A portion of $6600 is invested at a 5% annual return, while the remainder is invested at a...
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A portion of $6600 is invested at a 5% annual return, while the remainder is invested at a 3% annual return. If the annual income from the portion earning a 5% return is twice that of the other portion, what is the total income from the two investments after one year?
Show answer & explanation
Set up two equations: x + y = 6600 (total investment) and 0.05x = 2(0.03y) (income condition). Solving gives x = 3600 and y = 3000. Total income = 3600(0.05) + 3000(0.03) = 180 + 90 = $270.
Step-by-step Derivation:
Step 1: Define variables. Let x = amount at 5%, y = amount at 3%.
Step 2: Set up equations.
Equation 1: x + y = 6600
Equation 2: 0.05x = 2(0.03y) → 0.05x = 0.06y → 5x = 6y → x = 1.2y
Step 3: Substitute into Equation 1.
1.2y + y = 6600
2.2y = 6600
y = 3000
Step 4: Find x.
x = 6600 - 3000 = 3600
Step 5: Verify the income condition.
Income at 5%: 3600 × 0.05 = 180
Income at 3%: 3000 × 0.03 = 90
Is 180 = 2 × 90? Yes ✓
Step 6: Calculate total income.
Total = 180 + 90 = $270