Question 207776: Loren drove 200 miles at a certain rate, and his wife lois drove 100 miles at a rate 10 mph slower. If Loren had driven for the entire trip, they would have arived 30 minutes sooner. What was Loren's rate?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Loren drove 200 miles at a certain rate, and his wife lois drove 100 miles at a rate 10 mph slower. If Loren had driven for the entire trip, they would have arived 30 minutes sooner. What was Loren's rate?
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Basic formula is R * T = D where R is the rate of travel, T is the time it takes and D is the distance traveled.
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first formula is:
x * t1 = 200
where x is loren's rate of travel and t1 is the time it took loren to go 200 miles.
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second formula is:
(x-10) * t2 = 100
where (x-10) is lois' rate of travel and t2 is the time it took lois to go 100 miles.
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third formula is:
x * t3 = 300
where x is loren's rate of travel and t3 is the time it took loren to go 300 miles.
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fourth formula is:
t3 = t1 + t2 - .5
where t3 is the total time it would have taken loren to go 300 if he had driven the whole way.
note that t3 is 1/2 hour less than t1 + t2
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when they were both driving the total time it took them to go the 300 miles was x * t1 + (x-10) * t2
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if only loren drove, the total time it would have taken him was x * t3.
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you have 2 equations that are equal to 300.
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they are:
x * t3 = 300
and
(x * t1) + ((x-10) * t2) = 300
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since both these equations are equal to 300 then they are both equal to each other so you have:
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x * t3 = (x * t1) + ((x-10) * t2)
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simplifying this equation you get:
(x * t3) = (x * t1) + (x * t2) - (10 * t2)
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since t3 = t1 + t2 - .5, you can substitute in this equation to get:
(x * (t1 + t2 - .5) = (x * t1) + (x * t2) - (10 * t2)
you can simplify this equation to become:
(x * t1) + (x * t2) - (.5 * x) = (x * t1) + (x * t2) - (10 * t2)
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the (x * t1) and the (x * t2) on each side of this equation cancel out and you are left with:
- (.5 * x) = - (10 * t2)
if you multiply both sides of this equation by (-1) you get:
.5 * x = 10 * t2
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if you divide both sides of this equation by .5 you get:
x = 20 * t2
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if your remember in the beginning, we had a formula of:
(x - 10) * t2 = 100
this was lois traveling 100 miles at (x-10) miles per hour for t2 hours.
we can substitute 20 * t2 for x in this equation to get:
(20 * t2 - 10) * t2 = 100
simplifying this equation gets:
20 * t2^2 - 10 * t2 = 100
subtracting 100 from both sides of this equation gets:
20*(t2)^2 - 10*(t2) - 100 = 0
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dividing both sides of this equation by 10 gets:
2*(t2)^2 - (t2) - 10 = 0
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this factors out to be:
(2*(t2) - 5) * ((t2) + 2) = 0
which results in:
2 * (t2) = 5
or:
(t2) = -2
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since (t2) can't be negative, the only possible solution is:
2 * (t2) = 5
which results in:
t2 = 5/2
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since (x - 10) * t2 = 100 becomes (x - 10) * (5/2) = 100 we can solve for x to get:
x = 50 miles per hour
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loren's rate of travel is 50 miles per hour.
lois' rate of travel is 40 miles per hour.
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we have t2 = 5/2 hours
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since the formula of x * t1 = 200 miles becomes 50 * t1 = 200 miles, we can use this formula to solve for t1 to get:
t1 = 4 hours.
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since we know loren's rate of travel, we can use the formula for x * t3 = 300 to find t3.
this formula becomes 50 * t3 = 300 which becomes:
t3 = 300 / 50 = 6 hours.
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driving together it took them 4 hours plus 5/2 hours.
if loren drove alone, it would have taken him 6 hours.
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4 hours plus 5/2 hours = 13/2 hours = 6.5 hours
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the answers have been verified as correct.
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the answers are:
loren's rate of speed is 50 miles per hour.
lois' rate of speed is 40 miles per hours.
traveling together it took them 6.5 hours.
if loren had driven all the way it would have taken them 6 hours.
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since they only asked for loren's rate, the real answer is:
loren's rate of travel is 50 miles per hour.
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