Percents: Percent Translation

Every month, Harriet draws three paychecks from three different jobs. The largest paycheck incurs a 35% income tax, while the other two paychecks each incur income taxes of 25%. Is the total amount of tax deducted from the three paychecks greater than 30% of the total of the paychecks before taxes? >(1) Harriet's largest paycheck is $4,000 and her second-largest is $1,500. >(2) Harriet's smallest paycheck is $1,000.

Alternative method: Translate the question into percent language and try to simplify it: If Harriet has three salaries, $$x$$, $$y$$, and $$z$$ (where $$x$$ is the largest), then the taxes she pays can be translated as $$0.35 x$$ (35% of the largest), $$0.25 y$$, and $$0.25 z$$ (25% of the other two). Then 30% of her total salary can be written as $$0.3(x+y+z)$$. Now the question is if >$$0.35 x+0.25 y+0.25 z > 0.3(x+y+z)$$ which can be reduced to >$$0.05 x > 0.05 y + 0.05 z$$ and further reduced to >$$x > y+z $$. Now test the answer choices according to this inequality: Stat. (1) provides a definite answer, but Stat. (2) does not. Thus, the answer is A.

Incorrect. [[snippet]] Stat. (2): Consider the extremes. 1. __Maximum__: Since now there's no upper limit to the value of the largest and second-largest checks, you can obviously find an example where the tax is greater than 30%: $1,001 for the second check and $1,000,000 for the largest so that the smaller checks are negligible. >In this case, the tax will be very close to 35% of the total tax, and will definitely be greater than 30%. So the answer to the question stem is "Yes", but is it always "Yes"? 2. __Minimum__: Plug in the other extreme, with a minimal difference between the paychecks. For example, assume the small paycheck is $1,000, the medium paycheck is $1,001, and the large check is $1,002. You can even __Ballpark__ all three checks at $1,000 each. Now calculate 30% of the total: >>$$\text{Total} = \$1{,}000+\$1{,}000+\$1{,}000=\$3{,}000$$ >>$$30\% \text{ of Total} = 30\% \times \$3{,}000 = \$900$$ >Now calculate the total tax: >>$$\text{Tax for largest check} = 35\% \text{ of } \$1{,}000 = \$350$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}000+\$1{,}000) = \$500$$ >>$$\text{Total tax} = \$500+\$350 = \$850$$ >This is _not_ greater than $900, giving an answer of "No." Thus, there's no single answer, and **Stat.(2) → Maybe → IS → ACE**.

Incorrect. [[snippet]] Stat.(1): The second-largest check is $1,500, so the third-largest check can be anywhere from zero to $1,500. Plug in the extremes: 1. __Maximum__: If the third check is $1,500, then the total is >>$$\text{Total} = \$4{,}000+\$1{,}500+\$1{,}500=\$7{,}000$$. >Now calculate 30% of this using 10% blocks: >>$$30\% \text{ of Total} = 30\% \times \$7{,}000 = \$2{,}100$$ >So the question is now whether the total tax is greater than $2,100. >>$$\text{Tax for largest check} = 35\% \text{ of } \$4{,}000 = \$1{,}400$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}500+\$1{,}500) = \$750$$ >>$$\text{Total tax} = \$1{,}400+\$750 = \$2{,}150$$ >Since this is more than $2,100, the answer to the question stem is "Yes." Is it always "Yes"? Test it by plugging in the other extreme: 2. __Minimum__: Try zero dollars for the smallest check (or $1, close to zero). First find 30% of the total tax: >>$$\text{Total} = \$4{,}000+\$1{,}500+\$0=\$5{,}500$$ >>$$30\% \text{ of Total} = 30\% \times \$5{,}500 = \$1{,}650$$ >Now calculate the total tax: >>$$\text{Tax for largest check} = 35\% \text{ of } \$4{,}000 = \$1{,}400$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}500+\$0) = \$375$$ >>$$\text{Total tax} = \$1{,}400+$375 = \$1{,}775$$ >This is still greater than $1,650, the 30% target. Thus, the answer is always "Yes," so **Stat.(1) → Yes → S → AD**.

Correct. [[snippet]] Stat.(1): The second-largest check is $1,500, so the third-largest check can be anywhere from zero to $1,500. Plug in the extremes: 1. __Maximum__: If the third check is $1,500, then the total is >>$$\text{Total} = \$4{,}000+\$1{,}500+\$1{,}500=\$7{,}000$$. >Now calculate 30% of this using 10% blocks: >>$$30\% \text{ of Total} = 30\% \times \$7{,}000 = \$2{,}100$$ >So the question is now whether the total tax is greater than $2,100. >>$$\text{Tax for largest check} = 35\% \text{ of } \$4{,}000 = \$1{,}400$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}500+\$1{,}500) = \$750$$ >>$$\text{Total tax} = \$1{,}400+\$750 = \$2{,}150$$ >Since this is more than $2,100, the answer to the question stem is "Yes." Is it always "Yes"? Test it by plugging in the other extreme: 2. __Minimum__: Try zero dollars for the smallest check (or $1, close to zero). First find 30% of the total tax: >>$$\text{Total} = \$4{,}000+\$1{,}500+\$0=\$5{,}500$$ >>$$30\% \text{ of Total} = 30\% \times \$5{,}500 = \$1{,}650$$ >Now calculate the total tax: >>$$\text{Tax for largest check} = 35\% \text{ of } \$4{,}000 = \$1{,}400$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}500+\$0) = \$375$$ >>$$\text{Total tax} = \$1{,}400+$375 = \$1{,}775$$ >This is still greater than $1,650, the 30% target. Thus, the answer is always "Yes," so **Stat.(1) → Yes → S → AD**.

Stat. (2): Again, consider the extremes. 1. __Maximum__: Since now there's no upper limit to the value of the largest and second-largest checks, you can obviously find an example where the tax is greater than 30%: $1,001 for the second check and $1,000,000 for the largest so that the smaller checks are negligible. >In this case, the tax will be very close to 35% of the total tax, and will definitely be greater than 30%. So the answer to the question stem is "Yes", but is it always "Yes"? 2. __Minimum__: Plug in the other extreme, with a minimal difference between the paychecks. For example, assume the small paycheck is $1,000, the medium paycheck is $1,001, and the large check is $1,002. You can even __Ballpark__ all three checks at $1,000 each. Now calculate 30% of the total: >>$$\text{Total} = \$1{,}000+\$1{,}000+\$1{,}000=\$3{,}000$$ >>$$30\% \text{ of Total} = 30\% \times \$3{,}000 = \$900$$ >Now calculate the total tax: >>$$\text{Tax for largest check} = 35\% \text{ of } \$1{,}000 = \$350$$ >>$$\text{Tax for other checks} = 25\% \text{ of } (\$1{,}000+\$1{,}000) = \$500$$ >>$$\text{Total tax} = \$500+\$350 = \$850$$ >This is _not_ greater than $900, giving an answer of "No." Thus, there's no single answer, and **Stat.(2) → Maybe → IS → A**.

Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.

Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.

BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

EACH statement ALONE is sufficient to answer the question asked.

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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