In village A, 3/5 of the total population (male + females) are males. In village B, 2/3 of the total population (male + females) are males. The number of females in both villages a... In village A, 3/5 of the total population (male + females) are males. In village B, 2/3 of the total population (male + females) are males. The number of females in both villages are equal, and the total population of village A is 300 less than that of village B. What is the total population of village B?
Understand the Problem
The question involves two villages with specific male population ratios and a relationship between their total populations. We need to determine the total population of village B based on the information given about the populations and gender ratios of both villages.
Answer
The total population of village B is $1800$.
Answer for screen readers
The total population of village B is $1800$.
Steps to Solve
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Define the variables Let the total population of village A be ( P_A ) and the total population of village B be ( P_B ). According to the problem, ( P_A = P_B - 300 ).
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Express male and female populations in village A In village A, ( \frac{3}{5} ) of the population are males. Therefore, the number of males in village A can be expressed as: $$ M_A = \frac{3}{5} P_A $$ The remaining ( \frac{2}{5} ) of the population are females: $$ F_A = \frac{2}{5} P_A $$
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Express male and female populations in village B In village B, ( \frac{2}{3} ) of the population are males. Therefore, the number of males in village B can be expressed as: $$ M_B = \frac{2}{3} P_B $$ The remaining ( \frac{1}{3} ) of the population are females: $$ F_B = \frac{1}{3} P_B $$
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Set the number of females equal According to the problem, the number of females in both villages are equal: $$ F_A = F_B $$
Substituting the expressions for ( F_A ) and ( F_B ): $$ \frac{2}{5} P_A = \frac{1}{3} P_B $$
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Substitute ( P_A ) in terms of ( P_B ) Now, substitute ( P_A ) from Step 1 into the equation: $$ \frac{2}{5} (P_B - 300) = \frac{1}{3} P_B $$
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Clear the fractions To eliminate the fractions, multiply the entire equation by 15 (the least common multiple of 5 and 3): $$ 15 \cdot \frac{2}{5} (P_B - 300) = 15 \cdot \frac{1}{3} P_B $$
This simplifies to: $$ 6 (P_B - 300) = 5 P_B $$
- Distribute and solve for ( P_B ) Distribute on the left side: $$ 6 P_B - 1800 = 5 P_B $$
Now, isolate ( P_B ): $$ 6 P_B - 5 P_B = 1800 $$ $$ P_B = 1800 $$
The total population of village B is $1800$.
More Information
The answer obtained shows that village B has 1800 individuals. By finding the relationships between the populations of the two villages and their respective gender ratios, we were able to derive the total population for village B directly.
Tips
- Incorrectly equating populations: Sometimes, students might confuse the populations of the two villages or forget to account for the subtraction of 300 correctly. Always ensure to substitute values step by step.
- Misunderstanding ratios: Not accurately converting ratios to equivalent values can lead to errors. Be sure to follow each ratio through calculations.
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