Question

A man has 81 cows and each is marked with a number from 1 to 81. The number represents amount of milk produced by that cow. How can these cows be divided among nine persons such that all get an equal number of cows producing an equal amount of milk?

Answer 1

Answer:

• The total number of cows = 81.
• The cows are numbered from 1-81 according to how much milk is produced by them.
• For example: cow of number one will produce 1 litre of milk.
• The total cows that is 81 are to be divided among 9 people.
• The numbers of cows among the 9 people should be the same and the litres of milk produced by them should also be same.

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The total number of cows with all the nine people should be:- 81 ÷ 9 = 9 cows.

So, all the nine people should get nine cows each person.

Now, Finding how much total litres of milk is produced by all the cows:-

The cows are numbers from 1 to 81.

Hence, it makes an arithmetic progression.

So, here,

• a = 1
• d = 1
• n = 81

Now, Sn = n/2 (2a + (n - 1)d)

So, Sn = 81/2 (2 + (81 - 1)1)

Sn = 81/2 × (2 + 80)

Sn = 81/2 × 82

Sn = 81 × 41

Sn = 3,321 litres

The Total amount of milk produced by all the cows = 3,321 litres.

Total milk produced by the nine cows that will be given to all the nine people = 3,321 ÷ 9 = 369 litres.

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So, all the nine people will get nine cows that will totally produce 369 litres of milk.

Now, let's divide all the cows into group of 9 for each person.

Let's give the number 1 cow to the person 1, the number 2 cow to the person 2, number 3 cow to the person 3,.... and so on till number 9 Khao to the person 9.

Now, For the first person:-

1, 11, 21, 31, 41, 51, 61, 71, 81.

( By adding 10 to the first number of cow )

The total milk produced by all of them is 369 litres.

Now, for the second person:-

2, 12, 22, 32, 42, 52, 62, 72...

Here, we get 8 cows.

For the last one, add 1 to the last number of cow OR calculate the total of the litres of milk produced by all the 8 cows and subtracted from 369 which is the total litres of milk.

So, the list of cow numbers that will be given to second person:-

2, 12, 22, 32, 42, 52, 62, 72, 73.

Now, for the third person:-

Using the similar pattern,

3, 13, 23, 33, 43, 53, 63, ...

Now, here we have to cows remaining.

Now, adding 1 to the last number of cow.

3, 13, 23, 33, 43, 53, 63, 64..

Now, for the last cow that is 9th cow add 10 to the last number of cow.

3, 13, 23, 33, 43, 53, 63, 64, 74.

Similarly, using the same pattern we can find arrange the cows for all the nine people so that they get equal number of cows that is 9 cows which totally produce equal number of litres of milk that is 369 litres.

According to the pattern, the arrangement of cows is below:

Person 1 - 1, 11, 21, 31, 41, 51, 61, 71, 81.

Person 2 - 2, 12, 22, 32, 42, 52, 62, 72, 73.

Person 3 - 3, 13, 23, 33, 43, 53, 63, 64, 74.

Person 4 - 4, 14, 24, 34, 44, 54, 55, 65, 75.

Person 5 - 5, 15, 25, 35, 45, 46, 56, 66, 76.

Person 6 - 6, 16, 26, 36, 37, 47, 57, 67, 77.

Person 7 - 7, 17, 27, 28, 38, 48, 58, 68, 78.

Person 8 - 8, 18, 19, 29, 39, 49, 59, 69, 79.

Person 9 - 9, 10, 20, 30, 40, 50, 60, 70, 80.

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