Steve traveled 200 miles at a certain speed Had he gone 10mp

Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.

Solution

D = R*T
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where D = the distance traveled, R = the rate or speed, and T = the time
.
The rate and the time are the two unknowns. For the 200 miles we can write the equation for
the actual trip as:
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200 = R * T <--- call this the first equation
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For the proposed trip you know that the new rate equals the old rate plus 10 mph. You can
write this new rate as (R + 10).
.
And you also know that the new time is an hour less than the old time. You can write this
new time as (T - 1)
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Therefore you can write the equation for the new trip as:
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200 = (R + 10)*(T - 1)
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Multiply the right side out to get:
.
200 = R*T - R + 10*T - 10 <--- call this the second equation
.
Since you need to solve this second equation for R, solve the first equation for T in terms of
R and then substitute that into this second equation.
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Solving the first equation for T in terms of R you divide both sides by R and you get:
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200 = T * R
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200/R = (T*R)/R
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200/R = T
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Now substitute the left side of this equation into the second equation to get:
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200 = R*(200/R) - R + 10*(200/R) - 10
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Simplify this by multiplying out the right side:
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200 = 200 - R + (2000/R) - 10
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Subtract 200 from both sides and the equation becomes:
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0 = - R + 2000/R - 10
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Multiply both sides by -R and you get:
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0 = R^2 - 2000R/R + 10R
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The middle term on the right side simplifies to -2000 and the equation becomes:
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0 = R^2 - 2000 + 10R
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Transpose this equation (switch sides) and rearrange terms so it is in the more conventional
form of:
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R^2 + 10R - 2000 = 0
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This equation factors into:
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(R + 50)*(R - 40) = 0
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This equation will be true if either factor on the left side equals zero. So set each equal
to zero and solve for R:
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R + 50 = 0
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Subtract 50 from both sides and R becomes:
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R = -50 mph
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Then set the second factor equal to zero:
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R - 40 = 0
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Add 40 to both sides to get:
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R = 40 mph
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Ignore the first answer of -50 mph because a negative speed doesn\'t really make sense.
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So the answer is 40 mph as the speed.
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Check this out. At 40 mph you drive the 200 miles in 5 hours.
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Now increase the speed by 10 mph to 50 mph. If you drive 200 miles at 50 mph it will
take 4 hours. The answer checks ... increasing the speed to 50 mph reduces the time it
takes to drive 200 miles by 1 hour.
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So the answer to the original rate is 40 mph.