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A certain number of men are twice as many women and thrice as many boys; earn Rs. 5100 in 6 days. A woman earns one and a half times as a boy and a man as much as a woman and a boy earn together per day. How many women were there if a boy earned Rs. 25 daily?
A. $4$
B. $7$
C. $12$
D. $36$

Last updated date: 19th Jul 2024
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Hint: To obtain how many women there are in the given condition we will let the number of men, women and boys and we will also let money earn by each of them. Then we will form an equation by using the information given so that we can form a relation between the values we have let. Finally we will solve the equation and get our desired answer.

Complete step by step answer:
We have to find the number of women.
So let the number of men, women and boys be $x,y,z$ respectively.
Now according to the question the relation between each of them is as follows:
$x=2y$……$\left( 1 \right)$
$x=3z$……$\left( 2 \right)$
Next we will let that money earned by a man, woman and boy be Rs. $a$ Rs. $b$ Rs. $c$ per day respectively.
It is given that total men that is $x$ men earn Rs.5100 in 6 days so,
Money earned by $x$ men in 1 day $=\dfrac{5100}{6}$
$\therefore $ Money earned by 1 man in 1 day $=\dfrac{5100}{6\times x}$
As we have let that money earn by a man is $a$
$a=\dfrac{5100}{6\times x}$
$\therefore a=\dfrac{850}{x}$…..$\left( 3 \right)$
Now the relation between the earning of each of them is given as:
$b=\dfrac{3}{2}c$….$\left( 4 \right)$
$a=b+c$….$\left( 5 \right)$
Put value from equation (4) in equation (5) we get,
  & a=\dfrac{3}{2}c+c \\
 & \Rightarrow a=\dfrac{3c+2c}{2} \\
$\therefore a=\dfrac{5c}{2}$…..$\left( 6 \right)$
It is given that money earn by boy is Rs. 25
So $c=25$put it in equation (60 as follows:
  & a=\dfrac{5}{2}\times 25 \\
 & \therefore a=\dfrac{125}{2} \\
As one side of equation (3) and above equation is same we can say,
  & \dfrac{850}{x}=\dfrac{125}{2} \\
 & \Rightarrow 850\times \dfrac{2}{125}=x \\
 & \Rightarrow \dfrac{68}{5}=x \\
 & \therefore x=\dfrac{68}{5} \\
Put the above value in equation (1) we get,
  & \dfrac{68}{5}=2y \\
 & \Rightarrow y=\dfrac{68}{5\times 2} \\
 & \Rightarrow y=6.8 \\
$\therefore y=6.8\approx 7$
So the number of women is 7.
Hence there were 7 women.

So, the correct answer is “Option B”.

Note: In this type of question we should always try to get the relation between all the terms given. The mistake of calculation is very common and therefore it is necessary that we equate each equation so as to get a clear view of what all details we have. We can check whether our answer is correct or not by putting the obtained value in any equation and checking whether it satisfies the equation.