Question

Each year for $4$ years, a farmer increased the number of trees in a certain orchard by $\frac{1}{4}$ of the number of trees in the orchard the preceding year. If all of the trees thrived and there were $6,250$ trees in the orchard at the end of the $4$-year period, how many trees were in the orchard at the beginning of the 4-year period?

• A. $1250$
• B. $1563$
• C. $2250$
• D. $2560$
• E. $2752$

Answer 1

The correct option is D $2560$

Say the number of trees at the beginning of the $4$ year period was $x$, then

At the end of first year, the number of trees would be $x+\frac{1}{4}x=\frac{5x}{4}$

At the end of second year, the number of trees would be ${\left(\frac{5}{4}\right)}^2\times x$

At the end of third year, the number of trees would be ${\left(\frac{5}{4}\right)}^3\times x$

At the end of fourth year, the number of trees would be ${\left(\frac{5}{4}\right)}^4\times x$

At the end of $n^{th}$ year, the number of trees would be ${\left(\frac{5}{4}\right)}^n\times x$

So, we have that ${\left(\frac{5}{4}\right)}^4\times x=6250$

$\Rightarrow \frac{5^4}{4^4}=5^4\times 10$

$\Rightarrow x=4^4\times 10$

$\Rightarrow x=2560$