Problem 25 - 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. $$5 \%$$

Step-by-Step Solution

Answer

The approximate balance after 10 years would be $1,464.84.

Step 1: Identify the variables

Given: Principal amount (P) = $900; annual interest rate (r) = 5% or 0.05 (since percentage needs to be converted into decimal form); number of times compounded per year (n) = 1 (since it is compounded yearly); time (t) = 10 years.

Step 2: Plugging the values into the compound interest formula

Now, apply the values to the compound interest formula, \(A = P(1 + r/n)^{nt}\). The equation will be: \(A = 900(1 + 0.05/1)^{(1*10)}\).

Step 3: Calculate the Amount (A)

Solving the equation gives: \(A = 900(1.05)^{10} \). Calculate this to get the final amount.

Step 4: Round up to the nearest cent

The previous step gives an amount that might be in decimal form with more than two places after the decimal point. Since it refers to the amount of money, it should be rounded off up to two decimal places if necessary.

Key Concepts

Annual Interest Rate

The annual interest rate is a key component in determining how much your investment will grow over time. It represents the percentage of the principal amount that the bank pays you for keeping your money with them. The higher the interest rate, the more money you can earn on your deposit.

• The interest rate in our current example is 5%.
• To use it in calculations, convert the percentage into a decimal by dividing by 100 — so, 5% becomes 0.05.

This small change into a decimal is crucial because most formulas in mathematics, including the compound interest formula, require the interest rate in this form.
Understanding the decimal conversion allows easy participation in further calculations.

Principal Amount

The principal amount is the initial sum of money that you deposit into your account.
In our example, the principal amount is $900. This is the starting point for all compound interest calculations.
Here’s why it matters:

• The principal amount determines the base on which interest is calculated.
• In a compound interest scenario, both the principal and accumulated interest from previous periods earn interest themselves, increasing the total balance over time.

Knowing your principal amount helps start your calculations correctly and lets you forecast how much your savings could grow.

Compounded Yearly

When interest is compounded yearly, it means the interest is calculated and added to the principal at the end of each year.

• In our scenario, the compounding frequency is once per year (n = 1).

The compounding frequency affects how often the interest amount is recalculated and added to the principal. A higher frequency (e.g., quarterly or monthly) generally results in more interest because the interest component grows more frequently.
Understanding how often your interest is compounded helps you better plan your investment strategy.

Time in Years

The time in years refers to the duration for which the principal is left to earn interest.
In this problem, the time period is 10 years.
The time factor is essential because:

• Compound interest grows more substantial with time. As you give your money more time, the effect of compounding becomes more pronounced, and the overall amount increases significantly.
• Time dictates how many intervals you’ll calculate interest for, impacting the total interest accrued.

Having a clear understanding of your investment horizon helps you set realistic financial goals and expectations. Longer time frames usually mean a larger accumulation of wealth through compound interest.