get T = 2.5.

when T = 2.5 hours, the bus equation of (x + 6) * T = 90 becomes (x + 6) * 2.5 = 90.

this becomes 2.5 * x + 2.5 * 6 = 90 which becomes 2.5 * x + 15 = 90.

subtract 15 from both sides of this equation to get 2.5 * x = 75.

divide both sides of this equation by 2.5 to get x 30.

that's the speed of the trolley.

the speed of the bus is 30 + 6 = 36 miles per hours.

the time is 2.5 hours.
the speed of the bus is 36 miles per hour.
in 2.5 hours, the bus travels 36 * 2.5 = 90 miles.

the time is 2.5 hours.
the speed of the trolley is 30 miles per hour.
in 2.5 hours, the trolley travels 30 * 2.5 = 75 miles.

solution looks good.

solution is the speed of the bus is 36 miles per hours and the speed of the trolley is 30 miles per hour.

this could also have been solved a different way.

start with:

for the bus, the equation becomes (x + 6) * T = 90

for the trolley, the equation becomes x * T = 75

solve for T in each equation to get:

for the bus, T = 90 / (x + 6)

for the trolley, T = 75 / x

subtract the trolley equation from the bus equation to get:

0 = 90 / (x + 6) - 75 / x

multiply both sides of the equation by x to get:

0 = 90 * x / (x + 6) - 75

multiply both sides of the equation by (x + 6) to get:

0 = 90 * x - 75 * (x + 6)

simplify to get:

0 = 90 * x - 75 * x - 75 * 6

combine like terms to get:

0 = 15 * x - 75 * 6

add 75 * 6 to both sides of the equation to get:

75 * 6 = 15 * x

divide both sides of the equation by 25 to get:

5 * 6 = x

solve for x to get x = 30

that's the speed of the trolley.

the speed of the bus is 6 more to get the speed of the bus = 36 miles per hour.

same solution, only achieved using a different method.

this second method did not require you to solve for T because T canceled out of the system of equations when you subtracted one equation from the other.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

For the same amount of time, the ratio of distances traveled is 90/:75 = 6:5.

Since the times are the same, the ratio of the speeds is also 6:5.

The two speeds we are looking for are two numbers with a difference of 6 that are in the ratio of 6:5.

That is easily done with simple mental arithmetic.

In the ratio "6:5", the difference between the two numbers is 1. To get the difference of the two numbers to be 6, simply multiply both numbers in the ratio by 6.

6:5 = 36:30

The two speeds are 36 mph and 30 mph.
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