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Chapter 8: Problem 12

In Exercises 1-12, the principal represents an amount of money deposited in a savings account subject to compound interest at the given rate. a. Find how much money there will be in the account after the given number of years. (Assume 360 days in a year.) b. Find the interest earned. Round answers to the nearest cent.$$ \begin{array}{|l|l|l|l|} \hline \text { 12. } \$ 25,000 & 5.5 \% & \text { daily } & 20 \text { years } \\\ \hline \end{array} $$

Short Answer

Expert verified
You will need to apply the formula for compound interest, substituting the given values, and then evaluate the result. The total amount in the account after 20 years would be found in step 3. Interest earned is calculated in step 4 by subtracting the initial principal from the total amount after 20 years. Both answers should be rounded to the nearest cent.

Step by step solution

01

Identify and Convert all variables

We first identify and convert all variables for use in the calculation. The principal, P is $25,000. The annual interest rate, r is 5.5%. We convert this to a decimal by dividing by 100, which gives 0.055. The number of times the interest is compounded per year, n is determined by the frequency. Here, interest is compounded daily so n is 360 (given by the problem as the assumption for the number of days in a year). The time period is 20 years, so t is 20.
02

Apply the Compound Interest Formula

We now substitute the variables into the compound interest formula. So, \(A = 25000(1 + 0.055/360)^{(360*20)}\).
03

Calculate the Amount

Calculate the amount A after 20 years by evaluating the expression obtained in Step 2. Round your answer to the nearest cent.
04

Calculate Interest Earned

Now, subtract the initial principal (P) from the total amount (A) after 20 years. The result will be the earned interest.
05

Round the Interest Earned

Round the calculated interest to the nearest cent to get the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Financial Mathematics
At its core, financial mathematics is the application of mathematical methods to solve problems related to finance. In this context, one common problem is the calculation of how money grows over time through compound interest.

Understanding the time value of money is essential, and the principle behind it is that a certain amount of money today is worth more than the same amount in the future due to its potential earning capacity. This foundational concept helps determine the future value of an investment made today, after factoring in the interest it will accumulate over time. In exercises involving savings accounts and investments, knowledge of financial mathematics allows you to estimate not just how much an account will be worth, but also the income generated in the form of interest.
Savings Account Growth
The growth of a savings account over time is influenced by several factors, the most significant being the rate of interest and how that interest is compounded. Compound interest means that interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

Consider a savings account with an initial deposit, known as the principal. As time goes on, interest is added to this principal at certain intervals 鈥� daily, monthly, quarterly, or annually. Each time interest is added to the account, the account balance grows, and the next interest calculation includes the previously earned interest. This is the power of compounding; over time, even small rates can lead to significant growth of the initial investment.
Interest Rate Calculation
Interest rate calculation is integral to understanding how investments evolve over time. The interest rate is typically expressed as an annual percentage but can be calculated over different compounding periods.

To calculate the amount in a savings account subject to compound interest, we use the formula: \( A = P(1 + \frac{r}{n})^{(nt)} \)
where:\
  • \(A\) represents the future value of the investment or loan, including interest
  • \(P\) is the principal amount
  • \(r\) is the annual interest rate (in decimal form)
  • \(n\) is the number of times that interest is compounded per year
  • \(t\) is the time the money is invested or borrowed for, in years.

By applying this formula, as in our exercise, you can calculate not only the future value of your savings but also the interest earned by subtracting the original principal from the final amount.

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