Problem Understanding: Finding Village Male Population
The question asks us to determine the total number of males in a village based on its total population, the overall literacy percentage, the literacy percentage among males, and the literacy percentage among females. We are given specific conditions (\(x, y > z\)) which help ensure the result is meaningful.
Defining Variables for Calculation
Let's define the variables involved in the problem:
- Total village population = \(p\) persons
- Percentage of people who can read and write (literate) = \(x\)%$
- Percentage of males who can read and write (literate males) = \(y\)%$
- Percentage of females who can read and write (literate females) = \(z\)%$
- Let \(M\) represent the total number of males in the village.
- Let \(F\) represent the total number of females in the village.
Setting Up Equations for Literacy
We can express the number of literate individuals using the given percentages:
- Total number of literate people in the village = \((x/100) \times p\)
- Number of literate males = \((y/100) \times M\)
- Number of literate females = \((z/100) \times F\)
The total number of literate people is the sum of literate males and literate females:
\(\frac{x}{100} \times p = \left(\frac{y}{100} \times M\right) + \left(\frac{z}{100} \times F\right)\)To simplify, we can multiply the entire equation by 100:
\(xp = yM + zF \quad (*)\)Relating Males and Females
We know that the total population is the sum of males and females:
\(M + F = p\)From this, we can express the number of females in terms of the number of males and the total population:
\(F = p - M\)Solving for the Number of Males
Now, substitute the expression for \(F\) into the equation \((*)\):
\(xp = yM + z(p - M)\)Distribute \(z\) on the right side:
\(xp = yM + zp - zM\)Our goal is to find \(M\). Let's gather all terms containing \(M\) on one side and the other terms on the other side:
\(xp - zp = yM - zM\)Factor out \(p\) on the left side and \(M\) on the right side:
\(p(x - z) = M(y - z)\)Finally, isolate \(M\) by dividing both sides by \((y - z)\). Note that since we are given \(y > z\), the term \((y-z)\) is not zero and is positive.
\(M = \frac{p(x - z)}{y - z}\)Conclusion
The derived formula for the number of males in the village is \(M = \frac{p(x-z)}{y-z}\). This matches the first option provided.
Important Questions from Percentage
A number p increased by p% of 99 equals 99 increased by 99% of p. What is (p + 51)% of 928 + 72?
Amina saves 16% of her income. Now her income is increased by 20% but she still saves the same amount as before. What is the percentage increase in her expenditure?
What is 12% of 4% of 7% of 2 × 10 6 ?
Vignesh spends 42% of his monthly salary on food, 16% on house rent, 11% on entertainment and 7% on conveyance. But due to some family function, he has to borrow Rs. 12,000 from a money leader to meet the expenses of Rs. 18,000 What is his monthly salary?
If X is 12.25% more than Y. then Y is approximately_____ less than X.
