Data Sufficiency

A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Was the salesman's commission larger than his base salary last year?

(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.

(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.


OAA

Just need to understand the statement 2.

Ok, for anyone other than j_shreyans who might read this, I'll explain Statement 1 also.

Statement one gives us a way to figure out the relationship between the salary and the commission.

We know that if the commission goes up 30%, we get only a 10% increase. That's interesting because that means that the commission was only a relatively small part of the total pay package. If it were 100% of the package, an increase in commission would result in the same increase in total pay. If it were a very small part of the package, say 1%, then an increase in commission would give us almost no increase in total pay.

In this case, a 30% increase in commission results in only a 10% increase in total pay. Just to be clear about the commission's part of the total package and it's size relative to the salary, I'll calculate the exact numbers.

Calling commission C, salary S, and total T, we get .3C = .1T. So C = 1/3 T

Obviously if C = 1/3 T then S = 2/3 T. So C < S, and Statement 1 is sufficient.

Statement 2 tells us that the difference between his base salary and his commission was 50% of the base salary. Ok, so if the base salary were 100, then the commission could be 50% less, 50. The thing is the commission could also be 50% more, 150. So we can't tell from Statement 2 alone which is higher, base salary or commission.

So the answer is A.

j_shreyans wrote:A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Was the salesman's commission larger than his base salary last year?

(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.

(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.

Let C = commission, T = total income, and S - salary.

Statement 1:
Case 1: C = 100
30% increase in C = (30/100)(100) = 30.
Since this increase represents a 10% increase in T, we get:
30 = (1/10)(T)
T = 300.
Thus:
S = T-C = 300-100 = 200.
Since C = (1/2)S, C<S.

Case 2: C = 200
30% increase in C = (30/100)(200) = 60.
Since this increase represents a 10% increase in T, we get:
60 = (1/10)(T)
T = 600.
Thus:
S = T-C = 600-200 = 400.
Since C = (1/2)S, C<S.

In each case, C=(1/2)S, with the result that C<S.
SUFFICIENT.

Statement 2:
In math terms:
|S-C| = (1/2)S.

If S=100, we get:
|100-C| = (1/2)(100)
|100-C| = 50.

It's possible that C=50, in which case C<S.
It's possible that C=150, in which case C>S.
INSUFFICIENT.

The correct answer is A.

Let's call his base salary b and his commission c. We want to know if c > b.

S1::

If our commission had been 30% higher, it would've been equal to Commission + (30% of Commission).

Adding this to the base gives us (Base + Commission + 30% of Commission), or b + c + .3c.

If the salesman's salary had been 10% higher, it would've been (Base + Commission) + (10% of (Base + Commission), or b + c + .1*(b + c).

We're told these two equations are equal, so we have

b + c + .3c = b + c + .1*(b + c), or

.3c = .1b + .1c, or

2c = b. Hence b ≥ c, and we can answer the question definitively: NO, c is NOT > b.

S2::

|Base - Commission| = (50% of Base), or

|b - c| = .5b

If b - c >0, then |b - c| = b - c, and we have b - c = .5b, or .5b = c.

If b - c < 0, then |b - c| = -(b - c) = c - b, and we have c - b = .5b, or c = 1.5b.

Since we don't know whether (b - c) is positive or negative, we can't say if c = .5b or if c = 1.5b. This distinction is obviously important, since it could tell us which of these is bigger. Hence S2 is NOT sufficient.

Another way to evaluate 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.
Here, the commission increases by 30%, while the salary does NOT increase, implying the following percentages:
Commission increase = 30%.
Salary increase = 0%.
Increase in the MIXTURE of commission and salary = 10%.

Since the percentage for the mixture is closer to the percentage for the SALARY than to the percentage for the commission, the salary must constitute a greater proportion of the mixture than the commission.
In other words:
The salary must be GREATER than the commission.
SUFFICIENT.

Statement 2:
In math terms:
|S-C| = (1/2)S.

Thanks for your reply, just one question is that why do we use mode here?

Please advise.

Thanks..

j_shreyans wrote:

Statement 2:
In math terms:
|S-C| = (1/2)S.

Thanks for your reply, just one question is that why do we use mode here?

Please advise.

Thanks..

Difference is always nonnegative. When we don't know which of S and C is larger, we give the difference as |S - C| to ensure that we don't receive a negative result.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


1) b+1.3c=1.1I=1.1(b+c)=1.1b+1.1c ==> 0.2c=0.1b ==> b=2c ==> c>2c? NO, thus this is sufficient.
2) |b-c|=0.5b
case 1: b-c=0.5b ==> b=2c ==> c>b? NO
case 2: c-b=0.5b ==> c=1.5b ==> c>b? YES since we have both yes, no, the condition is NOT sufficient. Therefore the answer is A

This question is a pretty unique one, and in most of the cases the answer is C because in the original condition, the que is c>b? and we have three variables (l,b,c) in I(income)=b(base salary)+c(commission) with only one equation (l = b+c). We would need 2 more equations to match the number of variables, and for almost 70% of these questions the answer is normally C.


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