2013 AMC 10A Problem 2

The question is really "how many quarter cups make $2\tfrac12$ cups?" Count them piece by piece.

1. One whole cup holds $4$ quarter cups.
2. Two whole cups hold $2 \times 4 = 8$ quarter cups.
3. The extra half cup is $2$ more quarter cups, because two quarters make a half.
4. Altogether: $8 + 2 = 10$ fills.

The same thing with fractions. $2\tfrac12 = \tfrac52$ cups. Dividing by $\tfrac14$ is the same as multiplying by $4$:
$$\frac52 \div \frac14 = \frac52 \times 4 = 10.$$

Check. Ten quarter cups is $\tfrac{10}{4} = 2\tfrac12$ cups. That is exactly what the recipe needs.

Alice must fill the cup $\boxed{\textbf{(B)}\ 10}$ times.

"How many small pieces make the total?" is always a division: total $\div$ size of one piece. When the piece is a fraction, dividing by it means multiplying by its flip ($\div \tfrac14$ is the same as $\times 4$). If fractions feel shaky, count in the small unit directly, as in Steps 1 to 3. Both ways must agree.

Write the required amount as an improper fraction: $2\tfrac12 = \tfrac52$ cups. Each fill delivers $\tfrac14$ cup, so the number of fills is
$$\frac{5/2}{1/4} = \frac52 \cdot 4 = 10.$$

Check: $10$ quarter-cups make $\tfrac{10}{4} = 2\tfrac12$ cups.

Alice must fill the cup $\boxed{\textbf{(B)}\ 10}$ times.