A papaya tree was planted 2 years ago. It grows at the rate of 20% every year. If at present, the height of the tree is 540 cm, what was it when the tree was planted?

### Concept & Formula This is a reverse successive percentage problem. We are given the final value after a period of compound growth and need to work backward to find the initial base value. Using the standard compound growth formula: $$ \text{Final Value} = \text{Initial Value} \times \left(1 + \frac{R}{100}\right)^n $$ ### Step-by-Step Solution * **Given:** Final (present) height = $540\text{ cm}$. Growth rate = $20\%$ per year. Time = $2$ years. * **Let the initial height be $x$.** * **Determine Fractional Multiplier:** A $20\%$ increase equals a multiplier of $120\%$ or $\frac{120}{100} = \frac{6}{5}$. * **Set up the equation:** $$ x \times \left(\frac{6}{5}\right)^2 = 540 $$ $$ x \times \frac{36}{25} = 540 $$ * **Isolate $x$:** $$ x = 540 \times \frac{25}{36} $$ * **Simplify:** Notice that $540 \div 36 = 15$ (since $36 \times 10 = 360$ and $36 \times 5 = 180$). $$ x = 15 \times 25 $$ $$ x = 375\text{ cm} $$ ### Exam Strategy & Shortcut **The Ratio Method:** A $20\%$ increase means the ratio of old height to new height in one year is $5 : 6$. Over 2 years, the ratio compounds by squaring: $\text{Initial} : \text{Final} = 5^2 : 6^2 = 25 : 36$. We are given the final height is $540$. Set up the proportion: $36\text{ units} = 540$. $1\text{ unit} = \frac{540}{36} = 15$. The initial height is $25\text{ units}$. $25 \times 15 = 375$. This avoids algebraic rearrangement entirely. ### Common Pitfall A common trap is applying a $20\%$ decrease twice to the final value of $540$ to go backward (i.e., calculating $540 \times 0.8 \times 0.8 = 345.6$). This is mathematically incorrect because the $20\%$ growth was calculated on the *smaller initial height*, not on the current larger height. Always use the multiplier equation and isolate the unknown variable. ### Final Answer **Therefore, the correct answer is 375 cm.**