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**+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

#2**+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**+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