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 distance-rate-time 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.

1        1           1
+         =
5        4           x

1                 1                   1
(20x) +       (20x)  =          (20x)
5                4                    x

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

3                         2
=
10 + x                  10 - x

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

 Definition of the nth root of xWe define the nth root of x to be y if            yn = x

Examples What is the solution to

y2 = -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.