1. **Problem Statement:** A quantity increases by 200% over 60 units of time. At the end of each year, it becomes 2.5 times the previous amount. We want to find after how many years the quantity will reach a certain value. 2. **Understanding the problem:** - Increasing by 200% means the quantity becomes 3 times the original (100% + 200% = 300% = 3 times). - The quantity grows by a factor of 2.5 each year. 3. **Formula used:** The general formula for compound growth is: $$ A = P \times r^t $$ where: - $A$ is the amount after $t$ years, - $P$ is the initial amount, - $r$ is the growth rate per year (here 2.5), - $t$ is the number of years. 4. **Setting up the equation:** We want to find $t$ such that: $$ P \times 2.5^t = 3P $$ Dividing both sides by $P$ (assuming $P \neq 0$): $$ 2.5^t = 3 $$ 5. **Solving for $t$:** Take the natural logarithm on both sides: $$ \ln(2.5^t) = \ln(3) $$ Using logarithm power rule: $$ t \ln(2.5) = \ln(3) $$ Therefore: $$ t = \frac{\ln(3)}{\ln(2.5)} $$ 6. **Calculating the value:** Using approximate values: $$ \ln(3) \approx 1.0986, \quad \ln(2.5) \approx 0.9163 $$ So: $$ t \approx \frac{1.0986}{0.9163} \approx 1.199 $$ 7. **Interpretation:** It takes approximately 1.2 years for the quantity to increase by 200% at a growth rate of 2.5 times per year. --- **Final answer:** $$ t \approx 1.2 \text{ years} $$