the capital of 4 partners a,b,c and d are in ratio of 7:8:6:5.a's and c,s capital are there in the buisness for entire year.if each partner kept his money invested in the business for a period which is more than 6 months and b and d together get 111/267 of the total profit then for how many months is d capital invested? (a) 11 (b) 7 (c) 9 (d) cannot be determined

Let's break down the problem to solve it systematically.

### Step 1: Understanding the Capital Ratio
The capital contributions of the four partners A, B, C, and D are in the ratio:
\[
A : B : C : D = 7 : 8 : 6 : 5
\]

### Step 2: Time Periods
- A's and C's capital are in the business for the entire year, which is 12 months.
- B's and D's capital are invested for periods longer than 6 months, but we need to find out specifically how many months D's capital was invested.

### Step 3: Total Profit Share
We are given that B and D together receive \( \frac{111}{267} \) of the total profit.

### Step 4: Profit Distribution
The profit is distributed among the partners based on their capital contributions and the time for which the capital was invested.

Let's assume the capital of A, B, C, and D is \( 7x \), \( 8x \), \( 6x \), and \( 5x \) respectively, and the number of months B's and D's capital was invested is \( b \) and \( d \) months respectively.

The ratio of profits will be proportional to the product of the capital and the time invested. Hence:
\[
\text{Profit Ratio} = 7x \times 12 : 8x \times b : 6x \times 12 : 5x \times d
\]
This simplifies to:
\[
84x : 8bx : 72x : 5dx
\]

### Step 5: Expressing B's and D's Share of Profit
The total profit share for B and D is given by:
\[
\frac{8bx + 5dx}{84x + 8bx + 72x + 5dx} = \frac{111}{267}
\]

This can be simplified to:
\[
\frac{8b + 5d}{156 + 8b + 5d} = \frac{111}{267} = \frac{111}{267} = \frac{37}{89}
\]

### Step 6: Solve for \( d \)
Now, we cross multiply to solve for \( b \) and \( d \):
\[
89(8b + 5d) = 37(156 + 8b + 5d)
\]
Expanding and simplifying:
\[
712b + 445d = 5772 + 296b + 185d
\]
\[
416b + 260d = 5772
\]
Simplifying further:
\[
4b + 2.5d = 55.5
\]
\[
8b + 5d = 111 \quad \text{(Dividing both sides by 2)}
\]

Since B's capital was invested for more than 6 months, the values of \( b \) and \( d \) that satisfy this equation must also reflect this. Without knowing the exact investment period for \( b \), we cannot uniquely determine \( d \), given the multiple possible values of \( b \) that could satisfy this equation.

### Answer:
The correct answer is **(d) cannot be determined**.

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How would the solution change if B and D's investment periods were exactly 6 months?
2. What would the profit distribution be if A's capital was also reduced by 2 months?
3. How would the overall profit ratio change if C had invested only for 10 months?
4. If the profit shares were distributed equally, how long would D need to invest to earn the same as A?
5. How would the profit distribution change if D had doubled his investment amount?

**Tip:** When dealing with ratios and profit distribution problems, it's important to consider the impact of both capital amount and time invested on each partner's share.

Learn how to solve a ratio and profit distribution problem involving partners A, B, C, and D with detailed steps and solution. Understand how capital ratios and investment periods affect profit shares. Suitable for high school students studying ratio and proportion problems.

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