In a village hidden among the mountains, two thirds of the men are married to three fifths of the women. The tradition of this town is never to marry outsiders.

**What is the proportion of single people?**

#### Solution

If the proportion of married men is 2/3 and that of married women is 3/5, it means that 2/3 men in the town are the same number of people as 3/5 women in the town. Dividing we have that the proportion of men and women is 9/10, that is, 9 women for every 10 men.

Therefore, 3/5 of 10 women equals 2/3 of 9 men = 6 people.

So that of every 10 women, 6 are married and 4 are single and of every 9 men, 6 are married and 3 are single. Thus, **The proportion of singles is (4 + 3) / (10 + 9) = 7/19.**

We can also solve the problem by posing equations:

If M = number of women and H = Number of men, we have to:

(2/3) H = (3/5) M

H = (9/10) M

Total population: H + M = (19/10) M

Singles: (1/3) H + (2/5) M = (3/10) M + (2/5) H = (7/10) M

Proportion of singles: (7/10) M / (19/10) M = 7/19