Question

In a certain office, $ \dfrac{1}{3} $ of the workers are women, $ \dfrac{1}{2} $ of the women are married and $ \dfrac{1}{3} $ of the married women have children . If $ \dfrac{3}{4} $ of the men are married and $ \dfrac{2}{3} $ of the married men have children, what part of workers are without children ?

Answer 1

Hint:- Here we had to assume the total number of workers and with the help of this assumption we will be able to find the workers having children and once we know the workers having children we will easily find the workers without children by just subtracting it from the total workers.

Complete step by step solution:
Let us assume that the total workers work in that office as ‘x’.
If the total workers are ‘x’ and we are given that $ \dfrac{1}{3} $ workers are women then the women workers in the office = $ \dfrac{1}{3} $.
And if the women workers in office is $ \dfrac{1}{3} $ then men workers must be \[x - \dfrac{1}{3} = \dfrac{{2x}}{3}\].
Number of married women = $ \dfrac{1}{2} $ of total women
And the number of married women having a child =$ \dfrac{1}{3} $ of married women.
So, number of married women having child = \[\dfrac{1}{3} \times \dfrac{1}{2} \times \dfrac{x}{3} = \dfrac{x}{{18}}\]women. ---- ( 1 )
Similarly, number of married men = $ \dfrac{3}{4} $ of total men
And number of married men having child = $ \dfrac{2}{3} $ of married men
So, number of married men having child = \[\dfrac{2}{3} \times \dfrac{3}{4} \times \dfrac{{2x}}{3} = \dfrac{x}{3}\]men ---- ( 2 )
Now we had to find the total ( men and women both ) number of workers without children and for that we must know the total workers having children.
Adding equation ( 1 ) and ( 2 )
\[ \Rightarrow \dfrac{x}{{18}} + \dfrac{x}{3}\] taking L.C.M and solving it
\[ \Rightarrow \dfrac{x}{{18}} + \dfrac{x}{3} = \dfrac{{7x}}{{18}}\]
Now total workers having children = \[\dfrac{{7x}}{{18}}\].
So, total workers without children = \[x - \dfrac{{7x}}{{18}} = \dfrac{{11x}}{{18}}\].
Hence, we can say that total workers without children is \[\dfrac{{11}}{{18}}\] of total workers.

Note :- Whenever we come up with this type of problem it become very easy for us to solve the question if we assume the total number of workers because once we will know the total value than we can easily calculate the necessary results that we are asked to find required values that we need when we solve this type of question.