2008 AMC 10A Problem 4

1. First turn the given fact into plain numbers. Two thirds of $10$ bananas is $\frac{2}{3} \times 10 = \frac{20}{3}$ bananas. These are worth $8$ oranges.
2. Find the value of one banana, because then any number of bananas is easy. Divide the $8$ oranges among the $\frac{20}{3}$ bananas:
$$8 \div \frac{20}{3} = 8 \times \frac{3}{20} = \frac{24}{20} = \frac{6}{5}\ \text{orange per banana}.$$
3. Now the question. Half of $5$ bananas is $\frac{5}{2}$ bananas.
4. Each banana is worth $\frac{6}{5}$ of an orange, so
$$\frac{5}{2} \times \frac{6}{5} = \frac{30}{10} = 3\ \text{oranges}.$$

The answer is $\boxed{\textbf{(C)}\ 3}$.

"Worth as much as" is like an exchange rate between two currencies. Work out what one unit is worth, then multiply by how many units you have. Keeping fractions as fractions (instead of decimals) lets the numbers cancel neatly.

Scale the given fact straight to the amount asked about. We know $\frac{20}{3}$ bananas are worth $8$ oranges. The target is $\frac{5}{2}$ bananas, and $\frac{5}{2} \div \frac{20}{3} = \frac{5}{2} \times \frac{3}{20} = \frac{3}{8}$. So we want $\frac{3}{8}$ of the bananas, which are worth $\frac{3}{8}$ of the oranges: $\frac{3}{8} \times 8 = 3$.

Two thirds of $10$ bananas is $\frac{20}{3}$ bananas, and this equals $8$ oranges in value. So one banana is worth
$$8 \div \frac{20}{3} = \frac{24}{20} = \frac{6}{5}\ \text{oranges}.$$

Half of $5$ bananas is $\frac{5}{2}$ bananas, worth
$$\frac{5}{2} \cdot \frac{6}{5} = 3\ \text{oranges}.$$

The answer is $\boxed{\textbf{(C)}\ 3}$.