Question

You deposit \(\$ 1000\) in an account that pays \(3.5 \%\) annual interest compounded monthly. When is your balance at least \(\$ 1200\) ? \(\$ 3500\) ?

Short answer

The exact numbers of years to reach \$1200 and \$3500 can only be calculated by inserting the values from the exercise into the formula in the Steps 2 and 3. This number is usually non-integer and hence, rounded up to nearest higher number to reach desired amount in that year.

Step-by-step solution

Identify the variables

From the problem, the principal, P=\$1000, the interest rate, r=3.5\% or 0.035 (in the decimal form), and n=12 (since it's compounded monthly). The final amounts, A, are given as \$1200 and \$3500 which you need to find the corresponding time, t, to reach.

Apply formula for \$1200

Substitute A=\$1200, P=\$1000, r=0.035, and n=12 into the compound interest formula and isolate \( t \): \[ t = \frac{ \ln{(\frac{A}{P})} }{n \ln{(1 + \frac{r}{n})}} \] Substitute the given values and calculate t. This will be the number of years it takes for the account to reach \$1200 at given interest rate.

Apply formula for \$3500

Similarly, substitute A=\$3500 and the remaining variables into the compound interest formula and solve for t as in the previous step. This will be the number of years it takes for the account to reach \$3500.

Key concepts

Exponential Growth

Exponential growth refers to an increase in number or size, at a consistently rapid rate when compared to a base value. In the context of compound interest, this growth signifies how investments or loans can grow swiftly over time.

For instance, when you deposit money into a savings account, the amount doesn't just increase linearly by the same amount each year. Instead, the interest earned in one year is added to the principal, and in the following year, interest is earned on the new total, essentially earning 'interest on interest'. This compounding effect causes the account balance to grow exponentially.

The formula that captures this exponential growth due to compound interest is \( A = P \times \big(1 + \frac{r}{n}\big)^{n \times t} \), where \(A\) represents the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (initial sum of money), \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested or borrowed for, in years.

Logarithms

Logarithms are a critical mathematical concept, especially when dealing with exponential growth in financial mathematics. A logarithm answers the question: to what exponent must we raise a specific base number to produce a certain value?

In the scenario of our compound interest problem, logarithms are used to isolate the variable \(t\), which represents time. Because compound interest leads to an exponential equation in terms of time, we need to use logarithms to solve for \(t\). The formula used in the step by step solution is a rearranged version of the compound interest formula, solved for \(t\) using logarithms: \[ t = \frac{ \ln{(\frac{A}{P})} }{n \ln{(1 + \frac{r}{n})}} \].

This equation is derived using the natural logarithm (\ln) – which uses the constant \(e\) as the base – because it simplifies calculations involving continuous growth, and allows us to solve for time directly.

Financial Mathematics

Financial mathematics involves using mathematical techniques to solve financial problems. It includes concepts like valuation, investing, budgeting, and saving. Compound interest is a core concept within this area, and understanding it enables individuals to make more informed financial decisions.

The step by step solution provided for determining when the balance exceeds certain amounts at a given interest rate is an application of financial mathematics. The careful organization of the given information – principal, interest rate, and the number of times the interest is compounded – and the application of algebraic methods and logarithms, highlights the process of analyzing financial situations mathematically.

Practical knowledge of how to apply these concepts ensures that individuals are equipped to plan for goals such as retirement, saving for large purchases, or understanding the implications of taking on a loan. Moreover, this adeptness in financial mathematics can also lead to a greater understanding of more complex financial instruments and markets.