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Loan X has a principal of \(\$10,000\mathrm{x}\) and a yearly simple interest rate of \(4\%\). Loan Y has a principal of \(\$10,000\mathrm{y}\) and a yearly simple interest rate of \(8\%\). Loans X and Y will be consolidated to form Loan Z with a principal of \(\$(10,000\mathrm{x} + 10,000\mathrm{y})\) and a yearly simple interest rate of \(\mathrm{r}\%\), where \(\mathrm{r} = \frac{4\mathrm{x}+8\mathrm{y}}{\mathrm{x}+\mathrm{y}}\).
In the table, select a value for x and a value for y corresponding to a yearly simple interest rate of \(5\%\) for the consolidated loan. Make only two selections, one in each column.
21
32
51
64
81
96
We have two loans being consolidated:
Where \(\mathrm{r} = \frac{4\mathrm{x} + 8\mathrm{y}}{\mathrm{x} + \mathrm{y}}\)
Let's create a simple table to organize our information:
| Loan | Principal | Interest Rate |
| X | \(\$10,000\mathrm{x}\) | 4% |
| Y | \(\$10,000\mathrm{y}\) | 8% |
| Z (consolidated) | \(\$10,000(\mathrm{x}+\mathrm{y})\) | r% = \(\frac{4\mathrm{x}+8\mathrm{y}}{\mathrm{x}+\mathrm{y}}\) |
We need to select values for x and y such that the consolidated loan has an interest rate of 5%.
This means we need:
\(\frac{4\mathrm{x} + 8\mathrm{y}}{\mathrm{x} + \mathrm{y}} = 5\)
Let's solve this equation algebraically:
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%.
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:
Stop here - we found our answer.
Let's verify with \(\mathrm{x} = 96\) and \(\mathrm{y} = 32\):
\(\mathrm{r} = \frac{4(96) + 8(32)}{96 + 32}\)
\(\mathrm{r} = \frac{384 + 256}{128}\)
\(\mathrm{r} = \frac{640}{128}\)
\(\mathrm{r} = 5\) ✓
Final Answer:
These values satisfy our requirement that \(\mathrm{x} = 3\mathrm{y}\), resulting in a consolidated loan interest rate of exactly 5%.