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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

 Aug 5, 2015

Best Answer 

 #3
avatar+24093 
+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

 

 Aug 6, 2015
 #2
avatar+107367 
+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

 

 

 Aug 6, 2015
 #3
avatar+24093 
+10
Best Answer

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

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