no idea. Did you plug in the numbers and see what's the result?
But I will say that starting from 2000 it takes a lot more than 5 years to get to 221 million!
FV = PV(1 + i)^nt
FV = Future Value
PV = 2000
i = 3.15/4
n = 4
t = 5
He found that the future value of this account will be $221,794,618.84. Is Carmen’s solution correct?
If not, explain what he did wrong and provide the correct solution
But I will say that starting from 2000 it takes a lot more than 5 years to get to 221 million!
The formula Carmen used is the compound interest formula:
FV = PV(1 + i)^nt
Where:
FV = Future Value
PV = Present Value (initial investment)
i = Interest rate per compounding period
n = Number of compounding periods per year
t = Number of years
Carmen correctly plugged in the given values for PV, i, n, and t.
PV = $2000
i = 3.15% (which needs to be divided by 4 to get the quarterly rate: i = 3.15/4)
n = 4 (since it is compounded quarterly)
t = 5 years
However, Carmen made an error with the calculation of i. The interest rate per compounding period should be expressed as a decimal, not a percentage. So, i = 3.15/4 = 0.7875% or 0.007875 as a decimal.
Now, let's calculate the correct future value:
FV = PV(1 + i)^nt
FV = $2000(1 + 0.007875)^(4 x 5)
FV = $2000(1.007875)^20
FV ≈ $2000(1.163698126)
FV ≈ $2,327.40
The correct solution for the future value of the account after 5 years is approximately $2,327.40. Therefore, Carmen's solution of $221,794,618.84 is incorrect.
To calculate the interest rate per quarter (i), Carmen correctly divided the annual interest rate of 3.15% by 4, since interest is compounded quarterly. However, he should have converted it to a decimal by dividing it by 100. So, the correct value for i is 0.0315/4 = 0.007875.
For the exponent, Carmen used the formula correctly, multiplying the number of quarters (n) by the number of years (t) to get nt = 4 * 5 = 20. However, he should have used the correct values for n and t, which are given in the problem as n = 4 (compounded quarterly) and t = 5 (years). Therefore, nt should be 4 * 5 = 20.
Now, let's calculate the correct future value (FV) using the corrected values:
FV = PV(1 + i)^nt
FV = $2000(1 + 0.007875)^20
FV ≈ $2000 * 1.170949071 ≈ $2,341.90
Therefore, the correct future value of the account after 5 years is approximately $2,341.90, not $221,794,618.84 as Carmen calculated.