A shopkeeper charges sales tax of x percent on the first Rs. 2000 of a bill and y percent on the remaining amount. A customer pays a total tax of Rs. 320 on goods worth Rs. 6000 and a total tax of Rs. 680 on goods worth Rs. 12,000. What is the value of x - y?

Introduction / Context:
This question involves a piecewise or tiered tax system where different tax rates apply to different parts of the bill amount. We are told how much tax is paid for two different purchase values. Using this information, we must find the difference between the two tax rates x and y. It combines percentage calculation with solving simultaneous linear equations.

Given Data / Assumptions:

• For any bill, the first Rs. 2000 is taxed at x percent.
• The amount above Rs. 2000 is taxed at y percent.
• On goods worth Rs. 6000, the total tax paid is Rs. 320.
• On goods worth Rs. 12,000, the total tax paid is Rs. 680.
• We must find x - y.

Concept / Approach:
The tax on any bill can be written as the sum of two parts: tax on the first Rs. 2000 plus tax on the remaining amount. For a bill of Rs. 6000, the remaining amount is Rs. 4000. For a bill of Rs. 12,000, the remaining amount is Rs. 10,000. Writing equations for the total tax in each scenario in terms of x and y leads to two linear equations. Solving these equations simultaneously yields values of x and y and thus x - y.

Step-by-Step Solution:
Step 1: For a bill of Rs. 6000, tax on first Rs. 2000 is (x / 100) * 2000. Tax on remaining Rs. 4000 is (y / 100) * 4000.Step 2: Total tax for the Rs. 6000 bill is Rs. 320, so (x / 100) * 2000 + (y / 100) * 4000 = 320.Step 3: Multiply both sides by 100 to simplify: 2000x + 4000y = 32000. Divide entire equation by 100 to get 20x + 40y = 320.Step 4: For a bill of Rs. 12,000, tax on first Rs. 2000 is (x / 100) * 2000 and tax on remaining Rs. 10,000 is (y / 100) * 10,000.Step 5: Total tax for the Rs. 12,000 bill is Rs. 680, so (x / 100) * 2000 + (y / 100) * 10000 = 680.Step 6: Multiply both sides by 100: 2000x + 10000y = 68000. Divide by 100 to get 20x + 100y = 680.Step 7: Now we have the system: 20x + 40y = 320 and 20x + 100y = 680.Step 8: Subtract the first equation from the second: (20x + 100y) - (20x + 40y) = 680 - 320, giving 60y = 360.Step 9: Solve for y: y = 360 / 60 = 6.Step 10: Substitute y = 6 into 20x + 40y = 320: 20x + 40 * 6 = 320, so 20x + 240 = 320, thus 20x = 80 and x = 4.Step 11: Therefore x - y = 4 - 6 = -2.

Verification / Alternative check:
Verify using the found rates x = 4 percent and y = 6 percent. For Rs. 6000: tax on first Rs. 2000 is 4 percent of 2000 = 80. Tax on remaining Rs. 4000 is 6 percent of 4000 = 240. Total tax is 80 + 240 = 320, which matches the given amount. For Rs. 12,000: tax on first Rs. 2000 is again 80, and on Rs. 10,000 at 6 percent is 600. Total tax is 80 + 600 = 680, which matches the second given condition. So the values are correct.

Why Other Options Are Wrong:
If x - y were 0, then both rates would be equal, which does not satisfy the two different tax totals. Values -4, 5 or 2 do not result from the system of equations. Checking with those differences would lead to inconsistent tax amounts for one or both bills. Only x - y = -2 fits all the provided data and is supported by direct substitution.

Common Pitfalls:
Some students misinterpret the two slab rates and apply the higher rate to the entire bill instead of only the amount above Rs. 2000. Others may make algebraic errors while simplifying or subtracting the equations. Using clear stepwise manipulation and checking both scenarios with the found rates helps avoid these mistakes.

Final Answer:
The difference between the two tax rates is x - y = -2, so the correct option is -2.