### Instruction: A motel bills its customers by charging a flat fee for the first night and then adding on a fixed amount for every night thereafter. If it costs George $\$155$ to stay in the motel for 3 nights and Noah $\$290$ to stay in the motel for 6 nights, how much is the flat fee for the first night?### Response: I notice that the difference between George's and Noah's bills is $\$135$, and the difference between their number of nights is 3.
That means the motel charges $\$135/3 = \$45$ for each additional night after the first one.
To find the flat fee for the first night, I can subtract the cost of the extra nights from either George's or Noah's bill.
For example, George's bill is $\$155$, and he stayed for 3 nights, so he paid for 2 extra nights at $\$45$ each.
Therefore, his flat fee for the first night was $\$155 - 2 \times \$45 = \$65$.
I can check this by doing the same calculation for Noah's bill.
He paid $\$290$ for 6 nights, so he paid for 5 extra nights at $\$45$ each.
His flat fee for the first night was $\$290 - 5 \times \$45 = \$65$.
I get the same answer either way, so I'm confident that the flat fee for the first night is $\$65$.