Two Numbers in 5:3 Ratio

Date: 12/07/2002 at 00:15:59
From: Bryan Pierce
Subject: Ratios with an equation

"TWO FOR TWO"
Two numbers have a ratio of 5:3. The larger number is 36 more than 
half the difference of the two numbers. What are the two numbers?

NOTE: There are two pairs of numbers that will make this a true 
statement. You need to find both of them.

Your explanation must use algebra(variables and equations) to 
communicate the method you've used to solve the problems.

I think the equation is  36+{1(x+y)}     5x
                             2           3y

I don't know what the other equation is or how to find the numbers 
that make this a true statment.

If you could help me I would be very appreciative. Thanks.

Date: 12/08/2002 at 12:15:27
From: Doctor Mike
Subject: Re: Ratios with an equation

Hi Brian,  This may help.  
  
Let me change notation. L means the larger and S the smaller.
  
The first fact just says that L/S = 5/3. That's pretty simple.
  
BUT, the NOTE you are given hints that this is a TRICK question. Why?  
Because "half the difference of the two numbers" can mean two 
different things. The difference can be either L-S or S-L. The 
difference L-S will be positive, and S-L will be negative.
  
The second fact says (L-S)/2 = L-36 when you are looking for one of 
the solutions.
  
The second fact says (S-L)/2 = L-36 when you are looking for the other 
solution.
  
See what you can do with that. As the NOTE says, each solution will be 
a value for L and a value for S. Take the first equation together with 
ONE OF the forms of the second equation, and solve for L and S. All 
S and L values will be positive. Good luck.

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/