Loan X has a principal of $x and a yearly simple interest rate of 4%. Loan Y has a principal...

Phase 1: Owning the Dataset

Visualization: Weighted Average on Number Line

4% -------|----------- 5% -----------|------- 8%
    Loan X             r            Loan Y
    (weight: x)                    (weight: y)

The consolidated rate r is the weighted average of the two individual rates, where the weights are the loan principals.

Given Information:

• Loan X: principal = $x, rate = 4%
• Loan Y: principal = $y, rate = 8%
• Consolidated Loan Z: principal = $(x+y), rate = r%
• Formula: \(\mathrm{r = \frac{4x + 8y}{x + y}}\)

Phase 2: Understanding the Question

We need to find values of x and y from the given choices such that r = 5%.

Setting up the equation:

\(\mathrm{5 = \frac{4x + 8y}{x + y}}\)

Multiplying both sides by (x + y):

\(\mathrm{5(x + y) = 4x + 8y}\)
\(\mathrm{5x + 5y = 4x + 8y}\)
\(\mathrm{5x - 4x = 8y - 5y}\)
\(\mathrm{x = 3y}\)

Key insight: For a 5% consolidated rate, the principal of Loan X must be exactly 3 times the principal of Loan Y.

Phase 3: Finding the Answer

Since \(\mathrm{x = 3y}\), let's check which value from our choices can be y:

If \(\mathrm{y = \$32,000 \rightarrow x = 3 \times \$32,000 = \$96,000}\)

Checking if $96,000 is in our choices: Yes! ✓

? Stop here - we found our answer.

Verification (use calculator for precision):

\(\mathrm{r = \frac{4 \times 96,000 + 8 \times 32,000}{96,000 + 32,000}}\)

\(\mathrm{r = \frac{384,000 + 256,000}{128,000}}\)

\(\mathrm{r = \frac{640,000}{128,000}}\)

\(\mathrm{r = 5}\) ✓

Phase 4: Solution

Final Answer:

• x = $96,000
• y = $32,000

These values satisfy the relationship \(\mathrm{x = 3y}\) and produce a consolidated interest rate of exactly 5%.