You deposit $10,000 at 5% per year. What is the balance at the end of
one year if the interest paid is
(a) simple interest
(b) compounded monthly
Let
A=amount at the end of n periods
r=interest rate per annum, 0.06 for 6% p.a.
R=interest rate per period, in the form
(1+r/k) where k=number of periods per year
(1+0.06)=1.06 for 6% p.a., or
(1+0.06/12)=1.005 for 6% p.a. compounded monthly
Simple interest:
A=PRn
=10000*1.05=10500
Compound interest:
A=PR^n
=10000*(1+0.05/12)^12
=10000*1.051162
=10511.62
Compound interest, compounded monthly
Okay where did the 6% came in from?
The 6% is an example for the definition of the symbol R. Sorry if it misled you!
To calculate the balance at the end of one year with simple interest, you can use the formula:
Balance = Principal + (Principal * Rate * Time)
(a) For simple interest, the rate is given as 5% per year, or 0.05, and the time is 1 year. Plugging in these values into the formula:
Balance = $10,000 + ($10,000 * 0.05 * 1)
Balance = $10,000 + $500
Balance = $10,500
So, the balance at the end of one year with simple interest is $10,500.
Now let's calculate the balance at the end of one year with compounded monthly interest. Compound interest is calculated using the formula:
Balance = Principal * (1 + Rate / n)^(n * Time)
(b) In this case, the rate remains 5% per year or 0.05. However, since the interest is compounded monthly, we will divide the rate by 12 (the number of months in a year) and multiply the time by 12. Plugging in these values:
Balance = $10,000 * (1 + 0.05/12)^(12 * 1)
Calculating the values inside the brackets first:
(1 + 0.05/12) = 1.004167
Plugging in the values:
Balance = $10,000 * 1.004167^12
Balance = $10,000 * 1.0512
Balance = $10,512
So, the balance at the end of one year with compounded monthly interest is $10,512.