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Difficulty: MediumCaselets and Paragraph-Based Data

A municipal health department organized a vaccination drive across three different wards: Ward X, Ward Y, and Ward Z. The total population across the three wards is 12,00012,000, and this population is distributed among Ward X, Ward Y, and Ward Z in the ratio of 5:4:35:4:3, respectively.

In Ward X, 60%60\% of the population received Dose 1, and out of those who received Dose 1, 40%40\% also received Dose 2. The number of people who received Dose 1 in Ward Y is equal to the number of people who received Dose 1 in Ward X. In Ward Y, the number of people who received Dose 2 is half the number of people who received Dose 1 in the same ward. In Ward Z, the number of people who received Dose 1 is 40%40\% less than the number of people who received Dose 1 in Ward X. Furthermore, 50%50\% of the individuals who received Dose 1 in Ward Z went on to receive Dose 2.

Based on this information, what is the ratio of the total number of people who received Dose 2 in Ward X and Ward Z combined to the number of people in Ward Y who did NOT receive Dose 1?

  1. A
    7:57:5
  2. B
    27:1027:10
  3. 21:1021:10Answer
  4. D
    29:1029:10

Answer

The correct ratio is 21:1021:10.
Based on the given ratios and percentages, Ward X has 1,2001,200 people with Dose 2, and Ward Z has 900900 people with Dose 2, combining for 2,1002,100. Ward Y has a total population of 4,0004,000, and 3,0003,000 received Dose 1, leaving exactly 1,0001,000 people who did not receive Dose 1. The ratio is 2,100:1,0002,100 : 1,000, which simplifies to 21:1021:10.

Step-by-Step Solution

1
Calculate the total population for each ward based on the 5:4:35:4:3 ratio.
Total ratio parts = 5+4+3=125 + 4 + 3 = 12. Multiplier = 12,000÷12=1,00012,000 \div 12 = 1,000. Ward X population = 5×1,000=5,0005 \times 1,000 = 5,000. Ward Y population = 4×1,000=4,0004 \times 1,000 = 4,000. Ward Z population = 3×1,000=3,0003 \times 1,000 = 3,000.
Establishing the base population for each ward is necessary to apply the subsequent percentages.
2
Calculate Dose 1 and Dose 2 recipients for Ward X.
Dose 1 in X = 60%60\% of 5,000=3,0005,000 = 3,000. Dose 2 in X = 40%40\% of 3,000=1,2003,000 = 1,200.
The problem states 40%40\% of those who received Dose 1 went on to get Dose 2, meaning the base for Dose 2 is the Dose 1 count, not the total population.
3
Calculate the relevant statistics for Ward Y.
Dose 1 in Y = Dose 1 in X = 3,0003,000. People in Y who did NOT receive Dose 1 = Total Y population - Dose 1 in Y = 4,000−3,000=1,0004,000 - 3,000 = 1,000.
The question asks for the number of people in Ward Y who did not receive Dose 1 to be used as the denominator in the final ratio.
4
Calculate Dose 1 and Dose 2 recipients for Ward Z.
Dose 1 in Z is 40%40\% less than Dose 1 in X (3,0003,000), which is 3,000−(0.40×3,000)=1,8003,000 - (0.40 \times 3,000) = 1,800. Dose 2 in Z = 50%50\% of 1,800=9001,800 = 900.
This establishes the final required values needed for the numerator of the ratio.
5
Formulate and simplify the required ratio.
Ratio = (Dose 2 in X + Dose 2 in Z) : (No Dose 1 in Y) = (1,200+9001,200 + 900) : 1,000=2,100:1,000=21:101,000 = 2,100 : 1,000 = 21:10.
Combines the extracted data to answer the specific question asked in the stem.

Key Concept

Extracting and chaining percentage-based data across multiple entities to form relationships.
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Topics

Caselets and Paragraph-Based Data

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