8. Rosie needs to make $75. At her 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.

Remove parentheses.

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

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

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

Factor

Set the factors equal to zero.

Check:

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 nonsensical.