Loan X has a principal of $10,000x and a yearly simple interest rate of 4%. Loan Y has a principal...

Phase 1: Owning the Dataset

Understanding the Setup

We have two loans being consolidated:

• Loan X: Principal = \(\$10,000\mathrm{x}\), Interest rate = 4%
• Loan Y: Principal = \(\$10,000\mathrm{y}\), Interest rate = 8%
• Consolidated Loan Z: Principal = \(\$(10,000\mathrm{x} + 10,000\mathrm{y})\), Interest rate = r%

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

Visualization

Let's create a simple table to organize our information:

Phase 2: Understanding the Question

What We Need to Find

We need to select values for x and y such that the consolidated loan has an interest rate of 5%.

This means we need:
\(\frac{4\mathrm{x} + 8\mathrm{y}}{\mathrm{x} + \mathrm{y}} = 5\)

Setting Up the Equation

Let's solve this equation algebraically:

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

Key insight: \(\mathrm{x} = 3\mathrm{y}\)

This tells us that x must be exactly 3 times y for the consolidated rate to be 5%.

Phase 3: Finding the Answer

Systematic Check

Given our answer choices: [21, 32, 51, 64, 81, 96]

We need to find values where \(\mathrm{x} = 3\mathrm{y}\).

Let's check if y could be one of these values:

• If \(\mathrm{y} = 32 \rightarrow \mathrm{x} = 3(32) = 96\) ✓ (96 is in our choices!)

Stop here - we found our answer.

Verification

Let's verify with \(\mathrm{x} = 96\) and \(\mathrm{y} = 32\):
\(\mathrm{r} = \frac{4(96) + 8(32)}{96 + 32}\)
\(\mathrm{r} = \frac{384 + 256}{128}\)
\(\mathrm{r} = \frac{640}{128}\)
\(\mathrm{r} = 5\) ✓

Phase 4: Solution

Final Answer:

• X Value: 96
• Y Value: 32

These values satisfy our requirement that \(\mathrm{x} = 3\mathrm{y}\), resulting in a consolidated loan interest rate of exactly 5%.