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Zorluk: OrtaRatio and Proportion Word Problems

At a research foundation, annual grant funding was initially allocated among three divisions—Artificial Intelligence, Biotechnology, and Renewable Energy—in the ratio 7:4:37 : 4 : 3, respectively. At mid-year, the foundation redistributed the funding by transferring $10\$10 million from the Artificial Intelligence division to the Renewable Energy division, while the Biotechnology division's funding remained unchanged. After this transfer, the ratio of Artificial Intelligence funding to Renewable Energy funding became 3:23 : 2. What was the total annual grant funding, in millions of dollars, allocated across all three divisions?

Cevap: 140 million dollars

Cevap

The total annual grant funding allocated across all three divisions was 140140 million dollars.
By setting the initial funding amounts to 7x7x, 4x4x, and 3x3x million dollars, the total funding is 14x14x. The transfer alters Artificial Intelligence funding to 7x−107x - 10 and Renewable Energy funding to 3x+103x + 10. Setting their ratio to 32\frac{3}{2} yields the equation 2(7x−10)=3(3x+10)2(7x - 10) = 3(3x + 10), which simplifies to 5x=505x = 50, giving x=10x = 10. Thus, total funding is 14×10=14014 \times 10 = 140 million dollars.

Adım Adım Çözüm

1
Define variables for the initial allocation based on the ratio 7:4:37 : 4 : 3.
Artificial Intelligence funding =7x= 7x, Biotechnology funding =4x= 4x, Renewable Energy funding =3x= 3x, and total funding =7x+4x+3x=14x= 7x + 4x + 3x = 14x.
Ratios represent relative parts of a whole multiplier xx.
2
Model the redistribution of funds.
New Artificial Intelligence funding =7x−10= 7x - 10; New Renewable Energy funding =3x+10= 3x + 10.
Transferring $10\$10 million decreases Artificial Intelligence funding by 1010 and increases Renewable Energy funding by 1010.
3
Equate the new ratio to 3:23 : 2 and solve for xx.
7x−103x+10=32  ⟹  14x−20=9x+30  ⟹  5x=50  ⟹  x=10\frac{7x - 10}{3x + 10} = \frac{3}{2} \implies 14x - 20 = 9x + 30 \implies 5x = 50 \implies x = 10.
Cross-multiplication converts ratio relationships into a solvable linear equation.
4
Calculate the total annual grant funding.
Total funding =14x=14×10=140= 14x = 14 \times 10 = 140 million dollars.
Substituting x=10x = 10 into the total funding expression 14x14x gives the final answer.

Anahtar Kavram

Algebraic setup of multi-part ratio redistribution problems
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Konular

Ratio and Proportion Word Problems

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