# Looking for an appropriate algebraic equation for this problem

• Jun 11th 2012, 07:59 AM
jimdec23
Looking for an appropriate algebraic equation for this problem
Problem: In a town, two thirds of the males are married, and three fifths of the females are married. (Marriage between a man and woman) What fraction of the people (male and female) in the town is married?

Let P = population
M = males; F= females
Obviously M+F = P

Setting an equality, I get:
2/3*M + 3/5*F = P – (1/3*M + 2/5*F)
By using the LCD (15):
10M + 9F = 15P- (5M + 6F)
Collecting terms and simplifying gives back the original M+F = P

Intuitively it seems that:
2/3M + 3/5F would provide the fraction or per cent of the town’s married people.
P

Assuming this is correct, is there a better equation that relates more directly to the fraction of the town’s population that is married?

By substituting arbitrary numerical values for each of the variables, I obtain ~ 0.63 or 63/100 of the town being married. But is their a more appropriate way of algebraically stating this?
Any help will be appreciated.
• Jun 11th 2012, 08:29 AM
Soroban
Re: Looking for an appropriate algebraic equation for this problem
Hello, jimdec23!

Quote:

In a town, two thirds of the males are married, and three fifths of the females are married.
(Marriage between a man and woman.)
What fraction of the people (male and female) in the town is married?

I assume that the married men and women are married to each other.
That is: .(Married men) = (Married women).

Let .$\displaystyle \begin{Bmatrix}M &=& \text{no. of men} \\ W &=& \text{no. women} \end{Bmatrix}$

We are told: .$\displaystyle \tfrac{2}{3}M \,=\,\tfrac{3}{5}W \quad\Rightarrow\quad W \,=\,\tfrac{10}{9}M$ [1]

The ratio is: .$\displaystyle R \;=\;\frac{\text{married men}}{\text{men + women}} \;=\;\frac{\frac{2}{3}M}{M + W}$

Substitute [1]: .$\displaystyle R \;=\;\frac{\frac{2}{3}M}{M + \frac{10}{9}M} \;=\;\frac{\frac{2}{3}M}{\frac{19}{9}M} \;=\; \frac{6}{19}$
• Jun 11th 2012, 08:30 AM
emakarov
Re: Looking for an appropriate algebraic equation for this problem
Assuming there are no people in the town whose spouses live out of town, we have (2/3)M = (3/5)F. This allows expressing M through F. Therefore, the total population M + F can be expressed only through F. Next, the number of married people is twice the number of married females, i.e., 2(3/5)F. When this is divided by the total population expressed through F, the number F is canceled. The final answer I get is 12 / 19.
• Jun 11th 2012, 09:48 AM
jimdec23
Re: Looking for an appropriate algebraic equation for this problem
So, the ratio is expressed as the ratio of married men to the total population. We would then multiply by 2 (to account for the women) to get 12/19 as the faction of all married people in the population. am I correct?
• Jun 11th 2012, 10:26 AM
emakarov
Re: Looking for an appropriate algebraic equation for this problem
Quote:

Originally Posted by jimdec23
So, the ratio is expressed as the ratio of married men to the total population. We would then multiply by 2 (to account for the women) to get 12/19 as the faction of all married people in the population. am I correct?

I think so. The question asks for the "fraction of the people (male and female)" who are married, so we need to divide the number of all married people by the total population.