Problem: On selling a tea set at 5% loss and lemon set at 15% gain a crockery seller gains rupees 7. If he sells tea set at 5% gain and the lemon set at 10% gain, he gains rupees 13. Find the actual price of the tea set and the lemon set.

Solution:

Let the cost price of the tea set be x and the cost price of the lemon set be y.

First Case:

When tea set is sold at 5% loss and lemon set is sold at 15% gain, the seller gains Rs. 7.

Selling price of tea set = 0.95x (as it is sold at 5% loss)
Selling price of lemon set = 1.15y (as it is sold at 15% gain)

Total gain = Selling price of tea set + Selling price of lemon set - (Cost price of tea set + Cost price of lemon set)
= 0.95x + 1.15y - (x+y)
= -0.05x + 0.15y + 7

As per the question, Total gain = Rs. 7
So, -0.05x + 0.15y + 7 = 7
=> -0.05x + 0.15y = 0

Second Case:

When tea set is sold at 5% gain and lemon set is sold at 10% gain, the seller gains Rs. 13.

Selling price of tea set = 1.05x (as it is sold at 5% gain)
Selling price of lemon set = 1.1y (as it is sold at 10% gain)

Total gain = Selling price of tea set + Selling price of lemon set - (Cost price of tea set + Cost price of lemon set)
= 1.05x + 1.1y - (x+y)
= 0.05x + 0.1y + 13

As per the question, Total gain = Rs. 13
So, 0.05x + 0.1y + 13 = 13
=> 0.05x + 0.1y = 0

Solving Equations:

We have two equations from the above cases:
-0.05x + 0.15y = 0
0.05x + 0.1y = 0

On solving these equations, we get:
y = 3x (Multiplying first equation by 3 and adding it to second equation)
Substituting y = 3x in any of the above equations, we get:
-0.05x + 0.15(3x) = 0
=> 0.4x = 0
=> x = 0

This means that the cost price of the tea set is zero, which is not possible. Hence, this problem has no solution.

Conclusion:

After solving the equations, we found that the problem has no solution. This means that the given information is inconsistent and cannot be solved.