/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 Martha invested \(\$ 40,000\) in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 64

Martha invested \(\$ 40,000\) in a boutique 5 yr ago. Her investment is worth \(\$ 70,000\) today. What is the effective rate (annual effective yield) of her investment?

Short Answer

Expert verified
The effective annual interest rate for Martha's investment is approximately 11.94%.

Step by step solution

01

Identify the relevant information from the problem

For this problem, we have the following information: - Initial investment (P): $40,000 - Final value (A): $70,000 - Number of years (n): 5 years
02

Use the compound interest formula

The formula we will be using is: \[A = P(1 + r)^n\] Where: - A is the final value of the investment - P is the initial investment - r is the effective annual interest rate - n is the number of years
03

Plug in the given values to the compound interest formula

Now, we have: \(\$70,000 = \$40,000(1 + r)^5\)
04

Solve for the effective annual interest rate (r)

Divide both sides of the equation by $40,000: \(\frac{\$70,000}{\$40,000} = (1 + r)^5\) Next, take the fifth root of both sides: \((1 + r) = \sqrt[5]{\frac{\$70,000}{\$40,000}}\) Now, subtract 1 from both sides to solve for r: \(r = \sqrt[5]{\frac{\$70,000}{\$40,000}} - 1\) Now, simplifying the expression we have: \(r \approx 0.1194\)
05

Convert the decimal to a percentage

To convert the decimal to a percentage, multiply by 100: \(r \approx 0.1194 \times 100 = 11.94\%\) So the effective annual interest rate for Martha's investment is approximately 11.94%.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Compound Interest Formula
The compound interest formula is the mathematical expression used to calculate the value of an investment over time when interest is reinvested to earn additional interest. The standard formula is given by:

\[A = P(1 + r)^n\]

Here,
  • \(A\) represents the final amount of the investment,
  • \(P\) is the principal amount (initial investment),
  • \(r\) is the annual interest rate (expressed as a decimal), and
  • \(n\) is the number of compounding periods.
The beauty of this formula lies in its ability to depict the exponential growth of an investment due to compound interest. As the number of compounding periods increases, the investment's growth rate accelerates because interest is earned not just on the initial principal but also on the accumulated interest from previous periods. It's a powerful tool that investors use to estimate their investment's growth over time.
Initial Investment Calculation
The initial investment, also known as the principal, is the starting amount of money placed in an investment vehicle. Calculating it can be straightforward if you directly invest a lump sum of money, as was the case with Martha's \($40,000\). However, if your investment is through periodic payments, the initial investment would be the sum of all payments made towards the investment at the start.

In some scenarios, you might even need to reverse-calculate the initial investment if you know the final value and the rate of return. This involves rearranging the compound interest formula to solve for \(P\), that is, making \(P\) the subject of the formula. Understanding the initial investment is crucial because it serves as the reference point from which any returns or growth are measured.
Final Investment Value
The final investment value, noted as \(A\) in the compound interest formula, represents the total amount your investment is worth after interest is applied over a set period. This is the figure investors are typically most interested in as it indicates the payoff of their initial commitment.

To find the final investment value, as in Martha's scenario, you take her initial investment and apply the compound interest formula. The time value of money concept shows that, with compound interest, funds grow at a quicker rate than with simple interest because you earn interest on the interest. Reaching the final value is a key moment of satisfaction for investors, as it represents the fruition of their patience and strategic planning.
Rate of Return
The rate of return is a critical metric that represents the percentage increase or decrease in an investment over a period, usually expressed annually. In the case of Martha's investment, determining the effective annual interest rate鈥攅ssentially the rate of return鈥攚as the goal.

Calculated as a percentage, the rate of return is found by rearranging the compound interest formula to solve for the rate (\(r\)). After determining the rate, it is then converted from a decimal to a percentage to be more intuitively understood. A positive rate of return means an investment's value has increased, while a negative rate suggests a loss. Savvy investors look for opportunities that will yield a high rate of return, balancing the potential rewards with the inherent risks involved.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the periodic payment \(R\) required to accumulate a sum of \(S\) dollars over \(t\) yr with interest earned at the rate of \(r \% / y\) year compounded \(m\) times a year. S=350,000, r=7.5, t=10, m=12

Anthony invested a sum of money 5 yr ago in a savings account that has since paid interest at the rate of \(8 \%\) /year compounded quarterly. His investment is now worth \(\$ 22,289.22\). How much did he originally invest?

FiNANCING A HomE The Taylors have purchased a \(\$ 270,000\) house. They made an initial down payment of \(\$ 30,000\) and secured a mortgage with interest charged at the rate of \(8 \% / y\) ear on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over \(30 \mathrm{yr}\), what monthly payment will the Taylors be required to make? What is their equity (disregarding appreciation) after 5 yr? After 10 yr? After 20 yr?

Find the twentieth term and sum of the first 20 terms of the geometric progression \(-3,3,-3,3, \ldots\)

FINANGING CoLLEGE EXPENSES Yumi's grandparents presented her with a gift of \(\$ 20,000\) when she was 10 yr old to be used for her college education. Over the next \(7 \mathrm{yr}\), until she turned 17 , Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 5.5\%/year compounded monthly. Upon turning 17 , Yumi now plans to withdraw her funds in equal annual installments over the next \(4 \mathrm{yr}\), starting at age \(18 .\) If the college fund is expected to earn interest at the rate of \(6 \% /\) year, compounded annually, what will be the size of each installment?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.