+0

# 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

+5
288
2
+414

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

MaiaMitchell  Aug 5, 2015

#3
+18715
+10

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

heureka  Aug 6, 2015
Sort:

#2
+78719
+10

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

CPhill  Aug 6, 2015
#3
+18715
+10

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

heureka  Aug 6, 2015

### 4 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details