Enrico invested money at 3.95%, compounded weekly, while Paul invested money at 3.95%, compounded semi-annually. After 3.5 years, each investment is worth $4000. Who made the greater original investment, and by how much was it greater?
To find out who made the greater original investment and by how much, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (original investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
For Enrico's investment:
A = $4000
r = 3.95% = 0.0395 (in decimal form)
n = 52 (weekly compounding, 52 weeks in a year)
t = 3.5 years
$4000 = P(1 + 0.0395/52)^(52*3.5)
For Paul's investment:
A = $4000
r = 3.95% = 0.0395 (in decimal form)
n = 2 (semi-annual compounding, 2 times in a year)
t = 3.5 years
$4000 = P(1 + 0.0395/2)^(2*3.5)
Let's solve these equations to find P for both cases.
To determine who made the greater original investment, we need to find the initial principal amount for each investment.
Let's start by calculating the future value of Enrico's investment after 3.5 years using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Future value
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
For Enrico's investment, r = 3.95%, compounded weekly. Hence, we have:
A = 4000
r = 3.95% = 0.0395 (decimal form)
n = 52 (weekly compounding)
t = 3.5
Using these values, we can rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Calculating the value of P for Enrico:
P = 4000 / (1 + 0.0395/52)^(52 * 3.5) = $3675.02 (rounded to two decimal places)
Now, let's calculate the future value of Paul's investment after 3.5 years using the same formula. However, for Paul's investment, it is compounded semi-annually, so n = 2 (twice a year):
A = 4000
r = 3.95% = 0.0395 (decimal form)
n = 2 (semi-annual compounding)
t = 3.5
Calculating the value of P for Paul:
P = 4000 / (1 + 0.0395/2)^(2 * 3.5) = $3651.55 (rounded to two decimal places)
Now, we can compare the initial principal amounts, P. From the calculations, we can see that Enrico made the greater original investment of $3675.02, while Paul made an initial investment of $3651.55.
To determine the difference in the initial investment amounts, we subtract the smaller value from the greater value:
Difference = Enrico's investment - Paul's investment
Difference = $3675.02 - $3651.55
Difference = $23.47 (rounded to two decimal places)
Therefore, Enrico made a greater original investment by $23.47.
Enrico:
i = .0395/52 = .000759615 , (I stored all the decimals)
n = 52(3.5) = 182
E(1.000759615)^182 = 4000
E = 4000/(1.000759615)^182 = $ 3483.71
Repeat the same steps for Paul.