a ratio problem?

Question

3 people (x,y,z) share their profits such that x and y receive payment as a:b, y and z receive payments as c:d such that
ad-bc=0
the profit they make is $421. how much does each person receive?

Answer 1

Given
x/y=a/b
and
y/z=c/d

Since
ad-bc=0
ad=bc
or a/b=c/d

Thus three will get equal shares of the profit. Each will get $140.33

Answer 2

Given :
1) x : y = a : b which means x / y = a / b
2) y : z = c : d which means y / z = c / d
3) ad - bc = 0
4) x + y + z = 421

From 1) we have : x / y = a / b
From 2) we have : y / z = c / d or z / y = d / c
Multiplying these together gives : (x / y) * (z / y) = (a / b) * (d / c)
or, xz / y^2 = ad / bc

But from 3) we have ad = bc. So ad / bc = 1.
Therefore, xz / y^2 = 1 or xz = y^2.

So we have the 2 equations :

x + y + z = 421
xz = y^2

This is difficult to solve, even if we make a quadratic equation from them.
I did do that, and subsequently found through a little programming,
which I can't go into here, that there is only one solution.

But notice that if y = 20, then xz = 20^2 = 400.
If we let x = 1 and thus, z = 400, then the 3 of them add to 421.
This then, is the only solution :
(x, y, z) = (1, 20, 400), not necessarily in that order.

Checking back, we find that a / b = c / d = 1/ 20, just for interest.

Answer 3

People: x,y,z
Profits of x vs. y - a:b
Profits of y vs. z - c:d

Obviously this involves a matrix, ad-bc is the determinant of the matrix. You're also given the total profit as 421. Use the above ratios and the determinant to solve for a single variable, then the other two.

Answer 4

I think they all receive the same amount. $421/3. Not 100% sure though....

Answer 5

x and y get payment in ratio a:b; ie, x:y=a:b
y and z get payment in ratio c:d; ie,y:z=c:d

given ad-bc=0; or ad=bc;

rearranging, we get a/b=c/d or a:b=c:d;

so, we can write
x:y=a:b; or x:y= a^2:ab;
y:z=a:b or y:z=ab:b^2

combining, x:y:z=a^2:ab:b^2

hence,
x gets $421* a^2
y gets $421*ab
z gets $421*b^2

Answer 6

x/y=a/b
y/z=c/d such that ad-bc=0
i.e ad=bc or a/b = c/d
x/y = a/b so x=ay/b
y/z=c/d so z= dy/c
x+y+z=421
ay/b+y+dy/c=421
(a/b+1+d/c)y=421
(ac+bc+bd)y/bc=421
y=421bc/(ac+bc+bd)
since x=ay/b
x= 421ac/(ac+bc+bd)
since z=dy/c
z= 421bd/(ac+bc+bd)
alternatively
x=ay/b
since y/z=c/d=a/b
z= by/a
x+y+z=421
ay/b+y+by/a=421
(a/b+1+b/a)y=421
[(a^2+ab+b^2)/ab]y=421
y=421ab/(a^2+ab+b^2)
since x=ay/b
x= 421a^2/(a^2+ab+b^2)
since z= by/a
z=421b^2/(a^2+ab+b^2)

Answer 7

x get 421*ac/(ac+bc+bd)
y get 421*bc/(ac+bc+bd)
z get 421*bd/(ac+bc+bd),

and I believe it works for all a,b,c,d