Short Answer 
The investment proportions for a total of $15,000 were determined with the first person investing 3/5, the second 1/3, and the third contributing 1/15. Their respective earnings were calculated as $9,000, $5,000, and $1,000 based on these proportions.
Step 1: Identify Investment Proportions
First, determine how much each person invested out of the total capital of $15,000. The proportions are as follows:
- First person: invests 3/5 of the capital.
- Second person: invests 1/3 of the capital.
- Third person: invests the remaining part of the capital, which needs to be calculated.
Step 2: Calculate the Third Person’s Share
To find out how much the third person contributed, we need to subtract the proportions of the first two from the total (1, or 100% of the capital). This is done via the formula: 1 – (3/5 + 1/3).
- Combine the fractions, using a common denominator:
- 3/5 = 9/15 and 1/3 = 5/15, so, add them together: 9/15 + 5/15 = 14/15.
- Subtract this sum from 1: 1 – 14/15 = 1/15.
Step 3: Distribute the Total Amount
Now, calculate each person’s earnings based on their investment proportions. Multiply the total capital by their respective fractions:
- First person’s share: $15,000 x (3/5) = $9,000.
- Second person’s share: $15,000 x (1/3) = $5,000.
- Third person’s share: $15,000 x (1/15) = $1,000.
Thus, the total earnings are $9,000, $5,000, and $1,000 for each person respectively, distributed according to their investments.