Rosa works due north of home. Her husband Ray works due east. They leave for work at the same time. By the time Rosa is 5 miles from​ home, the distance between them is one mile more than Ray​'s distance from home. How far from home is Ray Ray​?

 Ray is 12 miles from home.

Let x be Ray's distance from home.  His distance is one leg of a right triangle; 5, Rosa's distance from home, is the other.  The hypotenuse, the distance between them, is 1 more than Ray's distance from home, or x+1.  Using the Pythagorean theorem,

5²+x²=(x+1)²
25+x²=(x+1)(x+1)

Multiplying the binomials, we have
25+x²=x*x+1*x+1*x+1*1
25+x²=x²+1x+1x+1
25+x²=x²+2x+1

Subtracting x² from both sides,
25 = 2x+1

Subtracting 1 from both sides,
24 = 2x

Divide both sides by 2, and we have
12 = x

Ray is 12 miles from home.