Question

A total salary of 15 men and 8 women is Rs.3050. The difference in salaries of 5 women and 3 men is 50. Find the sum of salaries of 3 men and 3 women?
\[\begin{align}
  & \text{A}.\text{ 75}0 \\
 & \text{B}.\text{ 85}0 \\
 & \text{C}.\text{ 95}0 \\
 & \text{D}.\text{ 1}000 \\
\end{align}\]

Answer 1

Hint: We are given the salary of 15 men and 8 women combined as 3050 while the difference of 5 women and 3 men as 50. We will start by taking the salary of woman as x and salary of man as y. Then, we will transform our equation in linear equations as
\[15y+8x=3050\text{ and }5x-3y=50\]
Then, we will use elimination method to find the value of x and y. Once we get the value of x and y, we will find the salary of 3 men and 3 women.

Complete step by step answer:
We are given that, salary of 15 men and 8 women is Rs.3050 while difference of salary of 5 women and 3 men is Rs.50
We are asked to find the sum of salaries of 3 men and 3 women.
We start our solution by considering that the salary of each woman as x and the salary of each man as y.
Now, the salary of 15 men and 8 women is 3050.
As the salary of 1 man is y. So, the salary of 15 men will be 15y.
Similarly, the salary of 1 woman is x. So, the salary of 8 women will be 8x.
We are given that the total of their salaries is 3050. So, we get:
\[15y+8x=3050\cdots \cdots \cdots \left( 1 \right)\]
Now, the salary of 5 women will be 5x and the salary of 3 men will be 3y.
We are given that their difference is 50. So, we get:
\[5x-3y=50\cdots \cdots \cdots \left( 2 \right)\]
Now, we have two linear equation:
\[\begin{align}
  & 15y+8x=3050 \\
 & 5x-3y=50 \\
\end{align}\]
Multiplying equation (2) with 5, we get:
\[25x-15y=250\cdots \cdots \cdots \left( 3 \right)\]
Now, adding equation (1) and equation (3), we get:
\[\begin{align}
  & 8x+15y=3050 \\
 & \underline{25x-15y=250} \\
 & 33x=3300 \\
\end{align}\]
Solving for x, we get:
\[\begin{align}
  & x=\dfrac{3300}{33}=100 \\
 & \therefore x=100 \\
\end{align}\]
Therefore, we have got the salary of a woman as Rs.100.
Now, putting x = 100 in equation (1) we get:
\[15y+8\times 100=3050\]
Simplifying we get:
\[\begin{align}
  & 15y=3050-800 \\
 & 15y=2250 \\
\end{align}\]
Solving for y, we get:
\[\begin{align}
  & y=\dfrac{2250}{15}=150 \\
 & \therefore y=150 \\
\end{align}\]
So we have got the salary of a man as Rs.150.
Now, the salary of 3 women and 3 men will be given as $3x+3y$. As x = 100 and y = 150, we get
\[\begin{align}
  & \text{Salary of 3 men}+\text{3 women}=3x+3y \\
 & \Rightarrow 3\times 100+3\times 150 \\
\end{align}\]
Solving we get:
\[\text{Salary of 3 women and 3 men}=\text{Rs}.\text{75}0\]

So, the correct answer is “Option A”.

Note: We are not asked about the salary of each woman and each man, so we cannot stop the solution after finding x and y. We are asked to find the salary of a particular number of men and women. So, we will use the value of x and y to get salaries asked. In elimination method, we eliminate one of the variables so that our equation remains with just one variable and we will easily be able to find the solution. Apart from this, we could also have used the substitution or the graphical method too. But these methods might take more time.