lambda in lagrangian != marginal utility of income?

[shawny boy]

so I have a simple maximization problem U(x,y) = x ^ 1/2 * y ^ 1/2, Px = 1, Py = 3, I = 100

my solution is x = 75, y = 8.333 and lambda = 0.0577

now here is where I am confused and would appreciate any help. the utility U(75,8.3333) = 25 @ I = 100 when I increase I to 101 I find that U(75.75, 8.41666) = 25.25 which would seem to indicate that the marginal utility of income for the system is in fact 0.25.

I see that the sum of the marginal utilities dU(x+y) = lambda, but they are both WAY different from the output of the function.

Please help me understand this.

Thanks in advance. Shawn