There are 10 children (aged 1 to 10 years) who have equal amounts of money. In the first transaction the eldest child gives one rupee to every child younger to him. In the second transaction, the second eldest child gives one rupee to every child younger to him. This type of distribution continues for the next two transactions. In the end the total sum with the children who have given money to other children is half the sum of money with the children who did not give any money.what was the original amount with EACH child?
Can we able to form an equation to find the amount each child had? Please help
If I understand correctly there would have been 4 transactions.
let the amount that each child has originally be x
first transaction:
1st child -- x-9
each of remaining 9 children has x+1
2nd transaction:
1st child -- x-9
2nd child -- x+1 - 8 = x-7
each of remaining 8 children has x+2
3rd transaction:
1st chld -- x-9
2nd child -- x-7
3rd child -- x+2 - 7 = x-5
each of the remaining 7 children has x+3
4th transaction:
1st child -- x-9
2nd child -- x-7
3rd child -- x-5
4th child -- x+3 - 6 = x-3
each of the remaining children --- x+4
"In the end the total sum with the children who have given money to other children is half the sum of money with the children who did not give any money"
---- x-9 + x-7 + x-5 + x-3 = (1/2)(6)(x+4)
4x - 24 = 3x+12
x = 36
each had 36
check:
distribution of money after each round
0: 36 36 36 36 ... 36
1: 27 37 37 37 ... 37
2: 27 29 38 38 ... 38
3: 27 29 31 39 39 ... 39
4: 27 29 31 33 40 40 .. 40
sum of those that gave money = 27+29+31+33 = 120
sum of those that received money = 6x40 = 240
so 120 = 1/2 of 240
All checks out
let the amount that each child has originally be x
first transaction:
1st child -- x-9
each of remaining 9 children has x+1
2nd transaction:
1st child -- x-9
2nd child -- x+1 - 8 = x-7
each of remaining 8 children has x+2
3rd transaction:
1st chld -- x-9
2nd child -- x-7
3rd child -- x+2 - 7 = x-5
each of the remaining 7 children has x+3
4th transaction:
1st child -- x-9
2nd child -- x-7
3rd child -- x-5
4th child -- x+3 - 6 = x-3
each of the remaining children --- x+4
"In the end the total sum with the children who have given money to other children is half the sum of money with the children who did not give any money"
---- x-9 + x-7 + x-5 + x-3 = (1/2)(6)(x+4)
4x - 24 = 3x+12
x = 36
Last step didn't understand
Sure, let's give it a try!
Let's assume that the original amount each child had is x rupees. As per the given conditions, in the first transaction, the eldest child gives one rupee to every child younger than him. So, the eldest child will give 9 rupees to the younger ones, and he will be left with x - 9 rupees. The other children (from 2nd to 10th) will each receive 1 rupee from the eldest child.
In the second transaction, the second eldest child gives one rupee to every child younger than him. So, the second eldest child will give 8 rupees to the younger ones, and he will be left with x - 8 rupees. The other children (from 3rd to 10th) will each receive 1 rupee from the second eldest child.
This pattern continues for the third and fourth transactions as well. At the end of the fourth transaction, the total sum with the children who have given money will be equal to half the sum of money with the children who did not give any money.
Let's summarize the amounts with each child after each transaction:
After the 1st transaction:
Eldest child: x - 9
Other children (2nd to 10th): 1
After the 2nd transaction:
Eldest child: x - 9 - 8
Other children (3rd to 10th): 1 + 1
After the 3rd transaction:
Eldest child: x - 9 - 8 - 7
Other children (4th to 10th): 1 + 1 + 1
After the 4th transaction:
Eldest child: x - 9 - 8 - 7 - 6
Other children (5th to 10th): 1 + 1 + 1 + 1
Now, the sum of the amounts with the children who gave money is equal to half of the sum of the amounts with the children who didn't give money:
(x - 9 - 8 - 7 - 6) + (1 + 1 + 1 + 1) = (9*1 + 8*1 + 7*1 + 6*1) / 2
Simplifying the equation:
x - 30 + 4 = (9 + 8 + 7 + 6) / 2
x - 26 = 30 / 2
x - 26 = 15
x = 15 + 26
x = 41
So, the original amount with each child is 41 rupees.
Yes, we can form an equation to find the amount each child had originally. Let's go step by step to solve the problem.
Let's assume that each child originally had x rupees. After the first transaction, the eldest child gives 1 rupee to each younger child. So, the eldest child now has x - 1 rupees, and each of the younger children has x + 1 rupees.
In the second transaction, the second eldest child gives 1 rupee to each younger child. At this point, the second eldest child has (x - 1) - 1 = x - 2 rupees, and each of the younger children has (x + 1) + 1 = x + 2 rupees.
This pattern continues for the next two transactions. So, after the third transaction, the third eldest child has (x - 2) - 1 = x - 3 rupees, and each of the younger children has (x + 2) + 1 = x + 3 rupees.
Likewise, after the fourth transaction, the fourth eldest child has (x - 3) - 1 = x - 4 rupees, and each of the younger children has (x + 3) + 1 = x + 4 rupees.
Based on the problem statement, the sum of money with the children who have given money is half the sum of money with the children who did not give any money. Therefore, we can form the following equation:
[(x - 1) + (x - 2) + (x - 3) + (x - 4)] = 1/2 * [(x + 1) + (x + 2) + (x + 3) + (x + 4)]
Now you can solve this equation to find the original amount with each child (x).