And deposits $20,000 into an account that pays simple interest at a rate of 2% per year Jim deposits $20,000 into an account that also pays 2% interest per year but it is compounded annually find the interest and and Jim earn during each of the first three years then decide who earns more interest for each year assume there are no withdrawals and no additional deposits
Let's calculate the interest earned by each account for the first three years.
1) Simple Interest Account:
Year 1:
Interest earned = Principal * Rate = $20,000 * 2% = $400
Year 2:
Interest earned = Principal * Rate = $20,000 * 2% = $400
Year 3:
Interest earned = Principal * Rate = $20,000 * 2% = $400
Total interest earned after three years = $400 + $400 + $400 = $1,200
2) Compound Interest Account:
Year 1:
Interest earned = Principal * Rate = $20,000 * 2% = $400
Year 2:
Interest earned = (Principal + Year 1 Interest) * Rate = ($20,000 + $400) * 2% = $408
Year 3:
Interest earned = (Principal + Year 2 Interest) * Rate = ($20,000 + $408) * 2% = $416.16
Total interest earned after three years = $400 + $408 + $416.16 = $1,224.16
Conclusion:
After comparing the interest earned by each account for the first three years, we can see that Jim earns more interest with the compound interest account.
To find the interest earned by each person during the first three years, we can use the formulas for simple interest and compound interest.
1. Simple Interest (Jim):
The formula for simple interest is: Interest = Principal x Rate x Time
For Jim's simple interest calculation:
Principal (P) = $20,000
Rate (R) = 2% = 0.02 (in decimal form)
Year 1:
Time (T) = 1 year
Interest = $20,000 x 0.02 x 1 = $400
Year 2:
Time (T) = 2 years
Interest = $20,000 x 0.02 x 2 = $800
Year 3:
Time (T) = 3 years
Interest = $20,000 x 0.02 x 3 = $1,200
2. Compound Interest (You):
The formula for compound interest is: A = P(1 + r/n)^(nt) - P
For your compound interest calculation:
Principal (P) = $20,000
Rate (r) = 2% = 0.02 (in decimal form)
Compounding Frequency (n) = 1 (compounded annually)
Year 1:
Time (t) = 1 year
A = $20,000(1 + 0.02/1)^(1*1) - $20,000
A = $20,000(1.02) - $20,000
A = $20,400 - $20,000
A = $400
Year 2:
Time (t) = 2 years
A = $20,000(1 + 0.02/1)^(1*2) - $20,000
A = $20,000(1.02)^2 - $20,000
A = $20,000(1.0404) - $20,000
A = $20,808 - $20,000
A = $808
Year 3:
Time (t) = 3 years
A = $20,000(1 + 0.02/1)^(1*3) - $20,000
A = $20,000(1.02)^3 - $20,000
A = $20,000(1.0612) - $20,000
A = $21,224 - $20,000
A = $1,224
Now let's compare the interest earned by each person for each year:
Year 1:
Jim's simple interest: $400
Your compound interest: $400
Year 2:
Jim's simple interest: $800
Your compound interest: $808
Year 3:
Jim's simple interest: $1,200
Your compound interest: $1,224
From the calculations, it can be observed that Jim earns the same interest as you during the first year. However, in the following years, your compound interest surpasses Jim's simple interest. Therefore, you earn more interest for each year after the first year.