Step 1: Understanding the Question:
This problem involves the concept of a business partnership where partners invest varying amounts for different durations.
The total profit percentage given at the end of the year allows us to calculate the "total effective investment" over the year, from which we can find individual initial investments.
Step 2: Key Formula or Approach:
Total Annual Profit = Rate of Profit \( \times \) Total Effective Annual Capital.
Effective Capital for a partner = Investment amount \( \times \) Time period (in years).
Total Effective Annual Capital = Sum of the effective capitals of all partners.
Step 3: Detailed Explanation:
Let the initial investment of friend P be \( 5x \) and that of friend Q be \( 6x \), since they invested in the ratio of 5:6.
They started the business together, so both P and Q kept their money invested for the full duration of 12 months (or 1 full year).
R joins after six months. The question states that R invests an amount equal to Q's investment, which means R also invests \( 6x \).
Since R joins after six months, R's money is invested for the remaining 6 months of the year (which is 0.5 years).
To calculate the 20% annual profit on the total investment, we need to find the total effective capital that was working throughout the year.
P's effective capital for the year = \( 5x \times 1 \text{ year} = 5x \).
Q's effective capital for the year = \( 6x \times 1 \text{ year} = 6x \).
R's effective capital for the year = \( 6x \times 0.5 \text{ years} = 3x \).
The total effective annual investment is the sum of these values: \( 5x + 6x + 3x = 14x \).
The problem states that a 20% profit was earned on this business investment, and this profit equates to Rs. 98,000.
We set up the equation: \( 20% \text{ of } 14x = 98,000 \).
This can be written as \( 0.20 \times 14x = 98,000 \).
Simplifying the left side gives \( 2.8x = 98,000 \).
To find \( x \), divide: \( x = \frac{98,000}{2.8} = 35,000 \).
The question asks for the amount invested by R, which we established is \( 6x \).
Substitute \( x \) back to find R's investment: \( 6 \times 35,000 = 210,000 \).
Therefore, R invested Rs. 2,10,000.
Step 4: Final Answer:
The amount invested by R was Rs. 2, 10,000.