finding area

Question

I have a piece of paper with a solid strip along the top and the left side. The stripes take up a third of the parper. The paper measures 5 inches by 8 inches. What will the widthe of the stripes be?

Answer 1

You don't say if the width of the stripe is the same at the top and at the side.  If it isn't, there isn't enough information to solve this, so let's assume it is.  Call that width w.

 

First of all, we know that the area of the whole sheet is 5 * 8 = 40.

 

Look at the area of the part of the paper NOT taken up by the stripes.  Since the full with is 5, the width of the part not in the stripe is 5 - w.  And similarly, the length of the part not in the stripe is 8 - w.  So the total area of the part not in the stripes is (5-w)(8-w) = 40 -13w +w². 

 

Since the stripe takes up 1/3 of the paper, this open area takes up the other 2/3.  So the area of this part is 2/3 * 40 = 80/3.

 

So now we have an equation:  40 -13w +w² = 80/3.

 

Subtract 80/3 from both sides and you get 40/3 -13w + w² = 0.

 

To get rid of the fraction, multiply both sides by 3:

40 - 39w + 3w² = 0

 

That's a quadratic equation that you have to solve using the quadratic formula -- the expression for w doesn't factor.

 

When you do this, you'll get two values.  Which one do you choose?  I bet that one of them won't make sense.  For example, it may be negative, or it may be bigger than the whole width of the paper.  That often happens when you have a word problem that leads to a quadratic equation.  If you can, you rule out the solution that doesn't make sense, and the actual answer is the other solution.