# Single and single

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