Problem Breakdown

Betty's interest is calculated quarterly based on the minimum balance during each quarter. She earns simple interest at a rate of 5% per annum, credited quarterly. Here’s a step-by-step approach to calculating the total interest Betty will earn and how much will be credited on December 31st.

Information:

• Annual interest rate: 5% per annum
• Quarterly interest rate: $\frac{5\%}{4} = 1.25\%$ per quarter
• Starting balance on January 1st: $80,000
• Transactions:
  • Jan 10: Deposit $2,500 (new balance = $82,500)
  • Mar 5: Withdrawal of $4,000 (new balance = $78,500)
  • Nov 1: Withdrawal of $13,000 (new balance = $65,500)

Plan:

We'll determine the minimum balance for each quarter, then calculate the interest for each quarter using simple interest formula $I = P \times r \times t$, where:

• $I$ is the interest,
• $P$ is the principal (minimum balance for the quarter),
• $r$ is the quarterly interest rate (1.25% or 0.0125),
• $t$ is time in years (for a quarter, $t = \frac{1}{4}$).

Step 1: Calculate the minimum balance for each quarter

1. First Quarter (Jan 1 to Mar 31):

   • Balance on Jan 1: $80,000
   • Deposit on Jan 10: New balance = $80,000 + $2,500 = $82,500
   • Withdrawal on Mar 5: New balance = $82,500 - $4,000 = $78,500
   • Minimum balance for Q1 = $78,500

2. Second Quarter (Apr 1 to Jun 30):

   • No transactions during this period.
   • Minimum balance for Q2 = $78,500

3. Third Quarter (Jul 1 to Sep 30):

   • No transactions during this period.
   • Minimum balance for Q3 = $78,500

4. Fourth Quarter (Oct 1 to Dec 31):

   • Withdrawal on Nov 1: New balance = $78,500 - $13,000 = $65,500
   • Minimum balance for Q4 = $65,500

Step 2: Calculate the interest for each quarter

Using the formula for simple interest $I = P \times r \times t$, where $r = 1.25\% = 0.0125$ and $t = \frac{1}{4} = 0.25$ year for each quarter:

1. Q1 (Jan 1 - Mar 31): $I_1 = 78,500 \times 0.0125 = 981.25$

2. Q2 (Apr 1 - Jun 30): $I_2 = 78,500 \times 0.0125 = 981.25$

3. Q3 (Jul 1 - Sep 30): $I_3 = 78,500 \times 0.0125 = 981.25$

4. Q4 (Oct 1 - Dec 31): $I_4 = 65,500 \times 0.0125 = 818.75$

Step 3: Calculate the total interest credited on December 31st

The total interest is the sum of the quarterly interest amounts: $I_{\text{total}} = I_1 + I_2 + I_3 + I_4 = 981.25 + 981.25 + 981.25 + 818.75 = 3,762.50$

Conclusion:

The total interest credited to Betty’s account on December 31st will be $3,762.50.

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Would you like further details or have any questions about the steps?

Here are 5 related questions:

1. How would the interest calculation differ if it were compounded quarterly instead of simple interest?
2. What happens to the interest calculation if Betty makes another deposit in December?
3. How does the calculation change if the interest rate increases during the year?
4. Can you explain how the minimum balance rule affects the interest calculation?
5. How would you calculate interest if the account had multiple deposits and withdrawals each quarter?

Tip: Always keep track of the minimum balance for interest calculations with a rule like this to avoid overestimating the amount!