The owner of two hotels is ordering towels. He bought 68 hand towels and 55 bath towels for his hotel in Stamford, spending a total of $686. He also ordered 66 hand towels and 80 bath towels for his hotel in Hampton, spending $932. How much does a hand towel and how much does a bath towel cost?
Let's represent the cost of a hand towel as "h" and the cost of a bath towel as "b".
From the first case, we can create the equation 68h + 55b = 686.
From the second case, we can create the equation 66h + 80b = 932.
To solve this system of equations, we can use the substitution method.
We solve the first equation for h: h = (686 - 55b) / 68.
We substitute this expression for h into the second equation: 66((686 - 55b) / 68) + 80b = 932.
We simplify the second equation: (44826 - 3630b) / 68 + 80b = 932.
We multiply each term in the second equation by 68 to eliminate the denominator: 44826 - 3630b + 5440b = 63376.
We combine like terms in the second equation: 1810b + 44826 = 63376.
We subtract 44826 from both sides of the equation: 1810b = 18550.
We divide both sides of the equation by 1810: b = 10.
We substitute this value for b into the first equation: 68h + 55(10) = 686.
We simplify the first equation: 68h + 550 = 686.
We subtract 550 from both sides of the equation: 68h = 136.
We divide both sides of the equation by 68: h = 2.
A hand towel costs $2 and a bath towel costs $10. Answer: \boxed{2, 10}.