For any purchase of less than $1000, a person doesn’t need to pay any tax. However, if the purchase value is between $1000 and $2000 included, then the tax is 5% of the excess value over $1000. Also, if the purchase value is more than $2000, then one needs to pay an additional tax of 10% on the excess value over $2000. Which of the following can be the possible purchase value, such that the total tax paid is not more than 4% of the total purchase value?
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This one took me a while. I set up the tax as 0.05*(x-1000) + 0.10*(x-2000) for x>2000, then required that to be ≤ 0.04x. Solving gave me x ≤ 2500, but I had to double-check the arithmetic because the choices are all close.
Same here, I almost picked 2499 but then realized I needed to check the inequality direction. The word 'not more than' means ≤, so the boundary is included.
The answer is A
Could you explain how you got that? I keep getting confused with the piecewise tax structure.
I found this tricky because the tax rate changes at 2000. For values just over 2000, the effective tax rate is 5% on the first 1000 over 1000, plus 10% on the rest. That quickly exceeds 4% of the total. The only way to keep the overall rate at or below 4% is if the purchase is not too far above 2000.
Wait, is the tax on the excess over 2000 additional? So for 2500, tax would be 5% of 1500? No, that doesn't seem right. Actually, it's 5% of (2500-1000) = 75, plus 10% of (2500-2000) = 50, total 125. 125/2500 = 5%, which is more than 4%. So 2499 gives 124.95/2499 ≈ 5.0%? That can't be right. Let me recalculate: 5% of 1499 = 74.95, 10% of 499 = 49.9, total 124.85, 124.85/2499 ≈ 4.996%. Still over 4%. I must be misunderstanding the tax structure.
I think you're right that the tax is on the excess over 1000 for the 5% part, and then additional 10% on the excess over 2000. But the question says 'the tax is 5% of the excess value over $1000' for purchases between 1000 and 2000. For purchases over 2000, it says 'one needs to pay an additional tax of 10% on the excess value over $2000.' So the total tax is 5% of (x-1000) + 10% of (x-2000). That's what I used. So for 2499, tax = 0.05*1499 + 0.10*499 = 74.95 + 49.9 = 124.85. That's 124.85/2499 = 0.04996, which is 4.996%, over 4%. So 2499 doesn't work. Maybe I need to solve 0.05(x-1000) + 0.10(x-2000) ≤ 0.04x. That gives 0.15x - 150 ≤ 0.04x => 0.11x ≤ 150 => x ≤ 1363.6. But none of the choices are that low. Something is off.
What it tests
Your ability to compute percentage change, percentage of a whole, and to work backwards from a final value.
Common trap
Treating successive percentage changes as additive — a 10% rise then a 10% fall is not back to the start.