Final answer:
To estimate the time it will take for John's $100,000 deposit to double at a 10% annual interest, the Rule of 72 can be applied, resulting in approximately 7.2 years.
Step-by-step explanation:
To find out how long it will take for John's initial deposit of $100,000 to double given a 10% annual interest rate, we can use the Rule of 72, a simple formula that gives a rough estimate of the time it will take for an investment to double at a fixed annual rate of interest. By dividing 72 by the interest rate, you get the approximate number of years it will take for the initial investment to double.
In John's case, since the interest rate is 10%, we'll divide 72 by 10. So, it takes roughly 7.2 years for John's initial deposit to double.
Here's the calculation: 72 / 10 = 7.2
However, to get the exact number of years, we would need to use the formula for compound interest:
A = P(1 + r/n)^nt
In this formula:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested or borrowed for, in years.
To find the exact time (t) to double the money, we solve for t when A is $200,000 (double of the initial $100,000), P is $100,000, r is 0.10 (10%), and n is 1 (compounded annually).
200,000 = 100,000(1 + 0.10/1)^1t
After simplifying and solving for t, we will get the exact number of years it will take for the money to double.