How much more interest will maria receive if she invests $1000 for one year at x% annual interest, compounded semianually,...

1. Translate the problem requirements: We need to find the difference between two scenarios: (1) \(\$1000\) invested at \(\mathrm{x}\%\) annual interest compounded semiannually versus (2) \(\$1000\) invested at \(\mathrm{x}\%\) annual interest compounded annually. The question asks for how much MORE interest Maria receives in the first scenario.
2. Calculate the final amount for annual compounding: Apply the basic compound interest principle where interest is added once at the end of the year.
3. Calculate the final amount for semiannual compounding: Apply compound interest twice - once at 6 months with half the annual rate, then again for the second 6 months on the new principal.
4. Find the difference in interest earned: Subtract the interest from annual compounding from the interest from semiannual compounding to determine the additional benefit.
5. Express the result in terms of x: Simplify the algebraic expression to match one of the given answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what Maria is comparing. She has \(\$1000\) to invest for exactly one year at the same interest rate \(\mathrm{x}\%\) in both scenarios. The only difference is how often the interest gets calculated and added to her account.

Scenario 1: Interest calculated and added once per year (annually)
Scenario 2: Interest calculated and added twice per year (semiannually)

We need to find how much MORE interest she earns in Scenario 2 compared to Scenario 1.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical scenarios

2. Calculate the final amount for annual compounding

With annual compounding, Maria's interest is calculated once at the end of the year.

Think of it this way: If the annual interest rate is \(\mathrm{x}\%\), then after one year, her \(\$1000\) becomes:

• Original amount: \(\$1000\)
• Interest earned: \(\$1000 \times (\mathrm{x}/100) = \$10\mathrm{x}\)
• Total amount: \(\$1000 + \$10\mathrm{x}\)

Using the compound interest approach:
Final Amount = \(\$1000 \times (1 + \mathrm{x}/100) = \$1000 \times (100 + \mathrm{x})/100\)

Interest earned annually = Final Amount - Principal = \(\$1000 \times (100 + \mathrm{x})/100 - \$1000 = \$1000 \times \mathrm{x}/100 = \$10\mathrm{x}\)

3. Calculate the final amount for semiannual compounding

With semiannual compounding, the interest is calculated and added twice during the year. Each time, Maria earns interest at half the annual rate \((\mathrm{x}/2)\%\) on whatever amount is in her account.

First 6 months:

• Starting amount: \(\$1000\)
• Interest rate for 6 months: \(\mathrm{x}/2\%\)
• Amount after 6 months: \(\$1000 \times (1 + \mathrm{x}/200) = \$1000 \times (200 + \mathrm{x})/200\)

Second 6 months:

• Starting amount: \(\$1000 \times (200 + \mathrm{x})/200\)
• Interest rate for 6 months: \(\mathrm{x}/2\%\)
• Final amount: \(\$1000 \times (200 + \mathrm{x})/200 \times (1 + \mathrm{x}/200)\)
• Final amount: \(\$1000 \times (200 + \mathrm{x})/200 \times (200 + \mathrm{x})/200\)
• Final amount: \(\$1000 \times (200 + \mathrm{x})^2/40000\)

Interest earned semiannually = \(\$1000 \times (200 + \mathrm{x})^2/40000 - \$1000\)
= \(\$1000 \times [(200 + \mathrm{x})^2 - 40000]/40000\)
= \(\$1000 \times [(200 + \mathrm{x})^2 - 200^2]/40000\)

4. Find the difference in interest earned

Now we need the difference between semiannual interest and annual interest.

Let's expand \((200 + \mathrm{x})^2\):
\((200 + \mathrm{x})^2 = 200^2 + 2(200)(\mathrm{x}) + \mathrm{x}^2 = 40000 + 400\mathrm{x} + \mathrm{x}^2\)

So: \((200 + \mathrm{x})^2 - 40000 = 400\mathrm{x} + \mathrm{x}^2\)

Interest from semiannual compounding = \(\$1000 \times (400\mathrm{x} + \mathrm{x}^2)/40000 = \$1000 \times (400\mathrm{x} + \mathrm{x}^2)/40000\)
= \(\$10\mathrm{x} + \$\mathrm{x}^2/40\)

Difference = Interest (semiannual) - Interest (annual)
= \((\$10\mathrm{x} + \$\mathrm{x}^2/40) - \$10\mathrm{x}\)
= \(\$\mathrm{x}^2/40\)

Process Skill: SIMPLIFY - Breaking down complex algebraic expressions step by step

5. Express the result in terms of x

The additional interest Maria receives from semiannual compounding compared to annual compounding is \(\$\mathrm{x}^2/40\).

Looking at our answer choices:

1. \(5\mathrm{x}\)
2. \(10\mathrm{x}\)
3. \(\mathrm{x}^2/20\)
4. \(\mathrm{x}^2/40\) ✓
5. \((10\mathrm{x}+\mathrm{x}^2/40)\)

Final Answer

The answer is D. \(\mathrm{x}^2/40\)

This makes intuitive sense: the benefit of more frequent compounding depends on the square of the interest rate, and becomes more significant as the interest rate increases. The \(\mathrm{x}^2\) term reflects that higher interest rates benefit more from frequent compounding.

Common Faltering Points

Errors while devising the approach

• Misunderstanding compound interest formula application: Students often confuse how to apply the compound interest formula when compounding frequency changes. They may incorrectly think that semiannual compounding means applying the full annual rate twice, rather than applying half the rate \((\mathrm{x}/2)\%\) twice during the year.
• Misinterpreting what the question is asking for: The question asks for the DIFFERENCE in interest earned, not the total interest amounts. Students may calculate both interest amounts correctly but then provide one of the individual amounts as their final answer instead of finding the difference.
• Confusion about time periods: Students may struggle with the concept that semiannual compounding means the interest is calculated and added every 6 months, not that the rate is applied over 6-month periods differently. This can lead to incorrect setup of the compound interest calculations.

Errors while executing the approach

• Algebraic expansion errors: When expanding \((200 + \mathrm{x})^2\), students commonly make errors such as forgetting the middle term \((2\times200\times\mathrm{x} = 400\mathrm{x})\) or making arithmetic mistakes in the expansion, leading to incorrect expressions like \((200 + \mathrm{x})^2 = 40000 + \mathrm{x}^2\) instead of \(40000 + 400\mathrm{x} + \mathrm{x}^2\).
• Fraction and decimal conversion mistakes: Students often struggle when converting percentages to decimals in compound interest formulas. For example, they might use \(\mathrm{x}\) instead of \(\mathrm{x}/100\), or incorrectly write \(\mathrm{x}/2\%\) as \(\mathrm{x}/2\) instead of \(\mathrm{x}/200\) in decimal form.
• Simplification errors in the final calculation: When subtracting the annual interest from semiannual interest, students may make arithmetic errors in combining like terms, such as incorrectly canceling the \(\$10\mathrm{x}\) terms or making mistakes when working with the denominator 40000.

Errors while selecting the answer

• Selecting total interest instead of the difference: Students may correctly calculate that semiannual compounding gives \(\$10\mathrm{x} + \$\mathrm{x}^2/40\) total interest but then select answer choice E \((10\mathrm{x} + \mathrm{x}^2/40)\) instead of recognizing that this represents the total interest, not the additional interest compared to annual compounding.
• Unit confusion: Students might calculate the difference correctly as \(\mathrm{x}^2/40\) but then second-guess themselves about whether this should be expressed in dollars or as a percentage, potentially leading them to select an incorrect answer choice that seems to better match their expected units.

Alternate Solutions

Smart Numbers Approach

We can solve this problem by choosing a convenient value for the interest rate x that makes calculations straightforward.

Step 1: Choose a smart value for x
Let's set \(\mathrm{x} = 10\) (representing 10% annual interest rate). This choice makes percentage calculations clean and avoids complex decimals.

Step 2: Calculate final amount with annual compounding
With annual compounding at 10%:
Final amount = \(\$1000 \times (1 + 0.10) = \$1000 \times 1.10 = \$1100\)
Interest earned = \(\$1100 - \$1000 = \$100\)

Step 3: Calculate final amount with semiannual compounding
With semiannual compounding, the rate per period is 10%/2 = 5%, applied twice:
After 6 months: \(\$1000 \times (1 + 0.05) = \$1000 \times 1.05 = \$1050\)
After 1 year: \(\$1050 \times (1 + 0.05) = \$1050 \times 1.05 = \$1102.50\)
Interest earned = \(\$1102.50 - \$1000 = \$102.50\)

Step 4: Find the additional interest
Additional interest = \(\$102.50 - \$100 = \$2.50\)

Step 5: Test answer choices with x = 10
A. \(5\mathrm{x} = 5(10) = 50 \neq 2.50\)
B. \(10\mathrm{x} = 10(10) = 100 \neq 2.50\)
C. \(\mathrm{x}^2/20 = (10)^2/20 = 100/20 = 5 \neq 2.50\)
D. \(\mathrm{x}^2/40 = (10)^2/40 = 100/40 = 2.50\) ✓
E. \((10\mathrm{x} + \mathrm{x}^2/40) = (100 + 2.50) = 102.50 \neq 2.50\)

Step 6: Verify with another value
Let's verify with \(\mathrm{x} = 20\) to ensure our answer is correct:
Annual: Interest = \(\$200\)
Semiannual: Final amount = \(\$1000 \times (1.10)^2 = \$1210\), Interest = \(\$210\)
Difference = \(\$210 - \$200 = \$10\)
Testing D: \(\mathrm{x}^2/40 = (20)^2/40 = 400/40 = 10\) ✓

The answer is D. \(\mathrm{x}^2/40\)