Question

40% of the population of a town are men and 35% are women. If the numbers of children are 12,600, find the number of men.

Answer 1

Hint: We assume the population of town as $x$. We use the given information of percentages to express the number of men $M$ and the number of women $W$in terms of $x$.We put the expression in the equation $x=M+W+C$ where $C=12600$ the number of children. We solve for $x$ and then get the number of men $M$.

Complete step-by-step solution
We know that when we say $a$ is $p \% $ of $b$, then we have,
\[a=\dfrac{p}{100}\times b\]
We are given the question that 40% of the population of a town are men and 35% are women. The number of children is 12,600. Let us denote the population of men, women, and children of the town as $M, W, C$ respectively. Let us assume that the population of the town is $x$. The size of the population is the sum of the numbers of men, women, and children in the town. So we have
\[x=M+W+C........\left( 1 \right)\]
The number of men is 40% of the total population which means 40% of $x$. So we have,
\[M=\dfrac{40}{100}\times x=0.4x\]
The number of women is 35% of the total population which means 35% of $x$. So we have,
\[W=\dfrac{35}{100}\times x=0.35\]
The number of children is 12,600. So we have
\[C=12600\]
We put the expressions for $M,W,C$ in equation (1) to have,
\[\begin{align}
  & x=0.4x+0.35x+12600 \\
 & \Rightarrow x=.75x+12600 \\
 & \Rightarrow x-0.75x=12600 \\
 & \Rightarrow 0.25x=12600 \\
\end{align}\]
We divide both side of above equation by 0.25 to have,
\[\begin{align}
  & \Rightarrow x=\dfrac{12600}{0.25} \\
 & \because x=50400 \\
\end{align}\]
So the number of men is
\[M=0.4x=0.4\times 50400=20160\]

Note: We can similarly obtain the number of women as $W=0.35x$. We can alternatively find $x$ using the rest percentage $100-\left( 40+35 \right)=25$as a percentage of children and then equating it to 12600. We note that the question presumes that the population does not change at the time of calculation which means no person has entered or left the town during the calculation.