To solve for [tex]\( P \)[/tex] in the given equation [tex]\( A = P(1 + k)^t \)[/tex], follow these steps:
1. Identify the given equation:
[tex]\[ A = P(1 + k)^t \][/tex]
2. Rearrange the equation to solve for [tex]\( P \)[/tex]:
We want to isolate [tex]\( P \)[/tex]. To do this, we need to move everything else to the other side of the equation. Divide both sides of the equation by [tex]\( (1 + k)^t \)[/tex]:
[tex]\[ P = \frac{A}{(1 + k)^t} \][/tex]
3. Substitute example values:
Consider the following example values:
- [tex]\( A = 1000 \)[/tex]: This is the amount after time [tex]\( t \)[/tex].
- [tex]\( k = 0.05 \)[/tex]: This is the interest rate per period.
- [tex]\( t = 10 \)[/tex]: This is the number of periods.
Substitute these values into the rearranged equation:
[tex]\[ P = \frac{1000}{(1 + 0.05)^{10}} \][/tex]
4. Calculate the denominator:
First, compute [tex]\( 1 + k \)[/tex]:
[tex]\[ 1 + k = 1 + 0.05 = 1.05 \][/tex]
Then raise this to the power of [tex]\( t \)[/tex]:
[tex]\[ (1.05)^{10} \approx 1.62889 \][/tex]
5. Perform the division:
Now, divide [tex]\( A \)[/tex] by the computed value:
[tex]\[ P = \frac{1000}{1.62889} \approx 613.91 \][/tex]
Therefore, the value of [tex]\( P \)[/tex] is approximately [tex]\( 613.91 \)[/tex]. This value represents the principal amount [tex]\( P \)[/tex] before the interest was applied for 10 periods at a rate of 5% per period.
