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Chapter 1: Problem 87

\(300\left(1+\frac{0.1}{12}\right)^{24}\)

Short Answer

Expert verified
This wouldn't have a simple answer since the calculation requires a calculator. The answer to the decimal place would be approximately 366.77.

Step by step solution

01

Understanding the problem

We have a compound interest problem with an initial amount of 300, an annual interest rate of 10% (which translates to 0.1), compounded monthly (which is 12 times in a year) over a period of two years (which is 24 months). Thus our problem is a direct application of the compound interest formula \(A=P\left(1+\frac{r}{n}\right)^{nt}\).
02

Substitution of values

Substituting the given values into the compound interest formula. Therefore, our formula becomes \(A=300\left(1+\frac{0.1}{12}\right)^{24}\).
03

Solve the given expression

Find the value of the fraction inside the parenthesis first, then add 1, take the result to the power of 24 and finally multiply by the principal, which is 300.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are fundamental in mathematics, as they allow us to model real-world problems using symbols and numbers. In the compound interest example, the expression \(300\left(1+\frac{0.1}{12}\right)^{24}\) is key to computing the final amount after interest is applied. The elements of this expression include:
  • Constants like 300, which represents the initial amount of money or principal.
  • The fraction inside the expression, \(\frac{0.1}{12}\), which depicts the monthly interest rate.
  • Exponents, as seen in \(^{24}\), which signify the number of compounding periods.
Breaking down complex expressions into simpler parts can help you understand and solve them efficiently. For example, calculate the monthly interest rate separately before incorporating it into the full expression.
Compound Interest Formula
The compound interest formula is a powerful tool for calculating how money grows over time when interest is added not just on the principal but also on the accumulated interest. The formula is expressed as:\[A = P\left(1 + \frac{r}{n}\right)^{nt}\] Where:
  • \(A\) is the amount of money accumulated after n years, including interest.
  • \(P\) is the principal amount (the initial amount of money).
  • \(r\) is the annual interest rate (decimal).
  • \(n\) is the number of times that interest is compounded per year.
  • \(t\) is the time in years.
This formula shows that the total amount grows exponentially, factoring in both the initial principal and the interest accrued over time.
Exponential Growth
In financial contexts, such as with compound interest, exponential growth refers to the increase in quantity according to a mathematical function where the rate of growth is proportional to the current quantity. Unlike simple interest, which adds the same amount of interest each period, compound interest reflects exponential growth where:
  • The interest earned or paid in one period is invested in the next, increasing the base on which the interest for future periods is calculated.
  • This creates a 'snowball effect,' where both the principal and accumulated interest earn more interest over time.
  • The mathematical expression \(\left(1 + \frac{r}{n}\right)^{nt}\) embodies exponential growth, increasing rapidly as time and compounding periods multiply.
Understanding exponential growth helps to grasp how small, consistent contributions or investments can lead to significant financial returns over a longer period.

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Most popular questions from this chapter

\(5(7+3) \neq 5(7)+3\)

\((-3)^{3}+\left(12 \div 2^{2}\right)\)

\(\frac{5+[(-12) \div 4]}{24}\)

Sales Tax You purchase a sweater for \(\$ 35.95\). There is a \(6 \%\) sales tax, which means that the total amount you must pay is \(35.95+0.06(35.95)\). (a) Use the Distributive Property to rewrite the expression. (b) How much must you pay for the sweater including sales tax?

\(100 \div 2 \times 50 \neq 1\)
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