Solving a Word Problem: Rancher Dividing Livestock

  • Thread starter RyokoTenchi
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In summary, the Western Rancher had at least 8 cows and at most 32 cows, and he had 4 sons. Each son received one more cow than the previous son, and each son's wife received one ninth of the remaining cows. In order to divide the remaining livestock equally, the rancher had to give each son one cow more than the previous son, and his wife received one ninth of those remaining cows. The youngest son received enough cows that his wife did not receive any, and the rancher also had 7 horses to divide amongst his sons.
  • #1
RyokoTenchi
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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 ^^
 
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  • #2
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 [tex]\frac{8}{9}[/tex] of a number of cattle.

So you could just try multiples of 8, with up to 4 sons. Untill you get to an answer.
 
  • #3


Based on the information given, we can determine that the rancher had at least four sons since each son was given one cow more than the previous son. However, we can also assume that there were more than four sons since the youngest son's wife did not receive any cows.

To solve the word problem, we can create a table to keep track of the number of cows each son and their wife received:

Son | Number of Cows | Wife's Share
---|---|---
John | x | x/9
Sam | x+1 | (x+1)/9
Third Son | x+2 | (x+2)/9
Fourth Son | x+3 | (x+3)/9
Fifth Son | x+4 | (x+4)/9

We can see that the number of cows each son received follows a pattern of adding one more cow than the previous son. So, we can represent the number of cows each son received as x, x+1, x+2, x+3, and x+4.

To find the total number of cows, we can add up all the cows received by the sons and their wives:

x + (x+1) + (x+2) + (x+3) + (x+4) + x/9 + (x+1)/9 + (x+2)/9 + (x+3)/9 + (x+4)/9 = 7x

We also know that the rancher had seven horses that were worth twice as much as a cow. So, the total value of the cows must be half of the total value of the horses. This can be represented as:

7x = (1/2)(7)(2x)
7x = 7x

This means that the value of the cows and horses are equal. Therefore, the rancher must have owned 7 cows and 7 horses. And since each son received one more cow than the previous son, we can determine that the rancher had 6 sons in total.

In conclusion, the rancher had 7 cows and 6 sons. This solution ensures that each son received one more cow than the previous son and that each family received an equal value of livestock.
 

1. How do I know how many animals to put in each pasture?

The number of animals to put in each pasture can be determined by dividing the total number of animals by the number of pastures available. For example, if there are 100 animals and 5 pastures, each pasture would have 20 animals.

2. How can I ensure that each pasture has an equal number of animals?

To ensure that each pasture has an equal number of animals, you can use the division method mentioned above. If there is a remainder after dividing, you can add or subtract an animal from each pasture to make the numbers equal.

3. What factors should I consider when dividing my livestock into pastures?

When dividing livestock into pastures, you should consider the size and capacity of each pasture, the type of animals and their specific needs, and the availability of resources such as water and food in each pasture.

4. How can I keep track of which animals are in each pasture?

You can keep track of which animals are in each pasture by using a numbering or labeling system. This can be done by assigning a number or color to each animal and marking the corresponding pasture with the same number or color.

5. What should I do if I need to add or remove animals from a pasture?

If you need to add or remove animals from a pasture, you can adjust the number of animals in each pasture accordingly by using the division method mentioned above. It is important to keep track of the changes to ensure that each pasture has an equal number of animals.

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