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Every month, Harriet draws three paychecks from three jobs. The greate

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Every month, Harriet draws three paychecks from three jobs. The greate [#permalink]

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New post 31 May 2015, 03:35
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  45% (medium)

Question Stats:

60% (01:43) correct 40% (01:22) wrong based on 90 sessions

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Every month, Harriet draws three paychecks from three jobs. The greatest paycheck incurs a thirty five percent income tax, while the other two paychecks incur twenty five percent income tax each. Is the total amount of tax deducted from the three paychecks greater than thirty percent of the total of the paychecks before tax deduction?

(1) Harriet's biggest salary is $4000, and the next largest salary is $1500.

(2) Harriet's smallest salary is $1000.
Stat.(1): The second largest check is $1500, so the third largest check can be anywhere between zero and $1500. Plug in the extremes: if the third check is the maximum $1500, then Harriet's salary is $4000+$1500+$1500=$7000. 30% of $7000 is $2100 (3 "10%" blocks of $700), so the question is now: is the tax greater than $2100?

Big check tax: 35% of $4000 = $1400 (3.5 × 10% blocks of $400)

Second and third check tax: 25% of ($1500+$1500) = 25%×3000 = $750

Total tax of $1400+$750 = $2150 , which is more than $2100, so the answer to the question stem is "yes".

Is it always "yes"? test it by plugging in the other extreme: what if you plug in zero dollars for the smallest check (or $1, close to zero)? The result is still greater than 30%:

Total salary = $4000+$1500+$0 = $5500

30% tax target = 30% × $5500 = $1650

Tax for large check = $1400

Tax for next largest check = $375.

Total tax: $1400+$375 = $1775, which is still greater than $1650, the 30% target.

Thus, the answer is always "yes", and Stat.(1)->Yes->S->AD.

Stat. (2): 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%: plug in $1001 for the second check and $1,000,000 for the largest salary, 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"? plug in the other extreme, with a minimum difference between the paychecks. For example,

small paycheck=$1000, medium check = $1001, large check = $1002. You can even ballpark all three checks at $1000 each.

Now the total salary is $1000+$1000+$1000 = $3000

Tax target of 30% = 30%×$3000 = $900

Tax for largest check: 35%×1000 = $350

Tax for medium and smaller check = 25%×($1000+$1000) = $250+$250 = $500.

Total tax = $500+$350 = $850, which is not greater than $900, giving an answer of "no".

Thus, there's no single answer, and Stat.(2)->Maybe->IS->A.



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, the tax she pays can be translated to 0.35·x (35% of the largest), 0.25·y and 0.25·z (25% of the other two). Thirty percent of her total salary can be written as 0.3(x+y+z). Now the question is

Is 0.35·x+0.25·y+0.25·z > 0.3(x+y+z) ?

which can be reduced to

Is 0.05·x > 0.05·y+0.05·z ?

and further reduced to the final question:

Is x > y+z ?

Now test the answer choices according to this inequality: (1) provide a definite answer, but (2) does not. Thus, the answer is A.

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Re: Every month, Harriet draws three paychecks from three jobs. The greate [#permalink]

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New post 31 May 2015, 23:00
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This problem can be solved quickly, and the trap answer noticed easily, by spending a little time on the question stem. A mistake that rookie test takers often make on data sufficiency is thinking that all of the value lies in the statements, and as a result not spending enough time on the question stem. Here we learn that she receives 3 pay checks each month, the lowest 2 each with a 25% tax rate and the highest a 35% tax rate. In order to know whether the overall tax rate ends up being higher or lower than 30%, we simply need to know whether the two lower checks together are worth more than the highest pay check or vice versa, or if they are even. It's a simple weighted average question.

Statement 1 gives us everything that we need to know, because if the highest pay check is worth $4,000, and the second highest is worth $1,500, then at most the third pay check is worth $1,500 also. That would mean that at most the two lowest pay checks are worth $3k, while the highest is $4k, and therefore the highest is definitely larger and we can conclude that indeed the tax rate is over 30%.

Statement 2 here presents us with our obvious trap, the C trap. This is likely to be fallen into by test takers who rushed to the statements without spending a little reflection time on the question stem.

Therefore the correct answer is A and the trap answer is C.

I hope this helps!
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Re: Every month, Harriet draws three paychecks from three jobs. The greate [#permalink]

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New post 08 Mar 2017, 00:01
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reto wrote:
Every month, Harriet draws three paychecks from three jobs. The greatest paycheck incurs a thirty five percent income tax, while the other two paychecks incur twenty five percent income tax each. Is the total amount of tax deducted from the three paychecks greater than thirty percent of the total of the paychecks before tax deduction?

(1) Harriet's biggest salary is $4000, and the next largest salary is $1500.

(2) Harriet's smallest salary is $1000.
Stat.(1): The second largest check is $1500, so the third largest check can be anywhere between zero and $1500. Plug in the extremes: if the third check is the maximum $1500, then Harriet's salary is $4000+$1500+$1500=$7000. 30% of $7000 is $2100 (3 "10%" blocks of $700), so the question is now: is the tax greater than $2100?

Big check tax: 35% of $4000 = $1400 (3.5 × 10% blocks of $400)

Second and third check tax: 25% of ($1500+$1500) = 25%×3000 = $750

Total tax of $1400+$750 = $2150 , which is more than $2100, so the answer to the question stem is "yes".

Is it always "yes"? test it by plugging in the other extreme: what if you plug in zero dollars for the smallest check (or $1, close to zero)? The result is still greater than 30%:

Total salary = $4000+$1500+$0 = $5500

30% tax target = 30% × $5500 = $1650

Tax for large check = $1400

Tax for next largest check = $375.

Total tax: $1400+$375 = $1775, which is still greater than $1650, the 30% target.

Thus, the answer is always "yes", and Stat.(1)->Yes->S->AD.

Stat. (2): 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%: plug in $1001 for the second check and $1,000,000 for the largest salary, 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"? plug in the other extreme, with a minimum difference between the paychecks. For example,

small paycheck=$1000, medium check = $1001, large check = $1002. You can even ballpark all three checks at $1000 each.

Now the total salary is $1000+$1000+$1000 = $3000

Tax target of 30% = 30%×$3000 = $900

Tax for largest check: 35%×1000 = $350

Tax for medium and smaller check = 25%×($1000+$1000) = $250+$250 = $500.

Total tax = $500+$350 = $850, which is not greater than $900, giving an answer of "no".

Thus, there's no single answer, and Stat.(2)->Maybe->IS->A.



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, the tax she pays can be translated to 0.35·x (35% of the largest), 0.25·y and 0.25·z (25% of the other two). Thirty percent of her total salary can be written as 0.3(x+y+z). Now the question is

Is 0.35·x+0.25·y+0.25·z > 0.3(x+y+z) ?

which can be reduced to

Is 0.05·x > 0.05·y+0.05·z ?

and further reduced to the final question:

Is x > y+z ?

Now test the answer choices according to this inequality: (1) provide a definite answer, but (2) does not. Thus, the answer is A.


Let the salaries be a,b,c (a>b>c)
As per the question,
0.35a+0.25(b+c)>0.30(a+b+c)
simplifying this, the question becomes, is a>b+c ?

As per statement 1, a=4,000 and b=1,500. So even if b=c, 4000>3000. So the answer is YES. Sufficient.

Statement 2: Clearly insuffcient.
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Re: Every month, Harriet draws three paychecks from three jobs. The greate [#permalink]

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New post 08 Mar 2017, 05:12
let the salaries be a,b,c where a>b>c.
tax = 0.35a +0.2b +0.2c

St 1: a =4000, b = 1500
tax = 0.35*4000 +0.2*1500 +0.2*c = 1700 +0.2c.
c can range from 0 to 1500 and tax will be 1700 to 2100 for the total salary 5500 to 7000. therefore the percentage will lie in the range of 2100/7000 x 100 to 30% to 1700/5500 x 100 = 31%, which is definitely greater or equal to 30%. ANSWER

St 2: c =1000. nothing can be concluded as we have no idea about a and b. INSUFFICIENT

Option A
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Re: Every month, Harriet draws three paychecks from three jobs. The greate [#permalink]

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New post 08 Mar 2017, 22:14
the answer should be A
we know that 2nd largest paycheck is 1500 so the 3rd could be anything between 1 to 1500 and can also be 1500 too.
so in all cases the tax amount will be greater than 30% of these there paychecks . so sufficient
Re: Every month, Harriet draws three paychecks from three jobs. The greate   [#permalink] 08 Mar 2017, 22:14
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