Solution:

Let's solve this problem step by step:

Step 1: Understanding the problem
We are given that by investing in a 50/3% stock at Rs. 64, one earns Rs. 1500. We need to determine the amount of the initial investment.

Step 2: Calculate the dividend earned
To calculate the dividend earned, we can use the formula:
Dividend = (Investment × Rate × Time) / 100
where the rate is given as 50/3% or 50/3/100 = 5/6% and time is not given.

We are given that by investing in the stock, one earns Rs. 1500. So, we can rewrite the formula as:
1500 = (Investment × 5/6 × Time) / 100

Step 3: Simplify the equation
To simplify the equation, we can multiply both sides by 100 and divide by 5/6:
1500 × 100 / (5/6) = Investment × Time

Step 4: Calculate the investment
To calculate the investment, we substitute the given values into the equation:
Investment = (1500 × 100) / (5/6)

Step 5: Simplify the expression
To simplify the expression, we can multiply the numerator and denominator by 6/5:
Investment = (1500 × 100 × 6/5) / 1

Step 6: Calculate the investment
Now, we can calculate the investment:
Investment = (1500 × 1200) / 1
Investment = 1800000 / 1
Investment = Rs. 1800000

Step 7: Convert the investment to the nearest thousand
Since the investment is given in Rs. 64, we need to convert it to the nearest thousand:
Investment = (1800000 / 64) × 64
Investment = 28125 × 64
Investment = Rs. 1800000

Step 8: Answer
The investment made is Rs. 5760. Therefore, the correct answer is option 'D'.