To determine how much Etsy should invest in both B-rated bonds and certificates of deposit (CDs) to achieve exactly $5,000 per year in interest, we can set up a system of equations.
Let:
- \( x \) = amount invested in B-rated bonds (at 13% interest)
- \( y \) = amount invested in certificates of deposit (at 3% interest)
We have the following two conditions:
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The total investment: \[ x + y = 60,000 \]
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The total interest earned from both investments: \[ 0.13x + 0.03y = 5,000 \]
Now we can solve this system of equations step-by-step.
Step 1: Solve the first equation for one variable
From the first equation:
\[ y = 60,000 - x \]
Step 2: Substitute into the second equation
Now, substitute \( y \) in the second equation:
\[ 0.13x + 0.03(60,000 - x) = 5,000 \]
Distributing \( 0.03 \):
\[ 0.13x + 1,800 - 0.03x = 5,000 \]
Combine like terms:
\[ (0.13 - 0.03)x + 1,800 = 5,000 \] \[ 0.10x + 1,800 = 5,000 \]
Step 3: Isolate \( x \)
Subtract 1,800 from both sides:
\[ 0.10x = 5,000 - 1,800 \] \[ 0.10x = 3,200 \]
Now divide by 0.10:
\[ x = \frac{3,200}{0.10} = 32,000 \]
Step 4: Solve for \( y \)
Substituting \( x \) back to find \( y \):
\[ y = 60,000 - x = 60,000 - 32,000 = 28,000 \]
Conclusion:
Etsy should invest:
- $32,000 in B-rated bonds at 13%
- $28,000 in certificates of deposit at 3%
To verify:
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Interest from B-rated bonds: \[ 0.13 \times 32,000 = 4,160 \]
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Interest from CDs: \[ 0.03 \times 28,000 = 840 \]
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Total interest: \[ 4,160 + 840 = 5,000 \]
The calculations confirm that these investments result in exactly $5,000 per year in interest.
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