8. Rosie needs to make \$75. If she were to get a \$1.25/hr raise, it would take her 2 hours less time to make that much money. How long would she have to work at her current rate?

Since this is a word problem, we first need to define our the unknown. In this problem, they are asking how long would it take ate her current rate. So we

let   x = how long it would take her at her current rate.

Her current rate would then be   75/x.   If she worked two hours less for the \$75 her rate would be   75/(x - 2).   You would need to add \$1.25 to her current rate to get the higher rate. This gives us the equation

where we express the \$1.25 as \$5/4.

This gives us a rational equation. Clear denominators. The smallest common denominator is   4x(x - 2).   Multiply both sides.

Since the left side has two terms, we must multiply both of them.

Assemble the surviving factors.

300(x - 2) + 5x(x - 2) = 300x

Remove parentheses.

300x - 600 + 5x2 - 10x = 300x

At this point we see that we have a quadratic. To get all of the terms on one side, all we have to do is to subtract   300x   from both sides. That gives us

5x2 - 10x - 600 = 0

First factor out a factor of 5 from all of the terms on the left.

5(x2 - 2x - 120) = 0

Since 5 is not 0, we can divide both sides of the equation by 5.

x2 - 2x - 120 = 0

Factor

(x - 12)(x + 10) = 0

Set the factors equal to zero.

x - 12 = 0     x + 10 = 0

x = 12       x = -10

Check: (In the English)

If   x = 12   then she worked for 12 hours and made   \$75/12hr = \$6.25/hr.   If she had worked 2 hours less she would have made the \$75 in only 10 hours, and   \$75/10hrs = \$7.50/hr,   which would be \$1.25/hr more, and it checks.

If   x = -10,   then she worked for -10 hours. There are several ways of interpreting that. One interpretation would be that she missed 10 hours of work. But if she made \$75 by missing 10 hours of work then it is costing her   \$7.50/hr   to go to work. One wonders what she does for a living. If she had worked 2 hours less then she would have missed 12 hours of work, and if she came out \$75 ahead after missing 12 hours of work, then she would have been losing only \$6.25/hr which is \$1.25/hr better than losing \$7.50/hr. Most authors would reject this solution as being too wierd, but that is not as much fun..