An amount of $9000 is divided among four persons A,B,C and D. The sum of shares of A,C and D is four times the share of B. The sum of shares of B and D is equal to four-fifths the sum of shares of A and C. Find the share of D.

Please help!! How to frame equation for share of each person?

just write the data as given:

a+b+c+d = 9000
a+c+d = 4b
b+d = 4/5 (a+c)

The first says that
a+c+d = 9000-b
using the second, that means that
9000-b = 4b
b = 1800

So, now we have
a+c+d = 7200
1800+d = 4/5 (a+c)

Now we know that
a+d = 7200-d, so
1800+d = 4/5 (7200-d)
1800+d = 5760 - 4/5 d
9/5 d = 3960
d = 2200

Thank you very much. Explained neat and wast. I was breaking my head for this.

typing error

this line a+d = 7200-d, so should be,

a+c=7200-d

*neat and good

Well, it seems like we've got a mathematical mystery on our hands! Let's try to solve it with some humor-infused equations:

Let's denote the share of person A as x, the share of person B as y, the share of person C as z, and the share of person D as w.

Now, we are given two clues to help us frame the equations:

Clue 1: The sum of shares of A, C, and D is four times the share of B.
Translated into an equation, that would be:
x + z + w = 4y ----- (Equation 1)

Clue 2: The sum of shares of B and D is equal to four-fifths the sum of shares of A and C.
Translated into an equation, that would be:
y + w = (4/5) * (x + z) ----- (Equation 2)

Now, to find the share of person D, we need to eliminate the other variables and solve for w.

To do that, let's rearrange Equation 1 to solve for x:
x = 4y - z - w ----- (Equation 3)

Now, we can substitute Equation 3 into Equation 2:
y + w = (4/5) * ((4y - z - w) + z)
y + w = (4/5) * (4y - w)
y + w = (16/5)y - (4/5)w
(1 - 16/5)y = - (4/5)w
(5/5 - 16/5)y = - (4/5)w
(-11/5)y = - (4/5)w
11y = 4w

Finally, we can substitute this result into Equation 1 to solve for w:
x + z + w = 4y
4y - z - w + z + w = 4y
4y = 4y

Oh, look at that! We ended up with 4y = 4y, which means "y" can have any value, and the value of "w" depends on that. Therefore, the share of person D, w, is not determined by the given information. It could be anything!

So, as it turns out, it's a clownishly tricky problem where the share of person D is unknown. We'll need some additional information or clever deduction to figure it out. Keep on clowning around with those numbers!

To solve this problem, we can first assign variables to represent the share of each person. Let's say the share of person A is represented by 'a', the share of person B is represented by 'b', the share of person C is represented by 'c', and the share of person D is represented by 'd'.

Based on the given information, we can form two equations:

1) The sum of shares of A, C, and D is four times the share of B:
a + c + d = 4b -- Equation (1)

2) The sum of shares of B and D is equal to four-fifths the sum of shares of A and C:
b + d = (4/5)(a + c) -- Equation (2)

Now, we can solve these equations simultaneously to find the share of person D.

Is there anything else you would like to know?