Question

You deposit \(\$1400\) in an account that pays 6% interest compounded yearly. Find the balance for the given time period. 8 years

Short answer

The balance of the account after 8 years will be approximately \$2094.46.

Step-by-step solution

Convert the Annual Interest Rate from Percentage to Decimal

The given annual interest rate is 6%. To convert it to a decimal, we divide by 100, which gives us \(0.06.\)

Insert Values into the Compound Interest Formula

Now that we know \(P = $1400\), \(r = 0.06\), \(n = 1\) (since the interest is compounded yearly), and \(t = 8\) years, we can insert these values into the formula: \(A = 1400(1 + 0.06/1)^{1*8}\).

Compute the Balance

Let's simplify and compute the balance: \(A = 1400(1 + 0.06)^{8} = 1400(1.06)^{8}\). Carrying out this calculation using a calculator, we find that \(A \approx \$2094.46\). Room4 is the correct choice.

Key concepts

Interest Rate to Decimal

When dealing with financial formulas, it's crucial to convert percentage interest rates into their decimal form. This is because percentage forms, like 6%, are not directly usable in mathematical formulas. To convert an interest rate from a percentage to a decimal, simply divide it by 100. For example, the given interest rate of 6% as a decimal becomes

\( \frac{6}{100} = 0.06 \).
This seemingly small step is critical for accurately calculating compound interest, as using the percentage directly in the formula would lead to incorrect results. Always remember to make this conversion before proceeding with further calculations.

Compound Interest Formula

The compound interest formula is essential for understanding how investments grow over time. It considers the principal amount, interest rate, frequency of compounding, and the number of periods the interest is applied. The formula is expressed as:
\[ A = P(1 + \frac{r}{n})^{n*t} \]
where

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial sum 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 for, in years.

Mastering this formula allows you to predict how your investment will grow over time, attributing to the 'magic' of compounding where you earn interest not only on your original investment but also on the interest it accrues.

Calculating Compound Interest

To calculate compound interest, you must first ensure all the variables are clearly defined and converted into the appropriate units, as seen in the formula section. After identifying the principal (P), annual interest rate (r), number of times interest is applied (n), and the time period (t), simply substitute these values into the compound interest formula.

For instance, with a principal of \$1400, a rate of 6% (decimal 0.06), compounded yearly (n=1), over 8 years (t=8), the calculation would be:
\[ A = 1400(1 + \frac{0.06}{1})^{1*8} = 1400(1.06)^8\]
Break down the formula step by step, simplifying the exponent first, then performing the multiplication. Using this structured approach ensures that calculating compound interest will be a systematic and error-free task.

Annual Interest Rate

The annual interest rate is the percentage increase per year that is applied to the principal balance in an account. It represents the cost of borrowing money or the profit from lending it. Understanding the annual interest rate is key to determining how much you will earn or owe over a period.

Interest rates are typically advertised on an annual basis, known as the nominal or stated rate. However, it is important to note that this rate might not reflect the actual earnings on an investment if the interest is compounded more frequently than once a year. Thus, the effective annual rate can sometimes provide a more accurate picture of the investment's growth potential, considering the effects of compounding within the year. Understanding the intricacies of interest rates and how they can vary with compounding frequency is vital for making informed financial decisions.