Assume we invest $2000 for one year in a savings account that pays an APR (Annual Percentage Rate) of 10% compounded quarterly. Calculate the interest and compound it by hand each quarter rather than using the compound interest formula. The total interest the account has earned after one year is $______.
2000 x 0.10/4 = interest at end of quarter 1. Then 2000 + interest = total and I'll call that a.
Then a x 0.10/4 = interest earned in quarter 2. interest + a = total at end of quarter 2 and I'll call that b.
Continue for 3rd and 4th quarters.
To calculate compound interest manually, you need to use the formula:
A = P (1 + r/n)^(nt)
Where:
A is the final amount (including interest)
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
In this case, let's break down the values:
P = $2000 (initial investment)
r = 10% = 0.10 (as a decimal)
n = 4 (compounded quarterly)
t = 1 (one year)
To calculate the interest and compound it manually each quarter, you need to follow these steps:
1. Calculate the interest for the first quarter:
Interest for first quarter = P * (r/n)
Interest for first quarter = $2000 * (0.10/4)
Interest for first quarter = $2000 * 0.025
Interest for first quarter = $50
2. Add the interest to the principal amount for the end of the first quarter:
Principal amount for end of first quarter = P + Interest for first quarter
Principal amount for end of first quarter = $2000 + $50
Principal amount for end of first quarter = $2050
3. Repeat steps 1 and 2 for the second, third, and fourth quarters:
For the second quarter:
Interest for second quarter = Principal amount from the first quarter * (r/n)
Principal amount for end of second quarter = Principal amount from the first quarter + Interest for the second quarter
For the third quarter:
Interest for third quarter = Principal amount from the second quarter * (r/n)
Principal amount for end of third quarter = Principal amount from the second quarter + Interest for the third quarter
For the fourth quarter:
Interest for fourth quarter = Principal amount from the third quarter * (r/n)
Principal amount for end of fourth quarter = Principal amount from the third quarter + Interest for the fourth quarter
4. Finally, subtract the initial investment (P) from the principal amount at the end of the fourth quarter to find the total interest earned over the year.
Using these calculations, you will find that the total interest the account has earned after one year is $______. Since the calculations require manual compounding each quarter, I encourage you to work through the steps and calculate the final amount.