Problem 46 - Algebra A Combined Function

Question

Solve. Use \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Round answers to two decimal places. Find the amount owed at the end of 5 years if \(\$ 3000\) is loaned at a rate of \(10 \%\) compounded quarterly.

Step-by-Step Solution

Answer

At the end of 5 years, the amount owed is approximately $4915.81.

Step 1: Identify the Values

In the formula \(A=P\left(1+\frac{r}{n}\right)^{nt}\), we're given:- Principal \(P = 3000\) dollars,- Annual Interest Rate \(r = 0.10\) (converted from \(10\%\)),- Number of compounding periods per year \(n = 4\) (quarterly),- Number of years \(t = 5\).

Step 2: Plug Values into the Formula

Input these values into the compound interest formula: \[A = 3000\left(1 + \frac{0.10}{4}\right)^{4 \times 5}\] This will help us calculate the amount owed.

Step 3: Calculate the Compound Interest

Simplify the expression inside the parentheses:\[1 + \frac{0.10}{4} = 1 + 0.025 = 1.025\]Then compute the exponent, which is \(4 \times 5 = 20\):\[A = 3000 \times (1.025)^{20}\].

Step 4: Compute the Final Amount

Using a calculator, compute \((1.025)^{20} \approx 1.6386\). Therefore, the final amount owed is:\[A = 3000 \times 1.6386 \approx 4915.81\].

Key Concepts

Principal

In the world of finance, the term "principal" plays a crucial role. Think of the principal as the initial sum of money you invest or borrow. It's the cornerstone upon which interest calculations are based in financial formulas such as those for compound interest.
When dealing with exercises involving compound interest, you'll often see the principal denoted by "\(P\)". For instance, if you start with \(\\(3{,}000\), this \(\\)3{,}000\) is your principal. It's the amount upon which interest will accumulate over time.

• **Initial Amount:** The principal is the starting point of your calculations and represents how much you start with before any interest is applied.
• **Investment or Loan:** Depending on the scenario, the principal can be money you invest or money you owe.
• **Base for Interest Calculation:** All interest calculations, whether simple or compound, use the principal as the foundation.

Understanding the principal is essential because it influences the total amount of interest you'll earn or pay. The higher the principal, the more you may earn (or owe!) at the end of the compounding period.

Annual Interest Rate

The annual interest rate is a critical factor in determining how much your money can grow or how much you might owe over time when subjected to compounding. It's usually expressed as a percentage and reflects the cost of borrowing money or the return on investment.

• **Expressed as a Percentage:** Typically, the rate is shown as a percentage; for example, \(10\%\) in our exercise translates to \(0.10\) when we use it in mathematical equations.
• **Influence on Growth:** A higher annual interest rate means your invested or borrowed amount will grow or accrue interest at a faster rate.
• **Conversion for Calculations:** Since the formula for computing compound interest requires the rate to be in decimal form, always convert percentages to decimals (e.g., \(10\% = 0.10\)).

It's important to remember that even small differences in the annual interest rate can significantly impact the final amount due, magnified over time, especially with frequent compounding.

Compounding Periods

Compounding periods refer to how often interest is calculated and added to the principal amount. The frequency of compounding has a direct effect on the total interest accumulated over the investment or loan duration.

• **Quarterly, Monthly, Daily:** Common compounding periods include quarterly (4 times a year), monthly (12 times a year), and daily (365 times a year).
• **Influence on Accrual:** The more frequently interest is compounded, the more opportunities the principal has to grow, because interest is calculated on an increasingly larger principal each time.
• **Impact on Formulas:** In our compound interest formula, \(n\) represents the number of compounding periods per year. In the example above, compounded quarterly means \(n = 4\).

Understanding compounding periods helps in predicting how much you can accrue over time. More compounding periods generally mean more interest accumulation, provided the rate and timeframe remain constant.