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Loan X has a principal of \(\$\mathrm{x}\) and a yearly simple interest rate of \(4\%\). Loan Y has a principal of \(\$\mathrm{y}\) and a yearly simple interest rate of \(8\%\). Loans X and Y will be consolidated to form Loan Z with a principal of \(\$(\mathrm{x} + \mathrm{y})\) and a yearly simple interest rate of \(\mathrm{r}\%\), where \(\mathrm{r} = \frac{4\mathrm{x} + 8\mathrm{y}}{\mathrm{x} + \mathrm{y}}\).
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.
21,000
32,000
51,000
64,000
81,000
96,000
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.
We need to find values of x and y from the given choices such that r = 5%.
\(\mathrm{5 = \frac{4x + 8y}{x + y}}\)
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.
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.
\(\mathrm{r = \frac{4 \times 96,000 + 8 \times 32,000}{96,000 + 32,000}}\)
\(\mathrm{r = \frac{384,000 + 256,000}{128,000}}\)
\(\mathrm{r = \frac{640,000}{128,000}}\)
\(\mathrm{r = 5}\) ✓
Final Answer:
These values satisfy the relationship \(\mathrm{x = 3y}\) and produce a consolidated interest rate of exactly 5%.