Investment Loss and Recovery Fraction Calculation

This problem involves calculating the final fraction of an initial investment after experiencing a loss on the first day and then recovering a portion of that loss on the second day.

Step 1: Determine the Investment Remaining After Day 1

Let the initial investment be represented by the variable $I$. On the first day, the trader lost two thirds of her investment. This loss can be written as:

Loss on Day 1 = $\frac{2}{3} \times I = \frac{2}{3}I$

The amount of investment remaining after the loss on Day 1 is the initial investment minus the loss:

Remaining Investment (after Day 1) = $I - \frac{2}{3}I$

To subtract these, we find a common denominator:

Remaining Investment (after Day 1) = $\frac{3}{3}I - \frac{2}{3}I = \frac{3-2}{3}I = \frac{1}{3}I$

So, after the first day, the trader is left with one third of her initial investment.

Step 2: Calculate the Amount Recovered on Day 2

On the second day, the trader recovered one third of the *previous day's loss*. The loss from the previous day was $\frac{2}{3}I$. The amount recovered on Day 2 is:

Recovery on Day 2 = $\frac{1}{3} \times (\text{Loss on Day 1})$

Recovery on Day 2 = $\frac{1}{3} \times \left(\frac{2}{3}I\right)$

Multiplying the fractions gives:

Recovery on Day 2 = $\frac{1 \times 2}{3 \times 3}I = \frac{2}{9}I$

This means the trader recovered two ninths of her initial investment amount on the second day.

Step 3: Calculate the Final Investment Amount

The final amount left is the investment remaining after Day 1 plus the amount recovered on Day 2.

Final Investment = Remaining Investment (after Day 1) + Recovery on Day 2

Final Investment = $\frac{1}{3}I + \frac{2}{9}I$

To add these fractions, we need a common denominator, which is 9.

Convert $\frac{1}{3}I$ to a fraction with a denominator of 9:

$\frac{1}{3}I = \frac{1 \times 3}{3 \times 3}I = \frac{3}{9}I$

Now, add the fractions:

Final Investment = $\frac{3}{9}I + \frac{2}{9}I = \frac{3+2}{9}I = \frac{5}{9}I$

Conclusion

The fraction of her initial investment that the trader is left with is $\frac{5}{9}$.