Forty eight weeks ago, Gabriella invested money at 3.9% interest, compounded weekly. Today, her investment is worth $225.50. How much interest has Gabriella's account earned in the past 48 weeks?
PV( 1.00075)^48 ) = 225.50
PV = $217.53
To find out how much interest Gabriella's account has earned in the past 48 weeks, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the initial principal (amount invested)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, the investment is compounded weekly, so n = 52 (since there are 52 weeks in a year). The interest rate is 3.9%, which is 0.039 as a decimal, so r = 0.039. The time period is 48 weeks, so t = 48.
We can rearrange the formula to find P:
P = A / (1 + r/n)^(nt)
Substituting the given values, we have:
P = $225.50 / (1 + 0.039/52)^(52*48)
Now we can calculate P:
P = $225.50 / (1 + 0.00075)^(2496)
P ≈ $225.50 / (1.00075)^(2496)
P ≈ $225.50 / 2.403041529
P ≈ $94.00
So, Gabriella initially invested $94.00.
To find out how much interest she earned, we can subtract the initial principal from the final amount:
Interest = A - P
Interest = $225.50 - $94.00
Interest ≈ $131.50
Therefore, Gabriella's account has earned approximately $131.50 in interest over the past 48 weeks.