There are a total of 110 men and women in a hall. If 3/7 of the men leave the hall and another 40 women enter the hall,the ratio of the numb

There are a total of 110 men and women in a hall. If 3/7 of the men leave the hall and another 40 women enter the hall,the ratio of the number of men to the number of women becomes 8:11. Find the number of women in the hall at first

$$\small{\text{ We set m = man, and w = women. In the hall are }}\\$$

$$\small{\text{$ \begin{array}{rrclr} & 110 &=& m + w \\ or &\mathbf{ m }& \mathbf{=} & \mathbf{110 - w} & (1)\\\\ \dfrac{3}{7} \text{ of the men leave the hall}: & m_{now} &=& m-\dfrac{3}{7}m\\ & m_{now} &=& \left( 1 - \dfrac{3}{7} \right)m \\ & m_{now} &=& \left(\dfrac{4}{7} \right)m \\\\ 40 \text{ women enter the hall}: & w_{now} &=& w+40 \\\\ \text{ The Ratio of the number of men to the number of women becomes }: & \dfrac{8}{11} &=& \dfrac{m_{now}}{w_{now}} \\\\ & \dfrac{8}{11} &=& \dfrac{ \left(\dfrac{4}{7} \right)m }{ w+40 } \qquad | \qquad m = 110-w \\\\ & \dfrac{8}{11} &=& \dfrac{ \left(\dfrac{4}{7} \right)(110-w) }{ w+40 } \\\\ & \dfrac{56}{44} &=& \dfrac{110-w}{ w+40 } \\\\ & 56\cdot(w+40) &=& 44\cdot (110-w) \\\\ & 56w+56\cdot 40 &=& 44\cdot 110 - 44w\\ & 100w &=& 44\cdot 110 - 56\cdot 40 \qquad | \qquad : 10\\\\ & 10w &=& 44\cdot 11 - 56\cdot 4\\ & 10w &=& 484- 224\\ & 10w &=& 260\\ & \mathbf{w} & \mathbf{=} & \mathbf{26} \end{array} $}}$$

The number of women in the hall at first is 26

We have....

M + W   = 110     →  M = 110 - W  .... and we know that

[(110 -W) - (3/7)(110 -W)]/ [W +40] = 8/11    simplify

[(4/7)(110 - W)] /[W + 40]  = 8/11      multiply both sides by [W + 40]

[(4/7)(110 - W)]  = (8/11)(W + 40]     multiply both sides by  (7/4)

110  - W   = (56/44) (W + 40)  simplify

110 - W  = (14/11)(W + 40)     multiply both sides by 11

1210 - 11W  = 14(W + 40) 

1210 -11W = 14W + 560

1210 - 560 = 25W

650  = 25W     didived both sides by 25

W = 26 women in the hall originally

Check......

[(4/7)(110 - 26)] / [ 26 + 40]  =    

(4/7)(84) / 66  =

48/66  =  8/11