### Instruction: The ratio of the length of a rectangular room to its width is $5:3$. The perimeter of the room is 48 feet. What is the area of the room?### Response: I want to find the area of the room, which is the product of the length and the width.
But I don't know the exact values of the length and the width, only their ratio.
Maybe I can use the ratio to write the length and the width in terms of a common variable.
Let's say the length is 5x and the width is 3x, where x is some positive constant.
Then, the perimeter of the room is the sum of twice the length and twice the width, or 2(5x) + 2(3x).
I know the perimeter is 48 feet, so I can set up an equation: 2(5x) + 2(3x) = 48.
Simplifying, I get 16x = 48, so x = 3.
Now I can plug in x = 3 into the expressions for the length and the width.
The length is 5x = 5(3) = 15 feet.
The width is 3x = 3(3) = 9 feet.
The area of the room is the product of the length and the width, or 15 * 9.