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Difference between revisions of "2007 UNCO Math Contest II Problems/Problem 2"

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== Solution ==
 
== Solution ==
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{{solution}}
 
  
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Let the number of men be equal to <math>m</math> and the number of women be equal to <math>w</math>, where <math>m</math> and <math>w</math> are whole numbers. Assuming a marriage consists of one man and one woman, we see that the number of married men is equal to the number of married women in the equation:
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Let the number of men be equal to <math>m</math> and the number of women be equal to <math>w</math>, where <math>m</math> and <math>w</math> are positive whole numbers. Assuming a marriage consists of one man and one woman, we see that the number of married men is equal to the number of married women in the equation:
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<math>\frac{4}{5}\times m = \frac{3}{7}\times w</math>
 
<math>\frac{4}{5}\times m = \frac{3}{7}\times w</math>
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<math>w = \frac{28}{15} m</math>
 
<math>w = \frac{28}{15} m</math>
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Dividing the the number of married persons by the entire adult population gives us:
 
Dividing the the number of married persons by the entire adult population gives us:
  
  
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<math>\%_{married} = \frac{\frac{3}{7}w + \frac{4}{5}m}{m + w}</math>
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<math>Fraction_{married} = \frac{\frac{3}{7}w + \frac{4}{5}m}{m + w}</math>
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 +
 
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<math>Fraction_{married} = \frac{\frac{8}{5}m}{\frac{43}{15}m}</math>
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 +
 
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<math>Fraction_{married} = \frac{24}{43}</math>
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 +
 
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Using the above statements, we can derive a generalized formula. If <math>k</math> is the fraction of married men and <math>p</math> is the fraction of married women, then:
  
  
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<math>\%_{married} = \frac{\frac{8}{5}m}{\frac{43}{15}m}</math>
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<math>Fraction_{married} = \frac{2 \times k m}{m + m \frac{k}{p}}</math>
  
  
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<math>\%_{married} = \frac{24}{43} \approx 55.8\%</math>
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<math>Fraction_{married} = \frac{2 p k}{p + k}</math>
  
 
== See Also ==
 
== See Also ==

Latest revision as of 17:49, 29 January 2018

Problem

In Grants Pass, Oregon $\frac{4}{5}$ of the men are married to $\frac{3}{7}$ of the women. What fraction of the adult population is married? Give a possible generalization.

Solution

Let the number of men be equal to $m$ and the number of women be equal to $w$, where $m$ and $w$ are positive whole numbers. Assuming a marriage consists of one man and one woman, we see that the number of married men is equal to the number of married women in the equation:


$\frac{4}{5}\times m = \frac{3}{7}\times w$


$w = \frac{28}{15} m$


Dividing the the number of married persons by the entire adult population gives us:


$Fraction_{married} = \frac{\frac{3}{7}w + \frac{4}{5}m}{m + w}$


$Fraction_{married} = \frac{\frac{8}{5}m}{\frac{43}{15}m}$


$Fraction_{married} = \frac{24}{43}$


Using the above statements, we can derive a generalized formula. If $k$ is the fraction of married men and $p$ is the fraction of married women, then:


$Fraction_{married} = \frac{2 \times k m}{m + m \frac{k}{p}}$


$Fraction_{married} = \frac{2 p k}{p + k}$

See Also

2007 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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