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Answer :
To solve this problem, we need to find out how much we should invest today to have [tex]$20,000 in 5 years with an average annual return of 6%.
This situation involves calculating the present value of a future amount of money, which is a standard problem in finance. Here's the step-by-step solution:
1. Understanding the Formula:
The formula to calculate the present value (the initial investment required) for a future value can be rearranged from the compound interest formula. The standard formula for the future value \( FV \) with compound interest is:
\[
FV = PV \times (1 + r)^t
\]
Where:
- \( FV \) is the future value (the amount we want in the future, which is $[/tex]20,000).
- [tex]\( PV \)[/tex] is the present value (the amount we need to invest today).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal, so 6% becomes 0.06).
- [tex]\( t \)[/tex] is the time in years (5 years in this case).
2. Rearranging the Formula:
To find the present value, we need to rearrange the formula to solve for [tex]\( PV \)[/tex]:
[tex]\[
PV = \frac{FV}{(1 + r)^t}
\][/tex]
So, to calculate how much we need to invest today to achieve $20,000 in 5 years at a 6% annual return, the formula becomes:
[tex]\[
PV = \frac{20000}{(1.06)^5}
\][/tex]
3. Matching the Provided Options:
We need to find which equation from the provided options matches the form [tex]\( FV = PV \times (1 + r)^t \)[/tex]:
- Option (A) is [tex]\( 20000 = x \times (1.06)^5 \)[/tex].
- Option (B) is [tex]\( y = 20000 \times (0.06)^5 \)[/tex].
- Option (C) is [tex]\( y = 20000 \times (1.06)^5 \)[/tex].
- Option (D) is [tex]\( 20000 = x \times (0.06)^5 \)[/tex].
The correct equation to calculate how much you should invest today is:
[tex]\[
20000 = x \times (1.06)^5
\][/tex]
This is option (A).
Therefore, the correct equation to find out the amount needed to invest today is [tex]\( 20000 = x \times (1.06)^5 \)[/tex], which is option (A).
This situation involves calculating the present value of a future amount of money, which is a standard problem in finance. Here's the step-by-step solution:
1. Understanding the Formula:
The formula to calculate the present value (the initial investment required) for a future value can be rearranged from the compound interest formula. The standard formula for the future value \( FV \) with compound interest is:
\[
FV = PV \times (1 + r)^t
\]
Where:
- \( FV \) is the future value (the amount we want in the future, which is $[/tex]20,000).
- [tex]\( PV \)[/tex] is the present value (the amount we need to invest today).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal, so 6% becomes 0.06).
- [tex]\( t \)[/tex] is the time in years (5 years in this case).
2. Rearranging the Formula:
To find the present value, we need to rearrange the formula to solve for [tex]\( PV \)[/tex]:
[tex]\[
PV = \frac{FV}{(1 + r)^t}
\][/tex]
So, to calculate how much we need to invest today to achieve $20,000 in 5 years at a 6% annual return, the formula becomes:
[tex]\[
PV = \frac{20000}{(1.06)^5}
\][/tex]
3. Matching the Provided Options:
We need to find which equation from the provided options matches the form [tex]\( FV = PV \times (1 + r)^t \)[/tex]:
- Option (A) is [tex]\( 20000 = x \times (1.06)^5 \)[/tex].
- Option (B) is [tex]\( y = 20000 \times (0.06)^5 \)[/tex].
- Option (C) is [tex]\( y = 20000 \times (1.06)^5 \)[/tex].
- Option (D) is [tex]\( 20000 = x \times (0.06)^5 \)[/tex].
The correct equation to calculate how much you should invest today is:
[tex]\[
20000 = x \times (1.06)^5
\][/tex]
This is option (A).
Therefore, the correct equation to find out the amount needed to invest today is [tex]\( 20000 = x \times (1.06)^5 \)[/tex], which is option (A).
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