A tree grows only $\frac{3}{5}$ as fast as the one beside it. In four years the combined growth of the trees is eight feet. How much does the shorter tree grow in 2 years?

### Concept & Logic This problem tests relative rates and linear equations. By defining the growth rate of the faster tree, we can express the growth rate of the slower tree as a fraction of it. Setting up an equation for their combined growth over a specific time period allows us to find their individual annual growth rates. ### Step-by-Step Solution * **Given:** Combined growth in 4 years = 8 feet. Shorter tree's rate is $\frac{3}{5}$ of the taller tree's rate. * **Calculation:** Let the annual growth rate of the taller tree be $x$ feet/year. The annual growth rate of the shorter tree is $\frac{3}{5}x$ feet/year. * Form an equation for their combined growth over 4 years: $$ 4x + 4\left(\frac{3}{5}x\right) = 8 $$ $$ x + \frac{3}{5}x = 2 \text{ (Dividing the entire equation by 4)} $$ $$ \frac{8}{5}x = 2 $$ $$ x = 2 \times \frac{5}{8} = \frac{10}{8} = 1.25 \text{ feet/year} $$ * Find the annual growth rate of the shorter tree: $$ \text{Shorter tree rate} = \frac{3}{5} \times 1.25 = 0.75 \text{ feet/year} $$ * Calculate the shorter tree's growth in 2 years: $$ \text{Growth} = 2 \times 0.75 = 1.5 \text{ feet} $$ * Compare 1.5 feet to the given options. 1.5 feet falls into the category of "Less than 2 feet". ### Exam Strategy & Shortcut Use the **Ratio Allocation Method**. The growth ratio of shorter to taller is $3 : 5$. Total combined growth in 4 years is 8 feet, meaning combined annual growth is $8 \div 4 = 2$ feet/year. Divide this 2 feet/year into the $3 : 5$ ratio (total 8 parts). 1 part = $\frac{2}{8} = 0.25$ feet. Shorter tree annual growth (3 parts) = $3 \times 0.25 = 0.75$ feet/year. In 2 years, it grows $2 \times 0.75 = 1.5$ feet. ### Common Pitfall Students often solve for the taller tree's growth or forget to multiply the final annual rate by the required 2 years, leading them to search the options for 0.75 or 1.25, and guessing incorrectly when they don't find an exact match. ### Final Answer Therefore, the correct answer is **Less than 2 feet**.