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Chapter 3: Problem 13

Linda has \(d\) dollars in an account that pays 3.4\(\%\) interest, compounded weekly. She withdraws \(w\) dollars. Express her first week's interest algebraically.

Short Answer

Expert verified
The algebraic expression that represents the first week's interest Linda earned after withdrawing \(w\) dollars from her account is \( (d - w)\frac{3.4}{100 \times 52} \).

Step by step solution

01

Define the variables

Let \(d\) be the initial amount of money Linda has in the account. Let \(w\) be the amount Linda withdraws. The remaining balance is then \(d - w\). The weekly interest rate is found by dividing the annual interest rate by the number of weeks in a year, so it is \(\frac{3.4}{100 \times 52}\).
02

Apply the formula for compound interest

The formula for compound interest is \(A = P (1 + r/n)^{nt}\), where \(A\) is the final amount, \(P\) is the principal (initial amount), \(r\) is the annual interest rate (as a decimal), \(n\) is the number of times compounded each year, and \(t\) is the time (in years). In this case, the initial amount \(P=d-w\), the rate \(r = 3.4/100\) is the annual interest rate, \(n = 52\) is the number of compounding periods in a year, and \(t = 1/52\) as we're interested in the interest for 1 week.
03

Simplify to find the interest

The formula for the future value \(A\) simplifies to \(A = (d-w)(1 + \frac{3.4}{100 \times 52})^{1}\). Hence, the first week鈥檚 interest can be found by subtracting the initial principal from this, giving interest = \(A - P = (d - w)(1 + \frac{3.4}{100 \times 52})^{1} - (d - w) = (d - w)\frac{3.4}{100 \times 52}\)

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Key Concepts

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

Interest Rate
Interest rate is the percentage at which interest is paid by a borrower for the use of money. In Linda's case, the bank provides an interest rate of 3.4% annually. However, since Linda's interest is compounded weekly, the annual interest rate needs to be converted into a weekly rate.
The weekly interest rate is calculated by dividing the annual rate by the number of compounding periods in a year. Since there are 52 weeks in a year, the weekly interest rate is \((\frac{3.4}{100 \times 52})\).
This conversion is crucial because compounding frequency impacts how often the interest is applied to the account, influencing the total interest earned.
Algebraic Expression
An algebraic expression consists of variables, numbers, and arithmetic operations. In the exercise, we use an algebraic expression to represent the interest Linda earns after the first week.
Initially, the account balance is represented as \(d - w\), where \(d\) represents the total dollars Linda initially had, and \(w\) is the amount she withdraws. Therefore, the remaining balance in the account after withdrawal is \(d - w\).
The interest for the first week is then given by \( (d-w) \cdot \frac{3.4}{100 \times 52} \). This expression provides a straightforward way to calculate the interest based on how much remains in the account after the withdrawal.
Compounding Frequency
Compounding frequency refers to how often interest is applied to the account balance. In Linda's scenario, the interest is compounded weekly, which means interest is calculated and added to the account balance every week.
This frequency plays a significant role in compound interest calculations. The more frequently interest is compounded, the more interest will be generated over time, due to the accumulation of previously earned interest.
  • With \(n = 52\), Linda鈥檚 account experiences 52 compounding periods per year.
  • The time for one period \(t\) is \(\frac{1}{52}\) since we want to find the interest for one week.
Understanding compounding frequency helps in realizing how often interest is accumulated and how it can affect overall earnings or costs in financial situations.

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Most popular questions from this chapter

Ridgewood Savings Bank charges a \(\$ 27\) per check overdraft protection fee. On July \(8,\) Nancy had \(\$ 1,400\) in her account. Over the next four days, the following checks arrived for payment at her bank: July \(9, \$ 1,380.15,\) July \(10, \$ 670\) and \(\$ 95.67 ;\) July \(11, \$ 130 ;\) and July \(12, \$ 87.60 .\) How much will she pay in overdraft protection fees? How much will she owe the bank after July 12\(?\)

Investigate the difference between compounding annually and simple interest for parts a-j. a. Find the simple interest for a one-year \(\mathrm{CD}\) for \(\$ 5,000\) at a 6\(\%\) b. Find the interest for a one-year CD for \(\$ 5,000\) at an interest rate of \(6 \%,\) compounded annually. c. Compare the results from parts a and b. d. Find the simple interest for a three-year \(\mathrm{CD}\) for \(\$ 5,000\) at an enterest rate of 6\(\% .\) e. Find the interest for a three-year CD for \(\$ 5,000\) at an interest rate f. Compare the results from parts d and e. g. Find the simple interest for a six-year \(\mathrm{CD}\) for \(\$ 5,000\) at an interest rate of 4\(\% .\) h. Find the interest for a six-year CD for \(\$ 5,000\) at an interest rate of 4\(\%\) , componded annually. i. Compare the results from parts \(\mathrm{g}\) and \(\mathrm{h} .\) j. Is interest compounded annually the same as simple interest? Explain.

Mike deposits \(\$ 5,000\) in a three-year \(\mathrm{CD}\) account that yields 3.5\(\%\) interest, compounded weekly. What is his ending balance at the end of three years?

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?

Gary and Ann have a joint checking account. Their balance at the beginning of October was 9,145.87 dollar . During the month they made deposits totaling 2,783.7 dollar, wrote checks totaling 4,871.90 dollar , paid a maintenance fee of 12 dollar, and earned 11.15 dollar in interest on the account. What was the balance at the end of the month?
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