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Let men=m, women=w, children=c

From the statement,

m=1.25w

0.8w=c

Children left resulted in equal number of men and women, women=122 at the end.

Can I assumed that the total number of people is 122+122=244?

m+w+c=244

1.25w+w+0.8w=244

3.05w=244

w=80

m=100

The number of men more than women m-w=100-80=20

Is it the correct way to solve this problem?

LTY
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  • there were $25%$ more men than women does not imply $1.25 m =w$ though may mean $m=1.25w$ i.e. $0.8m=w$, which is how you almost got to the right answer of $100-80=20$ more men than women in the beginning with offsetting errors – Henry Nov 01 '21 at 14:24
  • There are more men than women, and you multiply the number of men by 1.25. this number is even greater. So your equation is wrong. Similarly, you made the same error in the second equation. – Andrei Nov 01 '21 at 14:24
  • Thanks a lot for the suggestions. I already made the corrections on the equation, is it right way to answer this question? – LTY Nov 01 '21 at 14:37
  • Don't forget to answer the question, which is "How many more men than women were there in the beginning?" – N. F. Taussig Nov 01 '21 at 14:48
  • Yes, that is correct. – N. F. Taussig Nov 01 '21 at 14:59

2 Answers2

2

I think this question would benefit from the addition of indices to the variables, since the values are changing. The indices will be i for initial and f for final.

$$m_i=(1+0.25)w_i=1.25w_i$$ $$c_i=(1-0.20)w_i=0.8w_i$$ $$c_f=0$$ $$m_f+w_f-(m_i+w_i)=c_i-c_f$$ $$m_f=w_f$$ $$w_f=122$$ $$\text{Solve for: }m_i-w_i$$

Here we have six independent equations and six unknowns. We could use matrix math to solve this, or it is a simple enough problem to take the steps you did. Let's go the latter route.

$$w_f=122$$ $$m_f=w_f=122$$ $$m_f+w_f-(m_i+w_i)=122+122-(1.25w_i+w_i)=0.8w_i-0=c_i-c_f$$ $$244-2.25w_i=0.8w_i$$ $$244=3.05w_i$$ $$w_i=80$$ $$m_i=1.25w_i=100$$ $$m_i-w_i=20$$

There were 20 more men than women at the beginning of the book fair.

2Tasty
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Here's an alternative working, for your reference.

  1. Summarising the given information: $$M_1:W_1:C=125:100:80=25:20:16;\\\text{The children are all replaced with equally many adults;}\\M_2:W_2=1:1;\\W_2=122.$$
  2. Let the number of children be $u.$ Then $$\frac{25}{16}u+\frac{20}{16}u+u=122+122\\u=64;$$ therefore, at the beginning, there were $\displaystyle\frac{25}{16}u-\frac{20}{16}u=20$ more men than women.
ryang
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