/*! 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 3 On Olga's 16th birthday, her unc... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 3

On Olga's 16th birthday, her uncle invested \(\$ 2,000\) in an account that was locked into a 4.75\(\%\) interest rate, compounded monthly. How much will Olga have in the account when she turns 18\(?\) Round to the nearest cent.

Short Answer

Expert verified
Olga will have approximately \$2,205.66 in the account when she turns 18.

Step by step solution

01

Identify Variables for the formula

Let's first identify the numbers we know. The principal amount \( P \) is \$2000. The annual interest rate \( r \) is 0.0475 (4.75\% as a decimal). The time \( t \) is 2 years (from Olga's 16th to 18th birthday). And the interest is compounded monthly, so \( n \) is 12.
02

Substitute values into the formula

Now we substitute these values into the formula. \( A = 2000(1 + 0.0475/12)^{12*2} \).
03

Evaluate the formula

Evaluating and solving this calculation gives a future value of Olga's investment.
04

Round to the nearest cent

Finally, we round the result to the nearest cent, as the problem requires that.

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.

Interest Rates
Interest rates are a key concept in finance. They represent the cost of borrowing or the gain from lending money.
In this case, Olga's uncle invested money, so we're looking at the gain from the investment over a defined period.
  • Annual Interest Rate: This tells us how much interest will be paid or earned in one year. In Olga's case, the annual rate is 4.75%.
  • Compounded Monthly: Compounding means calculating interest on both the initial amount and the interest that has been added over time. Monthly compounding involves calculating interest 12 times per year.
The formula to calculate compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:
  • \( A \) is the amount of money accumulated after n years, including interest.
  • \( P \) is the principal amount \( \$2000 \)
  • \( r \) is the annual interest rate (0.0475 in Olga's case).
  • \( n \) is the number of times interest is compounded per year (12).
  • \( t \) is the number of years the money is invested for (2 years).
Financial Algebra
In financial algebra, we use mathematical formulas and equations to solve problems related to finance.
This discipline helps us understand, model, and manage financial situations involving money and risk.
  • Understanding Variables: Identifying and defining variables such as principal amount, interest rate, compounding frequency, and time is crucial.
  • Equation Application: By substituting known values into a financial formula, we calculate unknown values like future investment worth, payments, or savings.
Financial algebra is essential for calculating and predicting the financial outcomes of both short-term and long-term investments.
It assists in making informed decisions by showing potential growth based on varying conditions.
Investment Calculation
Investment calculation involves determining how much an investment will grow over a particular period. These calculations help investors decide where and how to invest their resources.
In Olga's case, her uncle鈥檚 investment will grow based on the compound interest formula.
  • Compound Interest Calculation: By substituting the principal amount, interest rate, and time into Olga鈥檚 formula, we find out how much money will be there by her 18th birthday.
  • Interpreting the Result: The result shows growth potential, allowing decision-making based on future value, which in Olga's case is how much she'll receive at 18.
  • Rounding Off: Since currency calculations involve cents, results are often rounded to the nearest cent for practical financial planning and transaction ease.
Investment calculations like these are vital to mapping financial growth over time and ensuring money is working effectively.

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

Rob deposits \(\$ 1,000\) in a savings account at New York State Bank that pays 4.4\(\%\) interest, compounded monthly. a. How much is in his account at the end of one year? b. What is the APY for this account to the nearest hundredth of a percent?

Hannah wants to write a general formula and a comparison statement that she can use each month when she reconciles her checking account. Use the Checking Account Summary at the right to write a formula and a statement for Hannah. $$\begin{array}{|l|l|}\hline \text { Checking Account Summary } \\ \hline \text { Ending Balance } & {B} \\ \hline \text { Deposits } & {D} \\ \hline \text { Checks Outstanding } & {C} \\ \hline \text { Revised Statement Balance } & {S} \\ \hline \text { Check Register Balance } & {R} \\\ \hline\end{array}$$

Create a check register for the transactions listed. There is a \(\$ 2.25\) fee for each ATM use. a. Your balance on 10\(/ 29\) is \(\$ 237.47\) b. You write check 115 on 10\(/ 29\) for \(\$ 182.00\) to Fox High School. c. You deposit a paycheck for \(\$ 162.75\) on 10\(/ 30 .\) d. You deposit a \(\$ 25\) check for your birthday on 11\(/ 4\) . e. On \(11 / 5,\) you go to a sporting event and run out of money. You use the ATM in the lobby to get \(\$ 15\) for snacks. f. Your credit card bill is due on \(11 / 10,\) so on 11\(/ 7\) you write check 116 to Credit USA for \(\$ 51.16 .\) g. Your sister repays you \(\$ 20\) on 11\(/ 10 .\) You deposit it. h. You withdraw \(\$ 25\) from the ATM to buy flowers on 11\(/ 12\) . i. You deposit your paycheck for \(\$ 165.65\) on 11\(/ 16\) . j. Your deposit a late birthday check for \(\$ 35\) on 11\(/ 17\) .

Assume \(\$ 20,000\) is deposited into a savings account. Bedford Bank offers an annual rate of 4\(\%\) simple interest for five years. Slick Bank offers a rate of 20\(\%\) simple interest for one year. Which earns more interest?

Albert Einstein said that compound interest was 鈥�. . .the most powerful thing I have ever witnessed.鈥� Work through the following exercises to discover a pattern Einstein discovered which is now known as the Rule of 72.. a. Suppose that you invest \(\$ 2,000\) at a 1\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? b. Suppose that you invest \(\$ 4,000\) at a 2\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? c. Suppose that you invest \(\$ 20,000\) at a 6\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? d. Albert Einstein noticed a very interesting pattern when an initial deposit doubles. In each of the three examples above, multiply the value of t that you determined times the percentage amount. For example, in a. multiply t by 1. What do you notice? e. Einstein called this the Rule of 72 because for any initial deposit and for any interest percentage, \(72 \div\) (percentage) will give you the approximate number of years it will take for the initial deposit to double in value. Einstein also said that 鈥淚f people really understood the Rule of 72 they would never put their money in banks.鈥� Suppose that a 10-year-old has $500 to invest. She puts it in her savings account that has a 1.75% annual interest rate. How old will she be when the money doubles?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.