1. **State the problem:** We have a 3 3/4-inch candle that burns down in 3 hours. We want to find how long a 6 1/4-inch candle of the same type will take to burn down. 2. **Formula and concept:** Since the candles burn at a rate proportional to their length, the time to burn is directly proportional to the candle's length. 3. **Convert mixed numbers to improper fractions:** - 3 3/4 inches = $3 + \frac{3}{4} = \frac{12}{4} + \frac{3}{4} = \frac{15}{4}$ inches - 6 1/4 inches = $6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4}$ inches 4. **Set up the proportion:** $$\frac{\text{time}_1}{\text{length}_1} = \frac{\text{time}_2}{\text{length}_2}$$ Substitute known values: $$\frac{3}{\frac{15}{4}} = \frac{t}{\frac{25}{4}}$$ 5. **Solve for $t$:** Multiply both sides by $\frac{25}{4}$: $$t = 3 \times \frac{25}{4} \div \frac{15}{4}$$ Rewrite division as multiplication by reciprocal: $$t = 3 \times \frac{25}{4} \times \frac{4}{15}$$ 6. **Simplify:** Cancel $4$ in numerator and denominator: $$t = 3 \times \frac{25}{\cancel{4}} \times \frac{\cancel{4}}{15} = 3 \times \frac{25}{15}$$ Simplify $\frac{25}{15}$ by dividing numerator and denominator by 5: $$\frac{25}{15} = \frac{\cancel{5} \times 5}{\cancel{5} \times 3} = \frac{5}{3}$$ So, $$t = 3 \times \frac{5}{3}$$ Cancel 3: $$t = \cancel{3} \times \frac{5}{\cancel{3}} = 5$$ 7. **Final answer:** It will take 5 hours for the 6 1/4-inch candle to burn down.