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Zorluk: OrtaAlgebraic Word Problems and Equation Modeling

An investment firm allocated a total of $120,000\$120,000 between two venture capital funds, Fund X and Fund Y. Fund X yielded an annual simple interest rate of 8%8\%, while Fund Y yielded an annual simple interest rate of 12%12\%. If the total interest earned from both funds combined after one year was $11,600\$11,600, how much money was invested in Fund X?

  1. $70,000\$70,000Cevap
  2. B
    $50,000\$50,000
  3. C
    $60,000\$60,000
  4. D
    $58,000\$58,000
  5. E
    $75,000\$75,000

Cevap

$70,000\$70,000
The correct answer is $70,000\$70,000. By modeling the investment in Fund X as xx, the investment in Fund Y becomes 120,000−x120,000 - x. Setting up the total annual interest equation 0.08x+0.12(120,000−x)=11,6000.08x + 0.12(120,000 - x) = 11,600 yields −0.04x=−2,800-0.04x = -2,800, giving x=70,000x = 70,000.

Adım Adım Çözüm

1
Define variables for the invested amounts
Let xx be the amount invested in Fund X. The remaining amount invested in Fund Y is $120,000−x\$120,000 - x.
The total capital is $120,000\$120,000, so the investments must sum to this total.
2
Set up the interest equation
0.08x+0.12(120,000−x)=11,6000.08x + 0.12(120,000 - x) = 11,600
Total interest is the sum of interest from Fund X (8%8\% of xx) and Fund Y (12%12\% of 120,000−x120,000 - x).
3
Expand and simplify the linear equation
0.08x+14,400−0.12x=11,600⇒−0.04x+14,400=11,6000.08x + 14,400 - 0.12x = 11,600 \Rightarrow -0.04x + 14,400 = 11,600
Distribute 0.120.12 across (120,000−x)(120,000 - x) and combine like terms.
4
Isolate the variable xx
−0.04x=11,600−14,400⇒−0.04x=−2,800⇒x=−2,800−0.04=70,000-0.04x = 11,600 - 14,400 \Rightarrow -0.04x = -2,800 \Rightarrow x = \frac{-2,800}{-0.04} = 70,000
Subtract 14,40014,400 from both sides and divide by −0.04-0.04 to find the investment in Fund X.

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Linear Equation Modeling for Investment Allocations
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Algebraic Word Problems and Equation Modeling

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