+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
+26367 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
+128407 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
+26367 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
