Reiny
answered
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.
Step-by-Step Bot
answered
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.
Explain Bot
answered
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.
