To find the return needed for the couple's investment to grow from $6,866.00 to $10,774.00 in one year, we can use the formula for calculating the future value of an investment based on its present value and return rate.
The formula for the future value (FV) can be expressed as:
\[ FV = PV \times (1 + r) \]
Where:
- \( FV \) is the future value ($10,774.00)
- \( PV \) is the present value ($6,866.00)
- \( r \) is the return rate (as a decimal)
We need to solve for \( r \):
- Plug in the values into the formula:
\[ 10,774 = 6,866 \times (1 + r) \]
- Divide both sides by $6,866 to isolate \( (1 + r) \):
\[ 1 + r = \frac{10,774}{6,866} \]
- Calculate the right side:
\[ 1 + r = 1.566 \]
- Subtract 1 from both sides to solve for \( r \):
\[ r = 1.566 - 1 \]
\[ r = 0.566 \]
- Convert \( r \) to a percentage by multiplying by 100:
\[ r = 0.566 \times 100 = 56.6% \]
Therefore, the mutual fund must generate a return of 56.6% over the year for the couple to reach their goal of $10,774.00.