<Question>
A worker receives an annual wage of $\$20{,}000$, which he always deposits into a savings account at the end of the year. By the end of the third year (when he makes the third deposit), he wants to have at least $\$66,200$ in the account to finance the purchase of a house. What is the minimal compound interest rate that the savings account must provide? Express your answer as a percentage, but do not include the percent sign.
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is about a worker who saves money annually in a savings account with compound interest. He wants to have at least $66,200 by the end of the third year. His annual wage is $20,000, which he deposits at the end of each year. We need to find the minimal compound interest rate required for this.

First, I need to recall how compound interest works.
The formula for compound interest is A = P(1 + r/n)^(nt), where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.

However, in this case, the worker is depositing $20,000 at the end of each year. So, this is actually an ordinary annuity, where payments are made at the end of each period. The formula for the future value of an ordinary annuity is different. Let me remember that.

The future value of an ordinary annuity is calculated by:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)

But since the interest is compounded annually, n would be 1.

<prompt_1.1> Determine the future value of an ordinary annuity with annual compounding. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Wait, the problem says "compound interest rate," but doesn't specify how many times it's compounded. Hmm. Maybe it's compounded annually? That's the most common default assumption if not specified. Let me check.

If it's compounded annually, then n = 1. So, the formula simplifies to FV = P * [(1 + r)^t - 1] / r. But wait, actually, the standard formula for the future value of an ordinary annuity with annual compounding is:

FV = P * [((1 + r)^t - 1)/r]

Yes, that's right. Because each deposit is made at the end of the year, so the first deposit earns interest for (t - 1) years, the second for (t - 2) years, etc. , and the last deposit earns no interest. So, the formula for the future value is the sum of each deposit's future value.

So, for three deposits (end of year 1, 2, 3), each $20,000:

FV = 20,000*(1 + r)^2 + 20,000*(1 + r) + 20,000

Alternatively, this can be written as:

FV = 20,000 * [(1 + r)^3 - 1]/r

Yes, that's the standard formula for the future value of an ordinary annuity. So, we can use that.

So, the equation we need is:

20,000 * [(1 + r)^3 - 1]/r ≥ 66,200

We need to solve for r, the interest rate. Then, express it as a percentage.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
This might not have a closed-form solution, so we might need to use numerical methods or trial and error to find the minimal r that satisfies the inequality.

Let me write that equation again:

<prompt_2.1> Set up the equation for the future value of the annuity and identify the need for numerical methods to solve it. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
20,000 * [(1 + r)^3 - 1]/r = 66,200

Divide both sides by 20,000:

[(1 + r)^3 - 1]/r = 66,200 / 20,000

Calculate 66,200 / 20,000:

66,200 ÷ 20,000 = 3.31

So:

[(1 + r)^3 - 1]/r = 3.31

So, the equation to solve is:

[(1 + r)^3 - 1]/r = 3.31
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me denote x = r. Then the equation becomes:

[(1 + x)^3 - 1]/x = 3.31

Let's expand (1 + x)^3:

(1 + x)^3 = 1 + 3x + 3x^2 + x^3

Subtract 1:

(1 + x)^3 - 1 = 3x + 3x^2 + x^3

Divide by x:

(3x + 3x^2 + x^3)/x = 3 + 3x + x^2

So, the equation simplifies to:

3 + 3x + x^2 = 3.31

Subtract 3.31 from both sides:

x^2 + 3x + 3 - 3.31 = 0

Simplify:

x^2 + 3x - 0.31 = 0

Wait, is that correct? Let me check again.

Wait, (1 + x)^3 - 1 divided by x is equal to 3 + 3x + x^2.

So, 3 + 3x + x^2 = 3.31

Therefore, x^2 + 3x + 3 - 3.31 = x^2 + 3x - 0.31 = 0

So, quadratic equation: x^2 + 3x - 0.31 = 0

Hmm, solving for x. Let's use quadratic formula:

<prompt_3.1> Solve the quadratic equation to find the interest rate. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
x = [-3 ± sqrt(9 + 1.24)] / 2

Because discriminant D = 9 + 4*0.31 = 9 + 1.24 = 10.24

sqrt(10.24) = 3.2

So, x = [-3 ± 3.2]/2

We can discard the negative solution because interest rate can't be negative. So:

x = (-3 + 3.2)/2 = 0.2/2 = 0.1

So, x = 0.1, which is 10%.

Wait, so r = 10%?
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Let me check if this is correct.

Let's plug r = 0.1 into the original equation:

<prompt_4.1> Verify the solution by checking if the calculated interest rate satisfies the original equation. </prompt_4.1>

But wait, the problem says "minimal compound interest rate". So, 10% is the answer.

But let me check if maybe there's a lower rate that also satisfies the equation. Because sometimes, when dealing with equations that are not linear, there might be multiple solutions.

<prompt_4.2> Consider the possibility of multiple solutions and confirm if 10% is the minimal rate. </prompt_4.2>

Wait, but maybe my approach is wrong here. Because when I expanded (1 + x)^3 -1 and divided by x, I got 3 + 3x + x^2. But is that correct?

Wait, let me verify.

<prompt_4.3> Verify the expansion and simplification of the equation to ensure the solution's correctness. </prompt_4.3>

But let's check with another rate. Suppose we take r = 0.09.

<prompt_4.4> Perform a final verification by calculating the future value at different interest rates to ensure 10% is the minimal rate. </prompt_4.4>

But let me check again. Maybe when we derived the quadratic equation, we assumed something that isn't true.

<prompt_4.5> Re-evaluate the derivation and assumptions to confirm the solution's correctness. </prompt_4.5>
</Plan_4>

<Execution_4>
<execution_4.1>
[(1 + 0.1)^3 - 1]/0.1 = [1.331 - 1]/0.1 = 0.331 / 0.1 = 3.31

Which matches the required 3.31. So, r = 0.1 or 10% is the solution.
</execution_4.1>

<execution_4.2>
Let's see.

Wait, the equation we set up was:

[(1 + x)^3 - 1]/x = 3.31

Which simplified to x^2 + 3x - 0.31 = 0, leading to x = 0.1.

But let's verify this. Let's plug x = 0.1 into the left-hand side:

[(1 + 0.1)^3 - 1]/0.1 = [(1.331) - 1]/0.1 = 0.331 / 0.1 = 3.31

Which matches the right-hand side. So, 10% is a solution. Since the quadratic equation only gives us one positive solution, that's the only feasible one. Therefore, 10% is the minimal rate.
</execution_4.2>

<execution_4.3>
If (1 + x)^3 - 1 = x*(3 + 3x + x^2). Let's compute x*(3 + 3x + x^2) = 3x + 3x^2 + x^3. Then, (1 + x)^3 - 1 = x^3 + 3x^2 + 3x. So yes, that's equal. Therefore, [(1 + x)^3 - 1]/x = 3 + 3x + x^2.

Therefore, the equation is correct. Then, solving 3 + 3x + x^2 = 3.31 gives x = 0.1. So, r = 10%. Therefore, the minimal rate is 10%.
</execution_4.3>

<execution_4.4>
Then:

FV = 20,000 * [ (1.09)^3 - 1 ] / 0.09

Calculate (1.09)^3:

1.09^3 = 1.09 * 1.09 * 1.09

1.09 * 1.09 = 1.1881

1.1881 * 1.09 ≈ 1.1881 + 0.107629 = 1.2957 (approximately)

Then, (1.2957 - 1)/0.09 ≈ 0.2957 / 0.09 ≈ 3.2856

Multiply by 20,000: 20,000 * 3.2856 ≈ 65,712, which is less than 66,200.

So, 9% gives about 65,712, which is less. At 10%, let's compute:

(1.10)^3 = 1.331

(1.331 - 1)/0.10 = 0.331 / 0.10 = 3.31

Multiply by 20,000: 20,000 * 3.31 = 66,200. Exactly.

So, 10% gives exactly 66,200. Therefore, the minimal rate is 10%.
</execution_4.4>

<execution_4.5>
Wait, perhaps not. Let's see.

Wait, when we have [(1 + x)^3 -1]/x = 3.31, and expanded (1 + x)^3 -1 as 3x + 3x^2 + x^3, then dividing by x gives 3 + 3x + x^2. Therefore, the equation becomes 3 + 3x + x^2 = 3.31, leading to x^2 + 3x - 0.31 = 0. So, the algebra seems correct. Solving gives x = 0.1. So, 10%.
</execution_4.5>
</Execution_4>

<Plan_5>
Based on execution_4.5:
Alternatively, perhaps check with r = 10%:

Year 1 deposit: $20,000 earns interest for 2 years: 20,000*(1.1)^2 = 20,000*1.21 = 24,200

Year 2 deposit: $20,000 earns interest for 1 year: 20,000*1.1 = 22,000

Year 3 deposit: $20,000

Total: 24,200 + 22,000 + 20,000 = 66,200. Exactly.

Therefore, 10% is the exact rate needed.
So, the minimal rate is 10%.
</Plan_5>

<Execution_5>
No prompt to execute.
</Execution_5>

<Final_answer>
Therefore, the minimal compound interest rate that the savings account must provide is $\boxed{10\%}$.
</Final_answer>
</think>