2009 AMC 10B Problem 2

A fraction built out of fractions is easiest if you simplify the top and the bottom on their own before dividing.

1. Top: $\frac13 - \frac14$. The common denominator is $12$, so this is $\frac{4}{12} - \frac{3}{12} = \frac{1}{12}$.
2. Bottom: $\frac12 - \frac13$. The common denominator is $6$, so this is $\frac{3}{6} - \frac{2}{6} = \frac{1}{6}$.
3. Now divide: $\frac{1}{12} \div \frac{1}{6}$. Dividing by $\frac16$ is the same as multiplying by $6$, so this is $\frac{6}{12} = \frac12$.
4. Quick sense check: a twelfth of a pizza is half of a sixth of a pizza, so the answer $\frac12$ makes sense.

The answer is $\boxed{\textbf{(C)}\ \frac12}$.

Stacked fractions look scary, but each layer is an ordinary subtraction. Simplify each layer to a single fraction, then divide by flipping the bottom fraction and multiplying. Keeping the numbers small at each step makes mistakes less likely.

Clear all the small fractions at once. Multiply the top and the bottom by $12$, which does not change the value:

• Top: $12 \times \frac13 - 12 \times \frac14 = 4 - 3 = 1$.
• Bottom: $12 \times \frac12 - 12 \times \frac13 = 6 - 4 = 2$.

So the whole thing equals $\frac12$.

Simplify numerator and denominator separately:
$$\frac13 - \frac14 = \frac{4 - 3}{12} = \frac{1}{12}, \qquad \frac12 - \frac13 = \frac{3 - 2}{6} = \frac16.$$
Dividing, $\dfrac{1/12}{1/6} = \dfrac{1}{12} \cdot 6 = \dfrac{1}{2}$.

The answer is $\boxed{\textbf{(C)}\ \frac12}$.