A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Did the salesman's base salary account for more than half of the salesman's yearly income last year?

Given: {Income} = {salary} + {commission}. Question basically asks: is {salary} > {commission}?


(1) If the amount of the commission had been 30 percent higher, the salesman's income would have been 10 percent higher last year:

1.1({salary} + {commission}) = {salary} + 1.3{commission};

{salary} = 2{commission};

Therefore, {salary} > {commission}. Sufficient.


(2) The difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year:

|{salary} - {commission}| = 0.5{salary}, notice that {salary} - {commission} is in absolute value sign ||, meaning that we can have two cases:

A. {salary} - {commission} = 0.5{salary};

0.5{salary} = {commission};

{salary} > {commission}, thus the answer would be YES;

Or:
B. {commission} - {salary} = 0.5{salary};

1.5{salary} = {commission};

{salary} < {commission}, thus the answer would be No.

Two different answers. Not sufficient.


Answer: A.

Hope it's clear.
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please explain why salary - commission is in absoulute value sign?

Because if {salary}>{commission} then {salary}-{commission}=0.5{salary}, since 0.5{salary}>0.

But if {salary}<{commission} then {commission}-{salary}=0.5{salary}.

So, the second statement, which says that "the difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year" should be expressed as |{salary}-{commission}|=0.5{salary}.
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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I don't know how answer choice A can be sufficient. See example below.

Using your statement:
1.1({salary}+{commission})={salary}+1.3{commission}

1.1 s + 1.1 c = s + 1.3 c ; for sake of simplicity, let's say that salary = 100 and commission = 100

1.1 (100) + 1.1 (100) = 100 + 1.3 (100)

110 + 110 = 100 + 130

220 < 230 ; Insufficient

^^ I'm confused, do we have to take a salary that's greater than commission to solve the question "Is salary > commission?"

Another way I thought of it was...

If instead of plugging in values, is you decide isolate salary (s) and commission (c) using your formula in bold, it would be:

.10 s = .02 c

In this case, for all positive values where salary > commission, it holds true. Sufficient.

Can someone please help explain how Bunuel got salary = 2 commission? What am I doing wrong above? Am I missing something?

~ Im2bz2p345

hi,

the above highlited part is wrong.
in that you are assuming salary = comission = 100
if both are equal how can you compare which one is bigger.

let say salary = \(s\)
comission =\(c\)

\(1.1(s + c) = s + 1.3 c\)
\(1.1s + 1.1c = s + 1.3 c\)
taking s items one side and c item one side
\(0.1s = 0.2c\)
ok now multiply both sides with 10
\(s = 2c\)

hope its clear now
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Dear Bunuel,

I'm not sure if you need to consider two cases here based on the question's second statement.
It is like stating " if the difference between A and B is 4, I would consider A-B = 4 and not |A-B| =4.

The similar question, which you used to merge the topic clearly specifies that the absolute difference between the base salary and the commission is..., there I can understand the two cases but not for the question above, where it does not state anything about the absolute difference.

Please correct if I'm wrong.

My 2 cents
from question stem: commission + base = 1

for statement 1,
-> 1.3base + commission = 1.1
plus 1base + commission = 1
-> 0.3 base = 0.1
-> base = 33%
-> sufficient

for statement 2,
-> base - commission = 0.5base
or
-> commission - base = 0.5base
-> base is equal 50% or 33%
-> not greater than 50%
-> not suffificent

-> Correct Answer is A

JMO, please correct me if there is any logical flaw. Thanks very much!

Hi santorasantu,

The prompt never stated whether the base salary was larger than the commission or the commission was larger than the base salary, so we CANNOT assume that the base salary is bigger just because it was mentioned first in the sentence. The word "difference" implies that one of them IS bigger, but we don't know which one. THAT is why Bunuel addressed it.

GMAT assassins aren't born, they're made,
Rich
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For statement 1, why can we assume that the base salary stays the same? i.e no percent change associated with base salary. If only the commission had been 30% higher, then the answer would be A, but what if the base salary could be, for example, 20% lower? Wouldn't the answer then be E?

Thanks

Hi stopchange,

The specific question that is asked refers to a base salary and a commission LAST YEAR, so we're dealing with 2 unknowns, NOT 2 variables. This means that the two numbers are constants, but we do NOT know what they are (and thus, we don't know which one is bigger).

Fact 1 uses a 'hypothetical' that points out that increasing JUST the commission (by 30%) would have led to an increase in income (of 10%). By extension, this assumes that the other pieces (in this case, the base salary) stay the same.

GMAT assassins aren't born, they're made,
Rich
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Can we also solve this problem via average weight logic?

Income increased by %10 when the commission is increased by %30. If commission and base salary were equal, then income would increase by %15 since base salary has %0 increase and the commision has %30 increase. (0+30)/2 = 15.

Since it is increased by %10, base salary (which has %0 increase) pulls it, meaning weight of S > C

I am not still satisfied with the explanation given above.How can we assume 2 cases when it is mentioned in statement (2) that difference between the amount of salesman's base salary and the the amount of commission was equal to 50 percent of the salesman's base salary last year.

Hi GouthamNandu,

The prompt never stated whether the base salary was larger than the commission or the commission was larger than the base salary, so we CANNOT assume that the base salary is bigger just because it was mentioned first in the sentence. The word "difference" implies that one of them IS bigger, but we don't know which one. Bunuel addresses this point in one of his posts.

GMAT assassins aren't born, they're made,
Rich
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I have an alternate solution based on pure number crunching.


Let's assume the total income was $100 with a $50:50 split between base and commission. If the commission goes up by 30% to $65, the total becomes $115 which is 15% of the total, but the (1) states it only went up by 10%, in which case the commission has to be less than 50% of the total income. So this answers the question.


Again, plugging numbers in this, let's assume $50 (B: Base Salary), $25 (C: Commission) and $25 (B-C). In this case Base Salary is $50/$75 = 67%.
However, the question states only the word difference. So we could have a C-B=(1/2)B scenario as well. In which case, assuming C to be $30, B would be $20 (<50% of the total) which is inconsistent with the 67% above, therefore insufficient and the answer is A.

Target question: Was the salesman's base salary larger than his commission last year?
This is a good candidate for rephrasing the target question.

Let B = base salary last year
Let C = commission last year
So, B+C = TOTAL income last year
REPHRASED target question: Is B greater than C?

Statement 1: If the amount of the commission had been 30 percent higher, the salesman's total income (salary plus commission) would have been 10 percent higher last year.
If we increase the commission by 30% the NEW commission = 1.3C, which means the TOTAL income = 1.3C + B
This NEW income is 10% greater than the actual TOTAL income (B+C)
We can write: 1.3C + B = 1.1(B + C)
Expand: 1.3C + B = 1.1B + 1.1C
Rearrange to get: 0.2C = 0.1B
Make "prettier" by multiplying both sides by 10 to get: 2C = 1B
Since C and B are both POSITIVE, we can see that B must be greater than C (since B is equal to C+C)
Another way say this is, B is greater than C
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: The absolute difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year.
We can write: |C - B| = 0.5B
This gives us two possible cases:
Case a: C - B = 0.5B. When we solve this for C, we get C = 1.5B, which means B is NOT greater than C
Case b: C - B = -0.5B. When we solve this for C, we get C = 0.5B, which means B IS greater than C
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Income - I
Commission - C
Base Salary - B

Originally last year - I = B + C

Option 1 -
B + 1.3C = 1.1I
Subtract this from the original equation
We get - 0.1I = 0.3C
Or C = 0.33I
This means that Base was 0.67 of I which is greater than 50%
Sufficient

Option 2 -
Since we are not given whether B > C - we should consider it both ways. First if B was greater than C -

B - C = 0.5B
C = 0.5B
Replacing in original equation
I = B + 0.5B
B = 0.66I

Hence this answers as yes, base salary accounts for more than 50% of the income

Now second if C was greater than B -

C - B = 0.5B
C = 1.5B

Replacing in original equation -

I = B + 1.5B
B = 0.4I

Hence this answers as no, base salary does not account for more than 50% of the income

Hence option 2 does not give us a definite answer. So not sufficient.

Answer A

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In Option A

How did u get {salary}=2{commission}. Please help

1.1({salary} + {commission}) = {salary} + 1.3{commission};

1.1{salary} + 1.1{commission} = {salary} + 1.3{commission};

0.1{salary} = 0.2{commission};

{salary}=2{commission}.

Hope it's clear.
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