Study the information carefully and answer the questions that follow.
A, B and C started a business by investing Rs. 800, Rs. 1600 and Rs. 2000 respectively. In the second quarter, they invested amounts in the ratio 1 : 4 : 2. In the next quarter again, they invested amounts in the ratio 3 : 2 : 3. In the last quarter, the ratio of their investments were same as in the 2nd quarter. Also, in the last quarter, the respective amounts of A, B and C was double than the respective amounts invested in `2^(nd)` quarter. The total investment of C before `4^(th)` quarter was Rs 1400 more than that of A during the same duration. Also, ratio of B’s share in profit to total profit at the end of year was 66 : 153. Please note: All the investments were for one quarter only.
If the investments of A, B and C in third quarter were changed and were now in the ratio 2 : 4 : 1 (other investments being the same), then what would be the total investment of all three in third quarter, if the average investment of A, B, and C was Rs. 3100 for whole year?

Options

A. Rs. 700
B. Rs. 800
C. Rs. 500
D. Rs. 900

Explanation

To solve the problem, we will break it down step by step.

### Step 1: Initial Investments
A, B, and C started a business with the following investments:
- A = Rs. 800
- B = Rs. 1600
- C = Rs. 2000

### Step 2: Second Quarter Investments
In the second quarter, their investments were in the ratio of 1:4:2. Let the investment amounts be:
- A's investment = x
- B's investment = 4x
- C's investment = 2x

The total investment in the second quarter can be expressed as:
\[ x + 4x + 2x = 7x \]

### Step 3: Third Quarter Investments
In the third quarter, the investments were in the ratio of 3:2:3. Let the investment amounts be:
- A's investment = 3y
- B's investment = 2y
- C's investment = 3y

The total investment in the third quarter can be expressed as:
\[ 3y + 2y + 3y = 8y \]

### Step 4: Fourth Quarter Investments
In the fourth quarter, the investments were the same as in the second quarter but doubled. Therefore, we have:
- A's investment = 2x
- B's investment = 8x
- C's investment = 4x

### Step 5: Total Investment of C and A
According to the problem, the total investment of C before the fourth quarter was Rs. 1400 more than that of A during the same duration.

Calculating total investments before the fourth quarter:
- Total investment of C = Rs. 2000 (1st quarter) + 2x (2nd quarter) + 3y (3rd quarter)
- Total investment of A = Rs. 800 (1st quarter) + x (2nd quarter) + 3y (3rd quarter)

Setting up the equation:
\[ 2000 + 2x + 3y = 800 + x + 3y + 1400 \]
This simplifies to:
\[ 2000 + 2x = 2200 + x \]
Thus,
\[ 2x - x = 2200 - 2000 \]
\[ x = 200 \]

### Step 6: Calculate Investments
Now substituting x back to find the investments:
- A's investment in the 2nd quarter = Rs. 200
- B's investment in the 2nd quarter = Rs. 800
- C's investment in the 2nd quarter = Rs. 400

### Step 7: Total Investment Calculation
Now we can calculate the total investments:
- Total investment of A = 800 + 200 + 3y + 2(200) = 800 + 200 + 3y + 400 = 1400 + 3y
- Total investment of B = 1600 + 800 + 2y + 8(200) = 1600 + 800 + 2y + 1600 = 4000 + 2y
- Total investment of C = 2000 + 400 + 3y + 4(200) = 2000 + 400 + 3y + 800 = 3200 + 3y

### Step 8: Average Investment
The average investment of A, B, and C for the whole year is given as Rs. 3100. Therefore:
\[ \frac{(1400 + 3y) + (4000 + 2y) + (3200 + 3y)}{3} = 3100 \]
This simplifies to:
\[ 8600 + 8y = 9300 \]
Thus,
\[ 8y = 9300 - 8600 \]
\[ 8y = 700 \]
\[ y = 87.5 \]

### Step 9: Total Investment in Third Quarter
Now substituting y back to find the total investment in the third quarter:
- A's investment = 3y = 3(87.5) = Rs. 262.5
- B's investment = 2y = 2(87.5) = Rs. 175
- C's investment = 3y = 3(87.5) = Rs. 262.5

Total investment in the third quarter:
\[ 262.5 + 175 + 262.5 = Rs. 700 \]

### Final Answer
The total investment of A, B, and C in the third quarter is Rs. 700.