Problem 18 - Algebra 1: Concepts and Skills

Question

You deposit \(\$1400\) in an account that pays 6% interest compounded yearly. Find the balance at the end of the given time period. $$12 years$$

Step-by-Step Solution

Answer

The balance in the account at the end of 12 years will be \$2798.72.

Step 1: Identify the variables

For the compound interest formula, \( A = P (1 + r/n)^{nt} \), we have the following values identified: \$P=1400\$, \(r=6\%=0.06\) (converted to decimal form), \(n=1\) (as interest is compounded yearly), and \(t=12\) years.

Step 2: Substitute into the formula

Next, plug these values into the compound interest formula: The equation becomes \(A = 1400 (1 + 0.06/1) ^{(1*12)}\).

Step 3: Solve the formula

Solve the equation to find the total balance after 12 years: \(A = 1400 (1+0.06)^{12}\). Simplifying, we find that \(A=1400 * ((1.06)^{12}) \) which equals \$2798.72 when rounded to the nearest cent.

Key Concepts

Simple Interest

Simple interest is a basic way to calculate the interest payable on a fixed sum for a specific time period. It's called "simple" because it uses a straightforward calculation where interest is applied only to the principal amount (the initial money invested or borrowed) and not on any accumulated interest.
The formula for simple interest is:

• \( I = P imes r imes t \)

where:

• \( I \) represents the interest earned or paid,
• \( P \) is the principal amount,
• \( r \) is the annual interest rate (as a decimal), and
• \( t \) is the time in years.

In contrast, compound interest calculates interest on both the initial principal and on the accumulated interest, making it often more favorable for growing savings over time.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In solving compound interest problems, algebra is essential as it involves substituting known values into formulas and solving for unknowns.
In our exercise, we use the compound interest formula:

• \( A = P (1 + r/n)^{nt} \)

The process involves:

• Identifying known variables such as principal \( P \), rate \( r \), compounding frequency \( n \), and time \( t \),
• Substituting these values into the algebraic formula, and
• Simplifying to find the final value.

Algebra helps in handling the exponents and variables that arise in more complex calculations, making it easier to determine future balances or amounts.

Interest Rate

Interest rate is crucial in financial calculations. It is the proportion of a loan or deposit that is charged as interest to the borrower or paid to the investor, usually expressed as an annual percentage of the principal. In our exercise, the interest rate is 6% per year, which needs to be converted into a decimal (0.06) to be used in the formula.
When interest is compounded, the rate at which you earn interest can significantly affect your total amount over time. A higher rate yields more earnings or costs more in terms of what must ultimately be paid back or invested.
It's important to distinguish between simple interest (applied only to the principal) and compound interest, where the rate is applied to both the principal and accrued interest periodically.

Exponents

Exponents are a mathematical operation used to represent repeated multiplication of a number by itself. In our context, they indicate how often interest is compounded over time.
In the compound interest formula:

• \( A = P (1 + r/n)^{nt} \)

The exponent \( nt \) signifies the number of times interest is applied. Here, since interest is compounded once per year over 12 years, the exponent becomes \( 12 \).
Understanding how to work with exponents is essential in calculating compound interest, as they determine how quickly your investments grow or how much your debt increases. Mastering this concept can help you make informed financial decisions that capitalize on this growth factor.