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Zorluk: OrtaRatio and Proportion Word Problems

A boutique investment fund allocates its capital among Technology, Healthcare, and Renewable Energy in the ratio 3:4:53 : 4 : 5, respectively. After receiving an additional $12\$12 million in new capital, the entire new amount is invested in Healthcare, changing the ratio of capital among Technology, Healthcare, and Renewable Energy to 3:6:53 : 6 : 5. What was the total initial capital, in millions of dollars, invested in the fund?

  1. A
    2424
  2. 7272Cevap
  3. C
    8484
  4. D
    3636
  5. E
    6060

Cevap

7272 million dollars
Let the initial capital allocations be 3x3x, 4x4x, and 5x5x million dollars for Technology, Healthcare, and Renewable Energy, respectively. The initial total capital is 3x+4x+5x=12x3x + 4x + 5x = 12x. When $12\$12 million is added to Healthcare, its new amount becomes 4x+124x + 12. Comparing the unchanged Technology amount to the updated Healthcare amount gives the proportion 3x4x+12=36=12\frac{3x}{4x + 12} = \frac{3}{6} = \frac{1}{2}. Cross-multiplying yields 6x=4x+126x = 4x + 12, so 2x=122x = 12 and x=6x = 6. The total initial capital is 12×6=7212 \times 6 = 72 million dollars.

Adım Adım Çözüm

1
Represent initial allocations using a multiplier xx
Technology =3x= 3x, Healthcare =4x= 4x, Renewable Energy =5x= 5x
Ratios represent proportional parts, so multiplying each term by a common constant xx expresses the actual amounts.
2
Set up the equation following the addition of $12\$12 million to Healthcare
New Healthcare allocation =4x+12= 4x + 12. The new ratio is 3x:(4x+12):5x=3:6:53x : (4x + 12) : 5x = 3 : 6 : 5.
Only the Healthcare amount changes, while Technology and Renewable Energy allocations remain 3x3x and 5x5x respectively.
3
Solve for xx using the ratio between Technology and Healthcare
\frac{3x}{4x + 12} = \frac{3}{6} \implies \frac{3x}{4x + 12} = \frac{1}{2} \implies 6x = 4x + 12 \implies 2x = 12 \implies x = 6
Equating the algebraic ratio of Technology to Healthcare to the new simplified ratio allows solving for xx.
4
Calculate the total initial capital
\text{Total initial capital} = 3x + 4x + 5x = 12x = 12(6) = 72
Summing the initial parts (12x12x) and substituting x=6x = 6 yields the initial total.

Anahtar Kavram

Ratio scaling and constant-part proportion adjustments
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Ratio and Proportion Word Problems

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