Question

The population of a certain town was 50,000. In year, the male population increased by 5% and the female population increased by 3%. Now the population becomes 52020. What was the number of male and female in the previous year?

Answer 1

Hint: We have been given population in two conditions. We can make two equations respectively using different variables for male and female populations. The solution of these equations will give the required number of males and females in the previous year.

Complete step-by-step answer:
For the given town, let the male and female populations be x and y respectively.
🡪 Male = x
🡪 Female = y
The population of the town was 50,000 initially, this population consists of both males and females. So, mathematically:
 $ x + y = 50000....(1) $
Now, the male population has increased by 5% while the female population has increased by 3%. This increase is given as:
🡪 Male = x + 5% of x
 $
   \Rightarrow x + \dfrac{{5x}}{{100}} \\
   \Rightarrow \dfrac{{105x}}{{100}} \\
  $
🡪 Female = y + 3% of y
 $
   \Rightarrow y + \dfrac{{3y}}{{100}} \\
   \Rightarrow \dfrac{{103y}}{{100}} \\
  $
The new population with the increase is 52020.
\[
   \Rightarrow \dfrac{{105x}}{{100}} + \dfrac{{103y}}{{100}} = 52020 \\
   \Rightarrow 105x + 103y = 5202000....(2) \;
 \]
(Taking 100 common)
We can find the values of x and y using (1) and (2) in order to get the value of respective populations for previous years:
Multiplying (1) by 103 and then subtracting from (2), we get:
 $
   \Rightarrow 105x + 103y - 103x - 103y = 5202000 - 5150000 \\
   \Rightarrow 2x = 52000 \\
   \Rightarrow x = \dfrac{{52000}}{2} \\
   \Rightarrow x = 26000 \;
  $
Substituting the value of x in (1) to get the value of y
 $
  26000 + y = c \\
   \Rightarrow y = 50000 - 26000 \\
  y = 24000 \;
  $
Therefore, the number of male and female in the previous year was $26000$ and $24000$ respectively.

Note: While adding the fractions, we need to confirm that they both have the same base. The fractions can be added or subtracted only when their bases are the same and if not, we make them the same using their LCM.
While finding the values of x and y, we first used elimination method, there we could also have found the value of y by multiplying equation (1) by 105 but we always prefer to eliminate a number which is smaller and thus eliminated with 103 as coefficient. Then the next method we used here is known as substitution method where known value is substituted to find the unknown.