Carmen is planning to invest $2000 in an account earning 3.15% interest, compounded quarterly, for 5 years. He used the following formula and variables to solve for the future value of the account after 5 years.
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
Carmen's solution is incorrect. To understand why, let's break down the formula used and identify the error.
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.
Carmen's solution is not correct. The error lies in the calculation of the interest rate (i) and the exponent (nt).
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.
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!