2023 AMC 10B Problem 4

The painted strip is a long, thin rectangle, so its area is width times length. The problem mixes three units, so turn everything into centimeters first.

1. Width. There are $10$ millimeters in a centimeter, so
$$6.5 \text{ mm} = 0.65 \text{ cm}.$$
2. Length. There are $100$ centimeters in a meter, so
$$25 \text{ m} = 2500 \text{ cm}.$$
3. Area. Multiply:
$$0.65 \times 2500 = 1625 \text{ square centimeters}.$$
(One easy way: $0.65 \times 2500 = 65 \times 25 = 1625$.)

The answer is $\boxed{\textbf{(C)}\ 1{,}625}$.

When the answer must be in a certain unit of area, put every length in the matching unit of length first. Then the area formula gives the right unit with no more work. Changing units after you already have an area is where mistakes happen, because $1$ square centimeter is $100$ square millimeters, not $10$.

Estimate. The strip is a little more than half a centimeter wide and $2500$ cm long, so the area is a bit more than half of $2500$, which is $1250$, but less than $2500$. Only $1{,}625$ is in that range.

The painted strip is a long thin rectangle. Convert both sides to centimeters before multiplying:
$$6.5 \text{ mm} = 0.65 \text{ cm}, \qquad 25 \text{ m} = 2500 \text{ cm}.$$
The area is
$$0.65 \times 2500 = 1625 \text{ cm}^2.$$

The answer is $\boxed{\textbf{(C)}\ 1{,}625}$.