Problem 27 - Algebra 1: Concepts and Skills

Question

You deposit \(\$900\) in an account that compounds interest yearly. Find the balance after 10 years for the given interest rate. $$7 \%$$

Step-by-Step Solution

Answer

The balance in the account after 10 years will be approximately \$1759.65.

Step 1: Convert Interest Rate to Decimal

The interest rate is given in percentage terms of 7%. We need to convert this to a decimal by dividing by 100. So, the rate in decimal form is \(0.07\).

Step 2: Define the variables for the compound interest formula

With the rate in decimal form, we can now identify and define all the variables we'll use in the compound interest formula. We have \(P = $900\), the initial principle, \(r = 0.07\), the interest rate in decimal form, \(n = 1\), the number of times interest is compounded per year (in this case yearly) and \(t = 10\), the time the money is to stay in the account.

Step 3: Substitution into the Compound Interest Formula

Next, substitute the appropriate values into the compound interest formula: \(A = P(1 + r/n)^{nt}\). By substituting those values, we get \(A = 900(1 + 0.07)^{10}\).

Step 4: Calculate the Balance

Finally, perform the calculations to find the total balance in the account. \(A = 900(1.07)^{10}\). Doing the math, we find that \(A ≈ 1759.65\). So, after 10 years, the balance in the account will be roughly \$1759.65.

Key Concepts

Interest Rate Conversion

Converting interest rates from percentage to decimal form is the first crucial step in working with financial formulas like compound interest. This conversion makes calculations simpler and is necessary since formulas require a decimal value for the rate of interest. To convert a percentage to a decimal:

• Divide the percentage by 100.

For example, with a 7% interest rate, you perform the conversion by calculating \[\text{Rate in decimal} = \frac{7}{100} = 0.07\]This simple conversion is important because a mistake here can affect the entire calculation, leading to an incorrect balance later on. Always double-check to ensure you have the correct decimal value before moving on to subsequent steps.

Compound Interest Formula

Compound interest is a powerful concept in finance that provides insights into how money grows over time when left to accumulate interest. The fundamental formula used to calculate the compound interest is:\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) represents the principal amount (initial deposit or loan amount).
• \(r\) stands for the annual interest rate (in decimal form).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) marks the time the money is invested for, in years.

Plugging in your values correctly into this formula is essential. In the given problem, with annual compounding, \(n\) is 1. It's helpful to double-check each variable's value and perform intermediate calculations step-by-step to avoid errors. Taking note of the compounding frequency is especially important as it can significantly affect the final balance.

Balance Calculation

After setting up your variables and formula, the final step is to calculate the balance. This involves substituting the values into the compound interest formula and conducting the arithmetic calculations. For the problem at hand, we substitute values as follows:\[A = 900 \left(1 + 0.07\right)^{10}\]This simplifies to:\[A = 900 \times (1.07)^{10}\]At this point, you calculate \((1.07)^{10}\) first. Use a calculator to raise 1.07 to the 10th power, then multiply the result by 900. Ensure accuracy in your computation to get the correct final result:\[A \approx 1759.65\]Therefore, after ten years, the account will have a balance of approximately \$1759.65. Double-checking your calculations is always a good practice, especially for the power and multiplication operations involved.