Hey there. I believe the way to handle this one is as follows:

A + B capital = 250,000 and you need to find out how much money is required by C to be admitted and obtain 1/5 or 20% of the total capital of the partnership. The standard equation is:

STEP 1: (A,Capital + B,Capital) + New Partner Contribution = Total Capital
STEP 2: Total Capital x 20% (or 1/5) = $150,000

Since you can't complete STEP 1 because you don't know how much A and B require that C contributes to join the partnership, you can't find the Total Capital and therefore you can't finish STEP 2 either. You must work backwards from STEP 2.

Total Capital x 20% = $150,000 ... can be reworked using basic algebra to be...
Total Capital = $150,000 x 20% ... complete the equation ...
Total Capital = $750,000 ... Now that you know the total capital, plug it into STEP 1

(A,Capital + B, Capital) + New Partner Contribution = Total Capital
(150,000 + 100,000) + New Partner Contribution = $750,000
250,000 + New Partner Contribution = $750,000
New Partner Contribution = $750,000 - $250,000
New Partner Contribution = $500,000

If Partner C has to contribute $500,000 to get into the partnership, but only receives $150,000 (or 20% or 1/5) total equity, then A and B each get a bonus according to the income sharing ratio of 3:2.

Partner A gets $300,000 ($500,000 x 3/5)
Partner B gets $200,000 ($500,000 x 2/5)

The journal entry should like like this:

Debit:
Cash 500,000

Credit
A, Capital 300,000
B, Capital 200,000
C, Capital 150,000

Memo: To admit new partner C into partnership with 20% or 1/5 ownership and give bonuses to A and B

Hope that helps!
Pete--