We are asked to find the principal amount that needs to be invested to obtain a future value of $17,500 in 10 months with a simple annual interest rate of 4%.

1. Problem Description

We are asked to find the principal amount that needs to be invested to obtain a future value of $17,500 in 10 months with a simple annual interest rate of 4%.

2. Solution Steps

Let $P$ be the principal amount.

Let $r$ be the annual interest rate. $r = 4\% = 0.04$.

Let $t$ be the time in years. Since the time is 10 months, $t = \frac{10}{12}$ years.

Let $A$ be the future value of the investment. A = \17,500$.

The formula for simple interest is

$A = P(1 + rt)$.

We want to find $P$, so we can rearrange the formula as

$P = \frac{A}{1 + rt}$.

Substituting the given values, we get

$P = \frac{17500}{1 + 0.04 \cdot \frac{10}{12}}$.

$P = \frac{17500}{1 + 0.04 \cdot \frac{5}{6}}$.

$P = \frac{17500}{1 + \frac{0.2}{6}}$.

$P = \frac{17500}{1 + \frac{1}{30}}$.

$P = \frac{17500}{\frac{31}{30}}$.

$P = 17500 \cdot \frac{30}{31}$.

$P = \frac{525000}{31}$.

$P \approx 16935.48387$.

Rounding to two decimal places, $P \approx 16935.48$.

3. Final Answer

$16935.48