An amount of $3000 was deposited in a bank at a rate of 2% annual interest compounded quarterly for 3yrs. The rate then increased to 3% annual interest and ws compounded quarterly for the next 3 yrs. If no money was withdrawn what was the balance at the end of this time?
$3000 at 2% for 3 years, followed by 3% for 3 years, all compounded quarterly.
The basic equation for the future value A (accumulated amount) of a principal P at r% interest per period for n periods is:
A=PR^n
where R=1+r
At the end of the three years at 2%, the amount is reinvested at 3%, so the calculated A becomes P for the second part.
First three years:
No. of periods, n = 4*3 = 12 quarters
interest per period, 1+2%/4=1.005
Principal = $3000
Amount at the end of three years:
A = PR^n = 3000*1.005^12
= 3000*1.06168
= $3185.03
For the second part,
P=$3185.03
R=1+3%/4=1.075
n=12
A=PR^n=3185.03*1.075^12=?
To calculate the balance at the end of the given time period, we will split it into two parts: the first three years with a 2% interest rate and the next three years with a 3% interest rate, both compounded quarterly.
Part 1:
Principal amount (initial deposit) = $3000
Interest rate per period = 2%/4 = 0.5%
Number of compounding periods = 3 years * 4 = 12
Using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final balance
P = the principal amount
r = the interest rate per period (as a decimal)
n = the number of compounding periods per year
t = the number of years
Substituting the given values into the formula, we get:
A1 = 3000(1 + 0.005)^(12)
A1 ≈ $3181.62
Therefore, the balance at the end of the first three years is approximately $3181.62.
Part 2:
Principal amount (initial deposit) = $3181.62 (balance from Part 1)
Interest rate per period = 3%/4 = 0.75%
Number of compounding periods = 3 years * 4 = 12
Using the same formula as before, we can calculate the balance for the next three years:
A2 = 3181.62(1 + 0.0075)^(12)
A2 ≈ $3481.02
Therefore, the balance at the end of the second three years is approximately $3481.02.
Adding the balances from both parts, we get the final balance:
Final balance = $3181.62 + $3481.02
Final balance ≈ $6662.64
Hence, the balance at the end of the given time period would be approximately $6662.64.
To calculate the balance at the end of the specified time period, we need to calculate the balance separately for each period and then combine them. Let's break it down step by step.
First, let's calculate the balance after 3 years with an initial deposit of $3000 at a rate of 2% annual interest compounded quarterly.
Step 1: Convert the interest rate to a quarterly rate
The annual interest rate of 2% needs to be converted to a quarterly rate. Divide the annual rate by 4 (since there are 4 quarters in a year):
Quarterly interest rate = 2% / 4 = 0.5% = 0.005 (decimal form)
Step 2: Calculate the balance after 3 years using the compound interest formula
The compound interest formula is given by:
Balance = Principal * (1 + Interest Rate/Compounding Period)^ (Compounding Period * Time)
In this case, the principal (initial deposit) is $3000, the interest rate is 0.005 (quarterly rate), and the compounding period is 4 (since interest is compounded quarterly). The time period is 3 years.
Balance after 3 years = $3000 * (1 + 0.005/4)^(4 * 3)
Calculating this expression will give you the balance after 3 years. Let's call this balance B1.
Now, let's move on to the second period:
- The balance at the beginning of the second period (after 3 years) is equal to B1.
- The interest rate for the second period is 3% annual interest, which we need to convert to a quarterly rate.
- The compounding period is still 4 (compounded quarterly).
- The time period for the second period is also 3 years.
Using the same compound interest formula, we can calculate the balance at the end of the second period. Let's call this balance B2.
Balance after 6 years = B1 * (1 + Quarterly Rate)^(Compounding Period * Time)
Finally, to find the total balance at the end of 6 years, we add the balances from both periods:
Total balance after 6 years = B1 + B2
By following these steps and performing the calculations, you can find the balance at the end of the 6-year period with the given conditions.