Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution)
m (→‎Solution)
 
(2 intermediate revisions by the same user not shown)
Line 5: Line 5:
  
 
== Solution ==
 
== Solution ==
−
{{solution}}
 
  
 
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:
 
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:
Line 26: Line 25:
  
 
<math>Fraction_{married} = \frac{24}{43}</math>
 
<math>Fraction_{married} = \frac{24}{43}</math>
 +
  
 
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:
 
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:
 +
  
 
<math>Fraction_{married} = \frac{2 \times k m}{m + m \frac{k}{p}}</math>
 
<math>Fraction_{married} = \frac{2 \times k m}{m + m \frac{k}{p}}</math>
 +
  
 
<math>Fraction_{married} = \frac{2 p k}{p + k}</math>
 
<math>Fraction_{married} = \frac{2 p k}{p + k}</math>

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_UNCO_Math_Contest_II_Problems/Problem_2&oldid=90058"