A woman suffers a loss of 20% on her first investment, makes a profit of 10% on her second investment, and earns a profit of 12.5% on her third investment. If the invested amounts are in the ratio 4 : 5 : 3, what is her overall profit percentage on the total investment?

Introduction / Context:
This question is about finding the overall profit or loss percentage when different portions of money are invested at different profit and loss rates. It is common in partnership and percentage topics to combine multiple investments or transactions and then determine the net effect on the total money. The key idea is to use weighted averages, where each individual percentage is weighted by the amount invested in that part.

Given Data / Assumptions:

Investments are in the ratio 4 : 5 : 3 for the first, second and third parts respectively.
First investment has a 20% loss (that is, minus 20%).
Second investment has a 10% profit.
Third investment has a 12.5% profit.
We need the overall profit or loss percentage on the total investment.

Concept / Approach:
We can assume convenient actual amounts matching the ratio, because only the relative proportions matter. Let the investments be 4x, 5x and 3x. Then we apply the respective profit or loss percentages to get the gain or loss on each part. The overall profit is the sum of all gains and losses divided by the total investment, expressed as a percentage. This is a weighted average of the individual percentages with weights 4, 5 and 3.

Step-by-Step Solution:
Step 1: Assume investments according to the ratio 4 : 5 : 3. Let first investment = 4x, second = 5x, third = 3x. Step 2: Compute gain or loss on each part. First part: 20% loss on 4x means loss = 0.20 * 4x = 0.8x. Second part: 10% profit on 5x means profit = 0.10 * 5x = 0.5x. Third part: 12.5% profit on 3x means profit = 0.125 * 3x = 0.375x. Step 3: Find net gain or loss. Net result = (profit on second + profit on third) minus loss on first. Net result = (0.5x + 0.375x) - 0.8x = 0.875x - 0.8x = 0.075x. This is a net profit of 0.075x. Step 4: Find total investment. Total investment = 4x + 5x + 3x = 12x. Step 5: Compute overall profit percentage. Overall profit percent = (net profit / total investment) * 100. = (0.075x / 12x) * 100. = (0.075 / 12) * 100. 0.075 / 12 = 0.00625. 0.00625 * 100 = 0.625%. Therefore, she makes an overall profit of 0.625%.

Verification / Alternative check:
We can choose a specific value for x, for example x = 100. Then the actual investments are 400, 500 and 300. The loss on 400 at 20% is 80. The profit on 500 at 10% is 50. The profit on 300 at 12.5% is 37.5. Net gain = 50 + 37.5 - 80 = 7.5. Total investment = 400 + 500 + 300 = 1200. Overall profit percentage = (7.5 / 1200) * 100 = 0.625%, which matches the previous calculation.

Why Other Options Are Wrong:
1.25% profit would require double the net gain relative to the total investment.
0.625% loss is wrong in sign because our net result is a gain, not a loss.
1.20% profit and 2% profit both imply much larger gains than the calculated 0.625%.

Common Pitfalls:
A common mistake is to average the percentages directly as (minus 20 + 10 + 12.5) / 3, which ignores the different investment amounts. Another error is to forget that the first percentage is a loss and treat it as positive. Learners may also mishandle the ratio and not convert it into actual proportional amounts, leading to incorrect weighted averages. Always multiply each amount by its own percentage first, then combine the results, and finally divide by the total investment.

Final Answer:
The woman earns an overall 0.625% profit on her total investment.