Norris currently has an account balance of $3,703.05. He opened the account 25 years ago with a deposit of $1,725.95. If the interest compounds monthly, what is the interest rate on the account?
My answer is 3.1%
That's what I get too
To calculate the interest rate on the account, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final account balance
P = the initial deposit
r = the interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the initial deposit (P) is $1,725.95, the final account balance (A) is $3,703.05, the number of times interest is compounded per year (n) is 12 (since interest compounds monthly), and the number of years (t) is 25.
Now, we need to solve for the interest rate (r). However, since the formula is non-linear, we need to use an iterative method or approximation techniques to find the interest rate. One such technique is the trial and error method.
We can start with an initial guess for the interest rate, such as 3%. We can then compute the final account balance using this interest rate and check if it matches the actual balance ($3,703.05). If it doesn't match, we can adjust the guess and repeat the process until we find a close approximation.
Let's calculate the final account balance using an interest rate of 3% (0.03 as a decimal):
A = $1,725.95(1 + 0.03/12)^(12*25)
A ≈ $1,725.95(1.0025)^300
A ≈ $1,725.95 * 0.746498
A ≈ $1,287.94
As we can see, the calculated balance of $1,287.94 is much lower than the actual balance of $3,703.05. So we need to increase our guess for the interest rate.
Let's try a higher interest rate of 3.1% (0.031 as a decimal):
A = $1,725.95(1 + 0.031/12)^(12*25)
A ≈ $1,725.95(1.002583)^300
A ≈ $1,725.95 * 0.765544
A ≈ $1,323.77
As we can see, the calculated balance of $1,323.77 is still lower than the actual balance of $3,703.05.
We continue this process of adjusting our guess for the interest rate by small increments until we find a rate that produces a calculated balance close to $3,703.05.
After several calculations and adjustments, a rate of around 7.14% seems to produce a calculated balance closer to the actual balance of $3,703.05.
Finally, let's calculate the final account balance using an interest rate of 7.14% (0.0714 as a decimal):
A = $1,725.95(1 + 0.0714/12)^(12*25)
A ≈ $1,725.95(1.00595)^300
A ≈ $1,725.95 * 1.85404
A ≈ $3,200.98
As we can see, the calculated balance of $3,200.98 is closer to the actual balance of $3,703.05, but it is still not an exact match.
To get a more precise interest rate, you can continue this trial and error method with smaller increments until you find a rate that produces a calculated balance very close to $3,703.05.