Problem Solving and Solving Roots
Work Problems Example Suppose that George can finish his homework in 5 hours and Carmen can finish her homework in 4 hours. How many hours will it take for them to finish their homework if they do it together?
Solution We can think of this problem as a distanceratetime problem where the distance for the whole test is 1. To find George's rate we write: d = rt 1 = (r)(5) or r = 1/5 To find Carmen's rate, write 1 = (r)(4) or r = 1/4 Hence the total rate is George's rate + Carmen's rate = 1/5 + 1/4 This is the amount of homework done in 1 hour. If together it takes x hours to do the homework then they will complete 1/x of the homework in 1 hour.
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1 1
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1 4x + 5x = 20 9x = 20 x = 20/9 = 2 2/9 It takes 2 2/9 hours for them to finish their homework together.
Example A boat travels 10 mi/hr in still water. If the boat takes the same amount of time to travel 3 miles upstream as 2 miles downstream, find the speed of the current. Let x = speed of the current. then the rate upstream is Rate Upstream = 10  x and the rate downstream is Rate Downstream = 10 + x since d = rt t = d/r Divide both sides by r For the two trips, the time is the same, so
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2 Cross multiply: 3(10  x) = 2(10 + x) 30  3x = 20 + 2x 10 = 5x x = 2. The current is going at 2 miles per hour.
Finding Roots
Examples
What is the solution to y^{2} = 3 ? Since the square of any number is positive, the above equation has no real solution. In general negative numbers do not have even roots.
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