Amanda deposits $10,000 into an account that pays 3% interest per year, compounded annually. Sam deposits $10,000 into an account that also pays 3% per year. But it is simple interest.
Find the interest Amanda and Sam 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.
Good grief!
At the same interest rate, compound interest will always give you more
after the first period, but if you insist .....
compound:
after 3 years,
amount = 10000(1.03)^3 = 10,927.27
simple:
after 3 years,
amount = 10000 + 10000(3)(.03) = 10,900.00
To find the interest earned by Amanda and Sam during each of the first three years, we can use the formula for compound interest and simple interest.
For Amanda:
Compound interest formula: A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
For Sam:
Simple interest formula: A = P(1 + rt)
Where:
A = final amount
P = principal amount (initial deposit)
r = annual interest rate (in decimal form)
t = number of years
Let's calculate the interest earned by Amanda and Sam for each year using these formulas.
Year 1:
Amanda:
A = 10000(1 + 0.03/1)^(1*1)
A = 10000(1 + 0.03)^1
A = 10000(1.03)
A = $10,300
Interest earned by Amanda in year 1 = $10,300 - $10,000 = $300
Sam:
A = 10000(1 + 0.03*1)
A = 10000(1 + 0.03)
A = 10000(1.03)
A = $10,300
Interest earned by Sam in year 1 = $10,300 - $10,000 = $300
Year 2:
Amanda:
A = 10000(1 + 0.03/1)^(1*2)
A = 10000(1 + 0.03)^2
A = 10000(1.0609)
A = $10,609
Interest earned by Amanda in year 2 = $10,609 - $10,000 = $609
Sam:
A = 10000(1 + 0.03*2)
A = 10000(1 + 0.06)
A = 10000(1.06)
A = $10,600
Interest earned by Sam in year 2 = $10,600 - $10,000 = $600
Year 3:
Amanda:
A = 10000(1 + 0.03/1)^(1*3)
A = 10000(1 + 0.03)^3
A = 10000(1.0927)
A = $10,927
Interest earned by Amanda in year 3 = $10,927 - $10,000 = $927
Sam:
A = 10000(1 + 0.03*3)
A = 10000(1 + 0.09)
A = 10000(1.09)
A = $10,900
Interest earned by Sam in year 3 = $10,900 - $10,000 = $900
Comparing the interest earned by Amanda and Sam for each year:
Year 1: Both Amanda and Sam earn $300 in interest.
Year 2: Amanda earns $609 in interest, while Sam earns $600 in interest. Amanda earns more interest.
Year 3: Amanda earns $927 in interest, while Sam earns $900 in interest. Amanda earns more interest.
Therefore, we can conclude that Amanda earns more interest than Sam for all three years.