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Chapter 6: Problem 8

APPLICATION Eight months ago, Tori's parents put \(\$ 5,000\) into a savings account that earns \(3 \%\) annual interest. Now, her dentist has suggested that she get braces. a. If the interest is calculated each month, what is the monthly interest rate? b. If Tori's parents use the money in their savings account, how much do they have? c. If Tori's dentist had suggested braces 3 months ago, how much money would have been in her parents' savings account? d. Tori's dentist says she can probably wait up to 2 months before having the braces fitted. How much will be in her parents' savings account if she waits?

Short Answer

Expert verified
a. 0.25%; b. $5101.01; c. $5062.95; d. $5125.57.

Step by step solution

01

Calculate the Monthly Interest Rate

First, convert the annual interest rate of 3% to a monthly interest rate. The formula to convert an annual interest rate to a monthly rate is to divide the annual rate by 12 months. Thus, the monthly interest rate is \( \frac{3\%}{12} = 0.25\% \).
02

Calculate the Total in Savings After 8 Months

Use the compound interest formula to find the total amount in the savings account after 8 months. The formula is \( A = P(1 + r)^n \), where \( A \) is the amount of money accumulated, \( P \) is the principal amount (\$5000), \( r \) is the monthly interest rate (0.0025 as a decimal), and \( n \) is the number of months. Calculate \[ A = 5000(1 + 0.0025)^8 \].
03

Calculation for Part b

Calculate \( 5000 \times (1 + 0.0025)^8 \). This equals \( 5000 \times 1.02020125 \approx 5101.01 \). So, after 8 months, they have approximately \$5101.01 in the account.
04

Calculate the Total in Savings After 5 Months

If the interest had been calculated 3 months ago, they would need to calculate the interest accrued over 5 months. Again, use the formula: \[ A = 5000(1 + 0.0025)^5 \].
05

Calculation for Part c

Calculate \( 5000 \times (1 + 0.0025)^5 \). This equals \( 5000 \times 1.0125906 \approx 5062.95 \). Thus, they would have had approximately \$5062.95 in the account 3 months ago.
06

Total in Savings After 10 Months

If Tori's family waits 2 more months, calculate for a total of 10 months of interest. Use the formula: \( A = 5000(1 + 0.0025)^{10} \).
07

Calculation for Part d

Calculate \( 5000 \times (1 + 0.0025)^{10} \). This results in \( 5000 \times 1.025113 \approx 5125.57 \). So, if they wait 2 more months, they will have approximately \$5125.57 in the account.

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Key Concepts

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

Savings Account
A savings account is a type of bank account where you can store your money securely while earning some interest over time. Think of it as a safe way to both save and grow your money. Typically, banks offer savings accounts to help people put away money for future needs, like Tori's parents did for her braces.
  • Earning interest means that the bank pays you for holding your money there.
  • Interest is added periodically (e.g., monthly or annually), increasing your total savings.
  • Easy access to your money, though some accounts might limit withdrawals.
Putting money in a savings account is a great first step towards building personal wealth and financial security. It allows your savings to grow over time with little to no risk.
Interest Calculation
Interest calculation is essential to understanding how money grows in a savings account. When money is placed in a savings account, it earns interest, which is basically a small percentage of the money paid to you for keeping your money with the bank. Interest can be calculated in different ways, but compound interest is one of the most powerful.
  • Compound interest means you earn interest on the initial amount (the principal) and also on any interest that has already been added to your account.
  • The formula for compound interest is: \[ A = P(1 + r)^n \]Where:
    • \( A \) is the total amount after interest,
    • \( P \) is the principal amount (initial deposit),
    • \( r \) is the monthly interest rate,
    • \( n \) is the number of times the interest is applied (months in this case).
By using the compound interest formula, you can see how quickly savings can grow over a number of months or years, making it an essential concept for managing money effectively.
Monthly Interest Rate
Understanding the monthly interest rate is key to calculating how much interest your savings account earns over time. Simply put, it is the fraction of the annual interest rate that applies to each month. To get the monthly rate from an annual rate, you simply divide by 12 months.
  • The formula is: \[ ext{Monthly Rate} = rac{ ext{Annual Rate}}{12} \]For example, if the annual interest rate is 3%, then the monthly rate comes out as \( rac{3 ext{ extperthousand}}{12} = 0.25 ext{ extperthousand} \).
  • This rate is essential as it determines how much additional money your account will earn each month, influencing how quickly your savings grow.
By converting the annual rate into a monthly rate, you can apply it more frequently in your compound interest calculations, giving a more precise view of growth from month to month.

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