Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex...

1. Translate the problem requirements: Alex makes two deposits of \(\mathrm{x}\) dollars each - one at the start and another after one year. The account earns \(8\%\) annual interest compounded annually. After two years total, the account has \(\mathrm{w}\) dollars. We need to express \(\mathrm{x}\) in terms of \(\mathrm{w}\).
2. Track each deposit separately through time: Follow the first deposit for two full years of compound growth, and the second deposit for one year of growth.
3. Set up the total value equation: Add the final values of both deposits to equal \(\mathrm{w}\) dollars.
4. Solve for x and match to answer choices: Isolate \(\mathrm{x}\) and verify it matches the given format in the options.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

Alex starts a savings account and makes two separate deposits of \(\mathrm{x}\) dollars each. The first deposit happens at the very beginning (let's call this "Year 0"). The second deposit happens exactly one year later ("Year 1"). The account pays \(8\%\) interest each year, and this interest gets added to the account balance so it can earn interest too (this is called compound interest).

After two full years have passed, Alex looks at the account and sees \(\mathrm{w}\) dollars total. Our job is to figure out how much each deposit (\(\mathrm{x}\)) was, if we know the final amount (\(\mathrm{w}\)).

Process Skill: TRANSLATE - Converting the problem's business language into a clear mathematical timeline

2. Track each deposit separately through time

The key insight is to follow each deposit's journey separately, since they earn interest for different amounts of time:

First deposit (made at Year 0):
• Starts with \(\mathrm{x}\) dollars
• After 1 year: earns \(8\%\) interest, so becomes \(\mathrm{x} + 0.08\mathrm{x} = 1.08\mathrm{x}\) dollars
• After 2 years: earns another \(8\%\) on the new balance, so becomes \(1.08\mathrm{x} \times 1.08 = 1.08^2 \times \mathrm{x}\) dollars

Second deposit (made at Year 1):
• Starts with \(\mathrm{x}\) dollars at the beginning of Year 1
• After 1 year (at end of Year 2): earns \(8\%\) interest, so becomes \(\mathrm{x} + 0.08\mathrm{x} = 1.08\mathrm{x}\) dollars

Notice how the first deposit gets two years of compound growth (\(1.08^2\)), while the second deposit only gets one year of growth (\(1.08\)).

Process Skill: VISUALIZE - Breaking down the timeline helps us see why each deposit grows differently

3. Set up the total value equation

Now we add up what both deposits are worth at the end of two years:

In plain English: (Value of first deposit after 2 years) + (Value of second deposit after 1 year) = Total account value

Mathematically: \(1.08^2 \times \mathrm{x} + 1.08 \times \mathrm{x} = \mathrm{w}\)

We can factor out \(\mathrm{x}\): \(\mathrm{x}(1.08^2 + 1.08) = \mathrm{w}\)

Let's calculate \(1.08^2\): \(1.08^2 = 1.1664\)
So our equation becomes: \(\mathrm{x}(1.1664 + 1.08) = \mathrm{w}\)
Or: \(\mathrm{x}(1.08^2 + 1.08) = \mathrm{w}\)

4. Solve for x and match to answer choices

To isolate \(\mathrm{x}\), we divide both sides by the coefficient:

\(\mathrm{x} = \mathrm{w} / (1.08^2 + 1.08)\)

We can factor the denominator: \(1.08^2 + 1.08 = 1.08(1.08 + 1) = 1.08 \times 2.08\)

But looking at our answer choices, we want the form that matches exactly. Our expression \(\mathrm{x} = \mathrm{w} / (1.08^2 + 1.08)\) can also be written as:

\(\mathrm{x} = \mathrm{w} / (1.08 + 1.08^2)\)

Looking at the choices, this matches Choice D: \(\mathrm{w} / (1.08 + 1.08^2)\)

Final Answer

The answer is Choice D: \(\mathrm{w} / (1.08 + 1.08^2)\)

This makes intuitive sense: to find the original deposit amount, we divide the final account value by the total "growth factor" - which is \(1.08\) (from the second deposit's one year of growth) plus \(1.08^2\) (from the first deposit's two years of compound growth).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the timing of deposits and interest calculations
Students often get confused about when each deposit earns interest. They might think both deposits earn interest for the same amount of time, or incorrectly assume the second deposit starts earning interest immediately when the first deposit is made. The key insight they miss is that the first deposit gets 2 full years of compound interest while the second deposit only gets 1 year.

2. Treating this as a simple interest problem instead of compound interest
Some students see "\(8\) percent annual interest" and apply simple interest formulas, calculating interest as a flat percentage each year without compounding. They fail to recognize that "compounded annually" means the interest itself earns interest in subsequent years, leading to exponential growth (\(1.08^n\)) rather than linear growth.

3. Attempting to combine deposits before tracking their individual growth
Students might try to add both \(\mathrm{x}\)-dollar deposits together first (getting \(2\mathrm{x}\)) and then apply some average interest rate. This approach fails because the deposits have different growth periods - they need to be tracked separately through their respective timelines before being combined at the end.

Errors while executing the approach

1. Incorrectly calculating compound interest powers
When calculating how much the first deposit grows over 2 years, students might write \(1.08 \times 2\) instead of \(1.08^2\). They confuse the number of compounding periods with multiplication, forgetting that compound interest requires raising the growth factor to the power of the number of periods.

2. Mixing up which deposit gets which growth factor
Students might assign \(1.08^2\) to the second deposit and \(1.08\) to the first deposit, reversing which deposit earns interest for how long. This happens when they lose track of the timeline - the first deposit (made at Year 0) should get \(1.08^2\) and the second deposit (made at Year 1) should get \(1.08\).

3. Arithmetic errors when expanding or factoring expressions
When setting up the equation \(\mathrm{x}(1.08^2 + 1.08) = \mathrm{w}\), students might incorrectly calculate \(1.08^2\) as \(2.16\) instead of \(1.1664\), or make errors when factoring expressions. These computational mistakes lead to wrong coefficients in the final answer.

Errors while selecting the answer

1. Choosing an answer with the wrong order of terms in the denominator
Students arrive at the correct expression \(\mathrm{x} = \mathrm{w}/(1.08^2 + 1.08)\) but then select an answer choice that has the terms flipped, like \(\mathrm{w}/(1.08 + 1.16)\) instead of \(\mathrm{w}/(1.08 + 1.08^2)\). They don't recognize that \(1.08^2\) should remain in exponential form to match the correct answer choice D.

2. Converting \(1.08^2\) to decimal form and matching wrong answer choices
After correctly deriving \(\mathrm{x} = \mathrm{w}/(1.08^2 + 1.08)\), students calculate \(1.08^2 = 1.1664\) and look for \(\mathrm{w}/(1.08 + 1.1664)\) or \(\mathrm{w}/(1.1664 + 1.08)\). When they don't find this exact form, they might incorrectly choose answer B or C which have similar-looking decimal values but represent completely different mathematical relationships.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a strategic value for w

Let's work backwards from the final amount. We need \(\mathrm{w}\) to be a value that will give us clean numbers when we solve for \(\mathrm{x}\). Looking at the answer choices, we see denominators involving \(1.08\) and \(1.08^2\). Let's choose \(\mathrm{w} = 2.24\), which will work nicely with these values.

Step 2: Apply the correct answer choice to find x

Since we know the correct answer is D, we have:
\(\mathrm{x} = \mathrm{w}/(1.08 + 1.08^2)\)
\(\mathrm{x} = 2.24/(1.08 + 1.1664)\)
\(\mathrm{x} = 2.24/2.2464\)
\(\mathrm{x} = 1\)

Step 3: Verify this works with the compound interest scenario

• Initial deposit: \(\mathrm{x} = \$1\)
• After 1 year: \(\$1 \times 1.08 = \$1.08\)
• Second deposit at end of year 1: \(\$1\)
• Total at start of year 2: \(\$1.08 + \$1 = \$2.08\)
• After year 2: \(\$2.08 \times 1.08 = \$2.2464\)

Wait, this doesn't equal our chosen \(\mathrm{w} = 2.24\). Let me recalculate more precisely.

Step 4: More precise calculation

• First deposit grows for 2 years: \(\$1 \times 1.08^2 = \$1.1664\)
• Second deposit grows for 1 year: \(\$1 \times 1.08 = \$1.08\)
• Total after 2 years: \(\$1.1664 + \$1.08 = \$2.2464\)

Step 5: Verify the formula

Using \(\mathrm{w} = 2.2464\) and \(\mathrm{x} = 1\):
\(\mathrm{x} = \mathrm{w}/(1.08 + 1.08^2) = 2.2464/(1.08 + 1.1664) = 2.2464/2.2464 = 1\) ✓

This confirms that answer choice D is correct using our smart numbers approach.