Question 74258
We're going to use the equation that says Distance (D) is equal to the Rate (R) times the Time (T)
or in equation form:
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D = R*T
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On the second day she drove 270 miles in the time T. In equation form this is:
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270 = R*T
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Solve this equation for R by dividing both sides by T to get R = 270/T
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On the first day she drove 360 miles in the time T+2.  In equation form this is:
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360 = R*(T+2)
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Solve this equation for R by dividing both sides by (T+2) to get R = 360/(T+2)
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The problem then says the R is the same on both days. Since it equals 360/(T+2) on the
first day and 270/T on the second day we can set these two equal to get:
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360/(T+2) = 270/T
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Multiply both sides by (T+2) to get rid of the denominator on the left side and you get:
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360 = 270*(T+2)/T
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Then multiply both sides by T to get rid of the denominator on the right side.  The equation becomes:
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360T = 270*(T+2)
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Multiply out the right side and the equation further becomes:
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360T = 270T + 540
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Next get rid of the 270T on the right side by subtracting 270 from both sides of the equation:
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90T = 540
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Now you can solve for T by dividing both sides of the equation by 90 and the result is:
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T = 540/90 = 6
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So the time (T) which is the driving time on the second day is 6 hours.  That means that
on the first day she drove 2 additional hours for a total driving time on the first day 
of 8 hours.
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Another and quicker way you could have done this problem is to say that that on the first
day she spent 2 more hours on the road and she drove 90 more miles (360 miles on that day
and 270 miles on the second day).  So in those 2 hours she averaged 45 miles per hour to cover
those 90 miles in 2 hours.  So at 45 miles per hour it would take 8 hours to cover the 
360 miles.  On the second day she still drove at 45 mph and to cover 270 miles divide 270 by 45
and find that it took 6 hours.  
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Hope these two approaches to solving the problem give you some insight into the relationship
among distance, rate, and time.