Assume x and y have 5:3 profit sharing ratio respectivly and a new member enters by taking 1/5 from x and 1/3 from y.what will be the new ratio of the three people?

The original proportion: x : y = 5:3 ---> x/y = 5/3 ---> x = 5y/3

after change:
first member : (4/5)x or 4x/5
2nd member: (2/3)y or 2y/3
new member: (1/5)x + (1/3)y or x/5 + y/3
new ratio : 4x/5 : 2y/3 : (x/5 + y/3)
= 4(5y/3)/5 : 2y/3 : (5y/3)/5 + y/3)
= 4y/3 : 2y/3 : 2y/3
= 4 : 2 : 2
= 2 : 1 : 1

check: suppose the original investments were $600 and $360
(notice 600:360 = 5:3)
first member left with 480
2nd member left with 240
new member has 120+120 = 240
ratio = 480 : 240 : 240 = 2:1:1
my answer is plausible

how do we get the 4x/5 and 2y/3

First member started with x

you took 1/5 away, leaving him with 4/5
(4/5)x = 4x/5

same steps for 2y/3

To find the new profit sharing ratio, we need to calculate the share of each person after the new member joins.

Let's start by determining the shares of x and y after the new member's deductions:
x's share = (5/5) - (1/5) = 4/5
y's share = (3/3) - (1/3) = 2/3

Now, let's calculate the new ratio of the three people by adding the new member's share:
New member's share = (1/5) + (1/3) = 8/15

To simplify the ratios, we need to ensure that the denominator is the same for all parts. Let's find the least common multiple (LCM) of 5 and 15, which is 15:

For x: (4/5) * (3/3) = 12/15
For y: (2/3) * (5/5) = 10/15

So, the new profit sharing ratio for the three people will be:
x : y : new member = 12/15 : 10/15 : 8/15

Finally, we can simplify the ratio:
x : y : new member = 4/5 : 2/3 : 8/15

Therefore, the new ratio of the three people is 4/5 : 2/3 : 8/15.