Step 1: Understanding the Concept:
Hidden Goodwill (Inferred Goodwill) occurs when the valuation of a firm is not explicitly given.
However, we can deduce the firm's total value based on the amount a new partner is willing to pay for a specific share.
If a partner pays \$40,000 for 25% (1/4) of the firm, the market implies that 100% of the firm is worth \$160,000.
If the actual recorded capitals (assets minus liabilities) are less than this implied value, the difference is considered the intangible reputation or Goodwill of the firm.
Key Formula or Approach:
1. Total Capitalized Value = New Partner's Capital $\times$ Reciprocal of his Share
2. Actual Combined Capital = Sum of capitals of all partners (Old + New)
3. Hidden Goodwill = Total Capitalized Value $-$ Actual Combined Capital
Step 2: Detailed Explanation:
1. Calculate the Implied (Capitalized) Value of the Firm:
Z brings in \$40,000 for a $\frac{1}{4}$th share.
Total value of the firm based on Z's capital = $\$40,000 \times \frac{4}{1}$
\[ \text{Total Capitalized Value} = \$160,000 \]
2. Calculate the Actual Total Capital of the new firm:
This is the sum of the capital of the existing partners plus the cash brought in by the new partner.
Capital of X = \$60,000
Capital of Y = \$40,000
Capital of Z = \$40,000
\[ \text{Actual Combined Capital} = \$60,000 + \$40,000 + \$40,000 = \$140,000 \]
3. Find the Difference (Goodwill):
The difference between what the firm "should be worth" based on Z's entry and what it "is actually worth" on paper is the goodwill.
\[ \text{Hidden Goodwill} = \$160,000 - \$140,000 \]
\[ \text{Hidden Goodwill} = \$20,000 \]
Step 3: Final Answer:
The total value of the Hidden Goodwill for the entire firm is \$20,000.
If the question asked for Z's share of goodwill specifically, it would be $\frac{1}{4}$ of \$20,000 = \$5,000.
Since the question asks for the "total value," the correct answer is \$20,000.