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A regional energy transition board reviewed the adoption of residential solar panels across three counties: County Alpha, County Beta, and County Gamma. At the beginning of 20252025, there were a total of 4,0004,000 households equipped with solar panels across these three counties. The number of such households in County Alpha was 1,5001,500. At that time, County Beta had 50%50\% more solar-equipped households than County Gamma.

Over the course of 20252025, the number of solar-equipped households in County Alpha increased by 20%20\%, while County Gamma saw a 40%40\% increase. By the end of 20252025, the total number of solar-equipped households across all three counties reached 5,0005,000.

Based on the information provided, which of the following statements are correct?

  1. The percentage increase in solar-equipped households in County Beta during 20252025 was 20%20\%.Cevap
  2. B
    At the end of 20252025, the number of solar-equipped households in County Beta was 2,1002,100.
  3. The ratio of the number of solar-equipped households in County Gamma to County Alpha at the end of 20252025 was 7:97:9.Cevap
  4. D
    The overall percentage growth in solar-equipped households across all three counties during 20252025 was 20%20\%.

Cevap

The statements confirming that County Beta saw a 20%20\% increase and that the final ratio of County Gamma to County Alpha was 7:97:9 are both correct.
The valid statements accurately reflect that County Beta's solar adoption increased by exactly 300300 households from a base of 1,5001,500 (a 20%20\% rise), and the final counts for County Gamma (1,4001,400) and County Alpha (1,8001,800) produce a ratio of 7:97:9.

Adım Adım Çözüm

1
Determine the initial number of solar-equipped households for all three counties at the start of 20252025.
Total = 4,0004,000. Alpha = 1,5001,500. Remaining for Beta and Gamma = 4,000−1,500=2,5004,000 - 1,500 = 2,500. Let Gamma = G. Beta is 50%50\% more, so Beta = 1.5G1.5G. Equation: 1.5G+G=2,500⇒2.5G=2,500⇒G=1,0001.5G + G = 2,500 \Rightarrow 2.5G = 2,500 \Rightarrow G = 1,000. Thus, Beta = 1,5001,500.
Establishing the baseline data for each category is necessary before applying growth percentages.
2
Calculate the growth and final number of households for County Alpha and County Gamma.
Alpha increase = 0.20×1,500=3000.20 \times 1,500 = 300, so Alpha final = 1,8001,800. Gamma increase = 0.40×1,000=4000.40 \times 1,000 = 400, so Gamma final = 1,4001,400.
These figures are explicitly given as percentage increases in the text and are required to find the unknown variables.
3
Deduce the final number and percentage increase for County Beta.
Total increase = Final Total - Initial Total = 5,000−4,000=1,0005,000 - 4,000 = 1,000. Beta increase = Total increase - (Alpha increase + Gamma increase) = 1,000−(300+400)=3001,000 - (300 + 400) = 300. Beta final = 1,500+300=1,8001,500 + 300 = 1,800. Beta percentage increase = (300/1,500)×100%=20%(300 / 1,500) \times 100\% = 20\%.
Using the overall total allows extraction of the missing piece of data through subtraction.
4
Evaluate all provided statements against the extracted data.
County Beta's growth was 20%20\% (True). County Beta's final count was 1,8001,800, not 2,1002,100 (False). Gamma to Alpha final ratio is 1,400:1,800=7:91,400:1,800 = 7:9 (True). Overall total growth is 1,000/4,000=25%1,000 / 4,000 = 25\%, not 20%20\% (False).
Comparing the mathematical findings directly to the given options determines the correct answers.

Anahtar Kavram

Extracting quantitative variables from paragraph text and executing multi-step percentage calculations.
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Caselets and Paragraph-Based Data

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