Introduction:
To find the selling prices of a shirt with a profit of 10% and 15% respectively, we are given that the cash difference between the two selling prices is ₹10.

Step 1: Understanding the Problem:
Let's assume the cost price of the shirt is ₹x.
To find the selling price, we need to add the profit percentage to the cost price. The selling price with a 10% profit is given by (100% + 10%) of the cost price, and the selling price with a 15% profit is given by (100% + 15%) of the cost price.

Step 2: Calculating Selling Prices:
Let's calculate the selling prices using the above formulas:

For a 10% profit:
Selling Price = Cost Price + (10% of Cost Price)
Selling Price = x + (10/100)x
Selling Price = x + 0.1x
Selling Price = 1.1x

For a 15% profit:
Selling Price = Cost Price + (15% of Cost Price)
Selling Price = x + (15/100)x
Selling Price = x + 0.15x
Selling Price = 1.15x

Step 3: Finding the Cash Difference:
Given that the cash difference between the two selling prices is ₹10, we can set up the following equation:

1.15x - 1.1x = ₹10
0.05x = ₹10
x = ₹10 / 0.05
x = ₹200

Step 4: Calculating Selling Prices:
Using the value of x, we can calculate the selling prices:

For a 10% profit:
Selling Price = 1.1x = 1.1 * ₹200 = ₹220

For a 15% profit:
Selling Price = 1.15x = 1.15 * ₹200 = ₹230

Conclusion:
The selling price of the shirt with a profit of 10% is ₹220, and the selling price with a profit of 15% is ₹230. The cash difference between the two selling prices is ₹10.