Solving a Word Problem: Rancher Dividing Livestock

[RyokoTenchi]

A Western Rancher finding himself well advanced in years, called his boys together and told them that he wished to divide his herds between them while he yet lived. "Now, John," He told the eldest, "you may take as many cows as you think you could conveniently care for, and your wife Nancy may have one ninth of all the cows left."
To the second son he said, "Sam, you may have the same number of cows that john took, plus one extra because john had the first pick. To your wife, Sally, i will give one ninth of what will be left."
To the third son he made a similar statement. he was to take one cow more than the second son, and his wife was to have one ninth of those left. the same applied to the other sons. Each took one cow more than his next oldest brother, and each son's wife took one ninth of the remainder.
After the youngest son had taken his cows, there were none left for his wife. Then the rancher said: "Since horses are worth twice as much as cows, we will divide up my seven horses so that each family will own livestock of equal value."
The problem is to tell how many cows the rancher owned and how many sons he had.

Please help me with this Problem ^^

 

[NateTG]

I'm not exactly sure how you'd make this particular leap, but if you set the number of cattle that the youngest son gets to x, then you can generate formulae for the number of cattle each couple gets in terms of x. For example, the youngest couple gets x cattle. Since each couple would have to get a different number of horses, and there are only seven horses, you can see that there can be no more than 4 sons. (It's impossible to distribute 7 horses in 5 different amounts.)

Once you've generated the formulae, you can determine the total difference in the number of cattle that each couple gets, and set that equal to half the number of horses that are needed.

An alternative approach is to notice that the number of cattle that the youngest of the brothers gets must be divisible by 8 since it is $$\frac{8}{9}$$ of a number of cattle.

So you could just try multiples of 8, with up to 4 sons. until you get to an answer.