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An investor opened a money market account with a single deposit of \(\$6000\) on Dec. 31, 2001. The interest earned on the account was calculated and reinvested quarterly. The compound interest for the first 3 quarters of 2002 was \(\$125\), \(\$130\), and \(\$145\), respectively. If the investor made no deposits or withdrawals during the year, approximately what annual rate of interest must the account earn for the 4th quarter in order for the total interest earned on the account for the year to be \(10\%\) of the initial deposit?
Let's break down what this problem is asking in plain English. We have an investor who put $6000 into an account at the end of 2001. During 2002, the account earned interest each quarter (every 3 months), and that interest got added back to the account.
The key requirement is that by the end of 2002, the total interest earned for the whole year should be 10% of the original $6000 deposit. Let's calculate what 10% of $6000 means:
\(10\% \text{ of } \$6000 = 0.10 \times \$6000 = \$600\)
So we need the account to earn exactly $600 in interest during the entire year 2002.
Process Skill: TRANSLATE - Converting the percentage requirement into a concrete dollar amount
Now let's see where we stand after the first three quarters of 2002:
• Quarter 1 (Jan-Mar 2002): $125 interest earned
• Quarter 2 (Apr-Jun 2002): $130 interest earned
• Quarter 3 (Jul-Sep 2002): $145 interest earned
Total interest earned in first 3 quarters = \(\$125 + \$130 + \$145 = \$400\)
Our target for the whole year is $600, and we've already earned $400. This means we're on track, but we need to see what the 4th quarter must contribute.
This is straightforward arithmetic:
Interest needed for Q4 = Total target - Interest already earned
Interest needed for Q4 = \(\$600 - \$400 = \$200\)
So the account must earn exactly $200 in interest during the 4th quarter (Oct-Dec 2002) to meet our goal.
To find what interest rate is needed, we need to know how much money is in the account at the beginning of Q4. This is the original deposit plus all the interest that has been added back:
Balance at start of Q4 = Original deposit + All interest earned through Q3
Balance at start of Q4 = \(\$6000 + \$400 = \$6400\)
So during Q4, this $6400 needs to earn $200 in interest.
Now we can find the quarterly interest rate for Q4:
Quarterly rate for Q4 = Interest earned ÷ Principal amount
Quarterly rate for Q4 = \(\$200 \div \$6400 = 0.03125 = 3.125\%\)
The question asks for the annual rate. Since there are 4 quarters in a year, we convert by multiplying by 4:
Annual rate = \(4 \times 3.125\% = 12.5\%\)
Process Skill: INFER - Recognizing that we need to convert quarterly rate to annual rate to match answer choices
The account must earn an annual rate of 12.5% in the 4th quarter for the total yearly interest to equal 10% of the initial deposit.
Looking at our answer choices, this matches choice E: 12.5%
Verification: If Q4 earns at 12.5% annually (3.125% quarterly), then \(\$6400 \times 0.03125 = \$200\), giving us total yearly interest of \(\$400 + \$200 = \$600 = 10\%\) of $6000 ✓
1. Misunderstanding the 10% target requirement: Students may incorrectly think they need to find what annual rate would make the 4th quarter interest alone equal to 10% of $6000 ($600), rather than understanding that the total yearly interest (including all 4 quarters) should equal 10% of the initial deposit. This leads them to set up the wrong equation entirely.
2. Confusing quarterly vs. annual rate terminology: Students may not realize the question is asking for an "annual rate" that the account must earn "for the 4th quarter." They might think this means finding a rate that applies to the whole year, rather than finding what annual rate (expressed as a yearly percentage) corresponds to the quarterly performance needed in Q4.
3. Forgetting about compound interest accumulation: Students may incorrectly use $6000 as the principal for calculating Q4 interest, forgetting that the previous quarters' interest has been "reinvested quarterly" and added to the account balance. This leads to using the wrong principal amount in their calculations.
1. Arithmetic errors in basic addition: When calculating the total interest earned in the first three quarters ($125 + $130 + $145 = $400), students may make simple addition mistakes, leading to an incorrect baseline for determining how much Q4 needs to contribute.
2. Incorrect principal calculation for Q4: Even if students understand that interest gets reinvested, they may make errors when calculating the account balance at the start of Q4. For example, they might add the quarterly interest amounts incorrectly or forget to add them to the original $6000 deposit.
3. Division and percentage conversion errors: When calculating $200 ÷ $6400 = 0.03125 and converting to 3.125%, students may make decimal placement errors or rounding mistakes that lead to an incorrect quarterly rate.
1. Selecting the quarterly rate instead of annual rate: Students may correctly calculate the quarterly rate of 3.125% but then select answer choice A (3.1%) thinking this is the final answer, forgetting that the question asks for the annual rate and they need to multiply by 4.
2. Confusion between 10% and the calculated rate: Some students may see that the problem mentions "10 percent" and automatically gravitate toward answer choice C (10.0%), thinking this must be correct without completing their calculations or verifying their work.