A certain income tax is graduated according to the table above. For

Question

https://gmatclub.com/forum/download/file.php?id=81561 A certain income tax is graduated according to the table above. For example, an individual with a taxable income of $40,000 pays (0.10)($10,000) + (0.15) ($20,000) + (0.25)($10,000) = $6,500 in taxes. Last year, Tim paid income tax based on the rates in the table and paid no other taxes. What was Tim’s taxable income last year? (1) Last year, Tim paid $9,000 in taxes. (2) Last year, Tim paid the equivalent of 18% of his taxable income in taxes. Capture.PNG GMAT-Club-Forum-ucs0kayx.png

Bunuel · Accepted Answer

The average tax rate is 18%. Ask yourself, could it still be 18% if the income is below $30,000, where the tax rates are 10% and/or 15%? No, becasue in this case the average tax rate would be less than 15%.

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=75233 A certain income tax is graduated according to the table above. For example, an individual with a taxable income of $40,000 pays (0.10)($10,000) + (0.15) ($20,000) + (0.25)($10,000) = $6,500 in taxes. Last year, Tim paid income tax based on the rates in the table and paid no other taxes. What was Tim’s taxable income last year? (1) Last year, Tim paid $9,000 in taxes.$9,000 in taxes implies that the income was more than $30,000. Thus, assuming the income was $x we'd have: (0.10)($10,000) + (0.15) ($20,000) + (0.25)($x - $30,000) = $9,000 We can solve for x. Sufficient. P.S. x comes out to be $50,000. (2) Last year, Tim paid the equivalent of 18% of his taxable income in taxes. The above also implies that the income was more than $30,000. Thus, we'd have: (0.10)($10,000) + (0.15) ($20,000) + (0.25)($x - $30,000) = 0.18 * $x And again, we can solve for x. Sufficient. Answer: D. 2024-02-27_23-14-02.png

Laracbo · Answer

Hi  Bunuel

Thank you for posting the solution to this problem. Question regarding statement (2):  Last year, Tim paid the equivalent of 18% of his taxable income in taxes.

You wrote: "The above also implies that the income was more than $30,000." how did you deduce from statement (2) that the income was more than 30,000?

Many thanks for your support.

Nishtapatel · Answer

Hi,

Why did you do .15(20,000) instead of .15(30,000)?

Bunuel · Answer

That's because a 15% tax rate applies to income exceeding $10,000, and similarly, a 25% tax rate applies to income exceeding $30,000. For example, if the income is $50,000, the initial $10,000 will be taxed at 10%, the subsequent $20,000 (from $10,000 to $30,000) will be taxed at 15%, and the remaining $20,000 (from $30,000 to $50,000) will be taxed at 25%.

KarishmaB · Answer

(1) Last year, Tim paid $9,000 in taxes. For each income amount, the tax will be different. Hence knowing the tax amount, we can find the income though we don't need to here since it is a DS question. If we were required to solve it (i.e. in a Quant question), we would simply note that the given example leads to $6500 in tax for a salary of 40k. If tax paid is extra $2500 (total tax paid is $9000), the income would be an amount X more than $40k such that 2500 = 25% of X. Hence X must be $10,000 which means total income of Tim is 50k. Sufficient alone (2) Last year, Tim paid the equivalent of 18% of his taxable income in taxes. Here think about weighted averages. We are given that the average tax rate on his income is 18%. This is the average of 10%, 15% and 25%. We know the weight of 10% is 10k and that of 15% is 20k. So the only unknown is the weight of 25% (the amount on which 25% tax was charged). We can find it using 18% = \frac{(10% * 10 + 15% * 20 + 25% * x)}{(10 + 20 + x)} where x is the income above 30k. This is a simple linear equation in x and we will get a single unique value of x. So we will get Tim's income. Sufficient alone Answer (D) Weighted Averages has been discussed here: https://anaprep.com/arithmetic-weighted-averages/

A certain income tax is graduated according to the table above. For

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