John deposited $10,000 to open a new savings account that earned 4% annual interest, compounded quarterly. If there were no...

1. Translate the problem requirements: John starts with \(\$10,000\), earns \(4\%\) annual interest compounded quarterly (meaning interest is calculated and added 4 times per year), and we need the account balance after exactly 6 months (which is 2 quarters).
2. Determine the quarterly interest rate: Since \(4\%\) annual interest is compounded quarterly, divide the annual rate by 4 to get the rate applied each quarter.
3. Apply compound interest for the first quarter: Calculate interest earned on the initial \(\$10,000\) and add it to get the balance after 3 months.
4. Apply compound interest for the second quarter: Calculate interest on the new balance from step 3 and add it to get the final balance after 6 months.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday terms:

• John starts with \(\$10,000\) in his savings account
• The bank pays him \(4\%\) interest per year
• But here's the key part: they don't wait until the end of the year to give him interest. Instead, they calculate and add interest to his account 4 times per year (every 3 months) - this is what "compounded quarterly" means
• We want to know how much money he has after exactly 6 months

Since \(6 \text{ months} = 2 \text{ quarters}\), the bank will calculate and add interest to his account exactly twice during this period.

Process Skill: TRANSLATE - Converting the compound interest terminology into concrete timing and calculations

2. Determine the quarterly interest rate

Since the bank adds interest 4 times per year instead of once, we need to figure out how much interest they add each time.

Think of it this way: if John earns \(4\%\) for the whole year, and the bank splits this into 4 equal payments throughout the year, then each quarterly payment \(= 4\% ÷ 4 = 1\%\)

So every 3 months, the bank calculates \(1\%\) of whatever money is in John's account at that time and adds it to his balance.

Quarterly interest rate \(= 1\% = 0.01\) (as a decimal)

3. Apply compound interest for the first quarter

After the first 3 months, here's what happens:

• John starts with: \(\$10,000\)
• The bank calculates \(1\%\) of this amount: \(1\% × \$10,000 = 0.01 × \$10,000 = \$100\)
• The bank adds this interest to his account: \(\$10,000 + \$100 = \$10,100\)

So after 3 months, John has \(\$10,100\) in his account.

4. Apply compound interest for the second quarter

After the second 3 months (which gets us to the 6-month mark), here's what happens:

• John now starts this quarter with: \(\$10,100\) (notice this is MORE than his original \(\$10,000\))
• The bank calculates \(1\%\) of his current balance: \(1\% × \$10,100 = 0.01 × \$10,100 = \$101\)
• The bank adds this interest to his account: \(\$10,100 + \$101 = \$10,201\)

This is the key insight of compound interest: in the second quarter, John earns interest not just on his original \(\$10,000\), but also on the \(\$100\) interest he earned in the first quarter. That extra \(\$1\) (\(\$101 - \$100 = \$1\)) comes from earning interest on previously earned interest.

Final Answer

After 6 months, John has \(\$10,201\) in his account.

Looking at our answer choices: (D) \(\$10,201\)

This matches our calculation perfectly. The answer is (D) \(\$10,201\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "compounded quarterly"

Students often confuse quarterly compounding with simple interest or annual compounding. They might think the \(4\%\) is applied once at the end of 6 months, or they might incorrectly assume that "quarterly" means the interest rate itself is \(4\%\) per quarter (which would be \(16\%\) annually). The key insight is that quarterly compounding means the annual rate is divided by 4 and applied 4 times per year.

2. Incorrect time period calculation

Some students struggle to convert 6 months into the correct number of compounding periods. They might think 6 months equals 6 quarterly periods instead of recognizing that \(6 \text{ months} = 2 \text{ quarters}\). This leads to applying the compound interest formula for the wrong number of periods.

3. Confusion about which interest rate to use

Students may incorrectly use the full \(4\%\) annual rate for each quarterly calculation instead of dividing it by 4 to get the \(1\%\) quarterly rate. This fundamental misunderstanding of how compound interest rates are divided across compounding periods leads to drastically incorrect calculations.

Errors while executing the approach

1. Arithmetic errors in percentage calculations

When calculating \(1\%\) of amounts like \(\$10,000\) and \(\$10,100\), students might make basic arithmetic mistakes. For example, they might calculate \(1\%\) of \(\$10,100\) as \(\$100\) instead of \(\$101\), or make decimal placement errors when converting percentages to decimals (using 1 instead of 0.01).

2. Forgetting to add interest to principal

Some students calculate the interest amounts correctly (\(\$100\) for first quarter, \(\$101\) for second quarter) but then forget to add these amounts to the running balance. They might just add the interest amounts together or forget to compound properly by using the updated balance for the second calculation.

3. Using simple interest instead of compound interest

Even after identifying the correct quarterly rate and time periods, students might apply simple interest logic. They calculate \(1\% × \$10,000 × 2 = \$200\) total interest, giving \(\$10,200\), because they fail to account for earning interest on previously earned interest in the second quarter.

Errors while selecting the answer

1. Selecting the simple interest result

After calculating both simple and compound interest scenarios (or getting confused during execution), students might select choice (C) \(\$10,200\), which represents the simple interest calculation rather than the correct compound interest result of \(\$10,201\).

2. Off-by-one errors in final selection

Students who perform most calculations correctly might make a final arithmetic error and select choice (B) \(\$10,101\) instead of (D) \(\$10,201\), essentially giving the result as if only one quarter of compounding occurred instead of two.