The following questions are accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/necessary to answer the questions.
Shivam and Deepak invested in a partnership business in the ratio of 4:5. Find the profit share of Shivam.
(I) Shivam invested Rs.12000 and period of investment of Shivam and Deepak is 10 months and 4 months respectively.
(II) Ratio of period of investment of Shivam andre Deepak is 5 : 2 and Deepaks profit share is Rs.12000 less than Shivam's profit share.

Options

A. Statement(I) alone is sufficient to answer the question but statement (II) alone is not sufficient to answer the question.
B. Statement (II) alore is sufficient to answer the question but statement (I) alone is not sufficient to answer the question.
C. Both the statements taken together are necessary to answer the question, but neither of the statements alone is sufficient to answer the question
D. Either statement (I) or statement (II) by itself is sufficient to answer the question.

Answer

To determine Shivam's profit share in the partnership business with Deepak, we need to analyze the two statements provided and see if they are sufficient to answer the question.

### Step 1: Understand the Investment Ratio
Shivam and Deepak invested in a partnership business in the ratio of 4:5. This means that if Shivam's investment is represented as 4X, then Deepak's investment is represented as 5X.

### Step 2: Analyze Statement (I)
**Statement (I):** Shivam invested Rs. 12000 and the period of investment of Shivam and Deepak is 10 months and 4 months respectively.

From this statement:
- We know Shivam's investment (4X) is Rs. 12000.
- Therefore, we can find the value of X:
\[ 4X = 12000 \implies X = \frac{12000}{4} = 3000 \]
- Deepak's investment (5X) can be calculated as:
\[ 5X = 5 \times 3000 = 15000 \]
- Now, we calculate the total investment made by both:
- Shivam's total investment over 10 months:
\[ 12000 \times 10 = 120000 \]
- Deepak's total investment over 4 months:
\[ 15000 \times 4 = 60000 \]
- The total investment is Rs. 180000.

### Step 3: Calculate the Profit Share Ratio
The profit share is directly proportional to the investment amount. We can find the ratio of their investments:
- Shivam's investment = Rs. 120000
- Deepak's investment = Rs. 60000

The ratio of their investments is:
\[ \text{Shivam : Deepak} = 120000 : 60000 = 2 : 1 \]

However, we still do not know the total profit amount, which is necessary to calculate Shivam's exact profit share. Therefore, **Statement (I) alone is not sufficient** to answer the question.

### Step 4: Analyze Statement (II)
**Statement (II):** The ratio of the period of investment of Shivam and Deepak is 5:2, and Deepak's profit share is Rs. 12000 less than Shivam's profit share.

From this statement:
- The ratio of the time of investment is given as 5:2, which can be used to determine the effective investment ratio.
- If we assume the investments are proportional to the time, we can express their effective investments as:
- Shivam's effective investment = 4 * 5 = 20
- Deepak's effective investment = 5 * 2 = 10
- The ratio of their effective investments is:
\[ \text{Shivam : Deepak} = 20 : 10 = 2 : 1 \]
- The statement also tells us that Deepak's profit share is Rs. 12000 less than Shivam's profit share.

Let Shivam's profit share be S and Deepak's profit share be D:
\[ D = S - 12000 \]
Given the ratio of their profit shares is 2:1, we can express this as:
\[ \frac{S}{D} = \frac{2}{1} \]
Substituting D from the previous equation:
\[ \frac{S}{S - 12000} = \frac{2}{1} \]
Cross-multiplying gives:
\[ S = 2(S - 12000) \]
Expanding and solving for S:
\[ S = 2S - 24000 \]
\[ 24000 = S \]
Thus, Shivam's profit share is Rs. 24000.

### Conclusion
From Statement (II) alone, we can determine Shivam's profit share. Therefore, **Statement (II) is sufficient** to answer the question.

### Final Answer
- Statement (I) alone is not sufficient.
- Statement (II) alone is sufficient.

The correct option is that **Statement (II) alone is sufficient to answer the question.**