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Bunuel
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A state legislator drafts an income tax proposal that imposes a tax of 16 Nov 2023, 02:30
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17% (02:45) correct 83% (02:24) wrong based on 58 sessions

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A state legislator drafts an income tax proposal that imposes a tax of 20% on the first $100,000 of an individual’s income, increases the tax to 25% on the next $50,000 in income, and increases the tax by an additional 5% on each additional $50,000 in income, to a maximum tax of 40% on the portion of income over $250,000. Under the proposal, would the state collect an average (arithmetic mean) of more than 30% of the income of the residents of County X ?

(1) The median income for a resident of County X is $65,000.
(2) No resident of County X earns an income of more than $350,000.


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B
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A state legislator drafts an income tax proposal that imposes a tax of 16 Nov 2023, 04:51
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Imo E because none of the statements really say anything about the distribution of incomes

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A state legislator drafts an income tax proposal that imposes a tax of 17 Nov 2023, 02:56
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Bunuel
A state legislator drafts an income tax proposal that imposes a tax of 20% on the first $100,000 of an individual’s income, increases the tax to 25% on the next $50,000 in income, and increases the tax by an additional 5% on each additional $50,000 in income, to a maximum tax of 40% on the portion of income over $250,000. Under the proposal, would the state collect an average (arithmetic mean) of more than 30% of the income of the residents of County X ?

(1) The median income for a resident of County X is $65,000.
(2) No resident of County X earns an income of more than $350,000.


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Why is it B and not E?
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A state legislator drafts an income tax proposal that imposes a tax of 01 Apr 2025, 10:06
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Quote:

A state legislator drafts an income tax proposal that imposes a tax of 20% on the first $100,000 of an individual’s income, increases the tax to 25% on the next $50,000 in income, and increases the tax by an additional 5% on each additional $50,000 in income, to a maximum tax of 40% on the portion of income over $250,000. Under the proposal, would the state collect an average (arithmetic mean) of more than 30% of the income of the residents of County X ?

(1) The median income for a resident of County X is $65,000.
(2) No resident of County X earns an income of more than $350,000.
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1st 100k tax is 20% = 20k tax
Next 50k is 25% = 12.5k tax
Next 50k is 30% = 15k tax
Next 50k is 35% = 17.5k tax
Anything above 250k is 40% tax.
A point to note is that total tax as a % of total income for any resident will be equal to or more than 20% and less than 40%.
20% ≤ tax rate < 40%. It can never go above 40% as some component of the income is taxed at 20-35%, so even if a person earns 100 billion, the tax rate will be some 39.9999999997% but never touch 40% due to weighted average of total tax.

1) Says that median income is 65k. So 1/2 of the residents will earn 65k, remaining can earn anything above 65k. Let's try 2 cases.
Case 1: Income of 3 residents = 65k, 65k, 65k
Tax for each is 13k, 13k, 13k
Taxes collected as a % of total income of the country = 13k x 3 / 65k x 3 = 20%.

Case 2: Income of 3 residents = 65k, 65k, 1000k.

Total tax = 13k + 13k + (20k + 12.5k + 15k + 17.5k + 750k*40%) = 26k + (65k + 300K) = 391k
Total income = 1130k
Taxes as a % of total income = 391k / 1130k = 34.6%

Taxes are 20% in case 1 and 34.6% in case 2, so can be above as well as below 30%

(1) is INSUFFICIENT

2) No one earns more than 350k.
Let's say country has 10 residents and all earn 350k, total income = 3500k, total tax = 10*(65k + 100k*40%) = 10*(105k) = 1050k
Tax as a % of total income = 105/350 = 30%
Let's say all 10 residents earn 340k, total income = 3400k, total tax = 10*(65k + 90k*40%) = 10*(101k) = 1010k
Tax as a % of total income = 101/340 = 29.7%

The more we lower the income, the portion of income that is taxed at 40% will also become lower, so overall tax rate will also reduce.
To make it clearer, lowest tax rate is 20% and highest is 40%. If the income has a bigger portion of the income that is taxed at 40% (i.e. above 250k) then the overall rate will tend to be closer to 40% but will never reach 40%.
And if the income has a bigger portion of the income that is taxed at lesser rates, then overall rate will reduce more and move toward 20%.

In our case, at 350k income, tax rate is 30%. But everyone's income is lower than 350k, so the overall rate for any income lower than 350k will always be lower than 30% (as more taxes are paid at lower rates than at 40% rates).

Maximum possible income for all residents can be <350k, so maximum possible tax rate for all residents <30%.
As overall tax rate is always lower than 30%, we have an answer.

(2) is SUFFICIENT

Answer: B
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