Question

The tax rates for various income slabs are given. There are 15 persons, and in each slab (four slabs) the count lies between 3 and 5. Which of the following is the correct {total tax range of the 15 persons?}

• A. 1350 to 7350, both excluded
• B. 1350 to 9800, both included
• C. 2175 to 7350, both excluded
• D. 2175 to 9800, both included
• E. None of the above

Hint:

When counts per slab are bounded, push people to lower slabs with lower endpoints for the minimum and to higher slabs with upper endpoints for the maximum. Watch strict/closed bounds to decide inclusion or exclusion.

Answer 1

The Correct Option is A

Progressive Tax Rates

Rates (progressive):

\[ \le 500:\,0,\qquad (500,2000]:\,5\%,\qquad (2000,5000]:\,10\%,\qquad (5000,10000):\,15\%. \]

Hence the tax (at key endpoints): \[ T(2000) = 0.05 \cdot 1500 = 75, \quad T(5000) = 75 + 0.10 \cdot 3000 = 375, \quad T(10000) = 375 + 0.15 \cdot 5000 = 1125. \]

Minimum Total Tax

Use the fewest people in the highest slabs and incomes just above the lower bounds. Feasible counts that minimize tax: \[ \text{S1} = 5, \quad \text{S2} = 4, \quad \text{S3} = 3, \quad \text{S4} = 3 \quad (5+4+3+3 = 15). \]

Then (with S3 just above 2000 and S4 just above 5000): \[ \text{Min total} > 3 \cdot 75 + 3 \cdot 375 = 225 + 1125 = 1350. \] (Strictly greater than 1350 because “\(>2000\), \(>5000\)” give slightly more than 75 and 375.)

Maximum Total Tax

Use the most people in the highest slabs and incomes near the upper bounds. Feasible counts maximizing tax: \[ \text{S1} = 3, \quad \text{S2} = 3, \quad \text{S3} = 4, \quad \text{S4} = 5. \]

With S2 at 2000, S3 at 5000, and S4 just below 10000: \[ \text{Max total} < 3 \cdot 75 + 4 \cdot 375 + 5 \cdot 1125 = 225 + 1500 + 5625 = 7350. \] (Strictly less than 7350 since the top slab is \(<10000\).)

Final Answer:

\[ \boxed{\text{Total tax range is } (1350,\ 7350) \ \text{(both excluded).}} \]