Mrs. K is paid at a reduced rate for contracts completed

Question

Mrs. K is paid at a reduced rate for contracts completed late, and the contract prices may vary. Her compensation for the first two late contracts in any month is reduced by 10%, and her compensation for any subsequent late contracts in the same month is reduced by 15%. If Mrs. K completed three contracts late, in the same month, was her total compensation for those three contracts reduced by more than 11%?

(1) Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.
(2) Without any reduction, she would have received $1500 for the first of the three late contracts.[/b]

(1) Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.
(2) Without any reduction, she would have received $1500 for the first of the three late contracts.

Let's call the original, unreduced prices of the 3 contracts -  x, y  and  z .

The reduction would then be:  0.1x, 0.1y  and  0.15z .

The percentage reduction may then be expressed as:

100 *\frac{(0.1x + 0.1y + 0.15z)}{(x + y + z)}

This is the value we need to establish as being on either side of 11% to get a sufficient answer.

Let's play around with it a bit to reduce the expression. Splitting  0.15z  into  0.1z + 0.05z , and factoring out  0.1  gives us:

100* \frac{ }{(x + y + z)}

This further reduces to:

10 + \frac{5z}{(x+y+z)}

So if we are able to establish, that the second term in this equation is always less than 1 or always more than 1, we have an answer.

Let's look at statement 1. We have a value here for  z , which gives us a clear idea of the numerator of the term.
We also have a minimum value of  x  and  y . We can further arrive at the inference that  (x+y)  is atleast  2400 . Let's look at that value of the term:

\frac{(5 * 550)}{(2400 + 550)}

=  \frac{55}{59}

This is obviously less than 1, which means the total percent will be some value between 10% and 11%.

If we continue to increase  x  and  y , over  1200 , it should be clear that since the numerator only depends on the value of  z , an increase in the denominator  (x + y + z ) only further reduces the value of this term, bringing the total percent closer to 10%.

Thus statement (1) is sufficient to conclude that the percentage will always be less than 11%.

Let's look at statement (2). All we have is a value of  x . Since nothing is indicated about  y  and more importantly  z , we can not conclude whether the term is less than 1 or more than 1. INSUFFICIENT.

Hope that clears things up, and that you guys find weighted averages to be simpler now!

EXTRA NOTE:
When would  \frac{5z}{(x+y+z)} = 1 ?
This would be when x + y = 4z 
i.e. when  z  = 20% of  (x + y + z)  or 25% of  (x +y)

arpanpatnaik · Accepted Answer

The wordings sure are confusing

But yea the answer is

For the question, Let x1, x2 and x3 be the contract rates for the 1st, 2nd and 3rd contracts respectively. Now the reduced rates would be as per the question, 90x1/100 , 90x2/100 and 85x3/100 . *This calculation is unnecessary but just love writing it down! Clarity

* Now to calculate the total reduction of her compensation for all the three contracts, we would need clearly x1+x2+x3 and if possible the individual values. Statement 1 states that "Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others." Hence, without any reduction, 550$ would have been the third compensation, i.e. x3 = 550 and finally, 1200$ for the other two, i.e. x1 = x2 = 1200 Since all the values are given, the percentage change can be calculated! Sufficient!

Statement 2 just provides the value of x1 = 1500, but does not talk about x2 or x3. Hence is insufficient. Hope my answer is accurate!

Regards, Arpan

mc · Answer

Answer is A:

(1) You know the non-reduced price for the first 2 (both discounted at 10%, so the individual payments don't matter) and the non-reduced price of the 3rd. From this you can calculate total reduction. Sufficient.

(2) You have no idea how the payments were distributed, so you don't know how to apply the different discount rates. Insufficient.

AbhayPrasanna · Answer

Actually, you only know what the minimum value for 2 of them is. So you really do have to explore whether the reduction is always going to be on one side of 11% for all possible values above 1200 for each of them.

mc · Answer

Oh, I guess I should try reading the question. You are correct.

AbhayPrasanna · Answer

Just as an illustration of that fact:

If the first were atleast 100, instead of 1200, you would have a percent increase of 13.67% for the value 100, but as we've described something just under 11% for the value 1200. Hope that tells you why jumping to an answer is a trap

srcc25anu · Answer

ST1: X1 = atleast 1200 and reduced payment = 0.9*1200 = 1080
X2 = atleast 1200 and reduced payment = 0.9*1200 = 1080
X3 = 550 (given) and reduced payment = 0.85*550 = 467.5
Total comp WITHOUT reduction = 1200 + 1200 + 550 = 2950
Total comp WITH reduction = 1080 + 1080 + 467.5 = 2627.5
is the reduction >11%?
1 - (2627.5/2950) = < 11% (YES)

Now X1 and X2 are atleast 1200. so if we increase them any further, lets say Z1 = X2 = 1500 and X3 = 550 then:
Total comp WITHOUT reduction = 1500 + 1500 + 550 = 3550
Total comp WITH reduction = 1350 + 1350 + 467.5 = 3167.5
% reduction from usual rate = 1 - (3167.5/3550) = Less than 11%

Hence SUFFICIENT

ST2: X1 = 1500 but we dont know about X2 and X3. so INSUFFICIENT

Ans A

jlgdr · Answer

Yeah I understand your logic but it is not sufficient to know that value cause you have an inequality. One needs to prove in the first statement that the weighted average will on the other side of 11% in order to reach a definite answer Statement 2 is clearly insufficient Would anybody have a try at setting up what the first statement looks like? I'm going to take a shot at it later today and see if I can post something Cheers! J

najaheya · Answer

I fumbled at the second one. Should have read it correctly

stne · Answer

This may help

Picking numbers , using extreme cases

statement 2

let the contracts be 1500 , 1, 1, for 1st, 2nd and 3rd respectively
then the reductions will be 150,.1,.15, respectively so total reduction 150.25

now 11% of 1502 =165.22 so we can see that the actual reduction is  less  than 11% of the total.

let the contracts be 1500, 20000,30000
then reductions will be 150,2000,4500- total reduction 6650

11% of 56500= 6215 here original reduction is  more  than 11% of the total.

Since 2 cases this is insuff.

In the same way we can see that A is suff.
For any combination such that the first two is at least 1200 and the last is 550, actual reduction will be less than 11% of the total.

jlgdr · Answer

As I mentioned in my post above this question is strictly dealing with weighted average. Let's see how to solve this

We have that first two contracts 10% and the last one 15% reduction. We need to know if the weighted average is higher than 11%.
Therefore we need the ratio of the quantities of the contracts

Statement 1

We are told that 550 is 15%
and 2400 is 10%, let's see what we get out of here

3400/550=48/11=more than 4.

1(15)+10(4) / 5 = More than 11

I'm just calculating weighted average here, but this seems sufficient.

Statement 2

We only have that 1500 = 10% therefore without knowing the amounts for the other contracts it is impossible to answer the question.

Therefore A is the correct answer

Holla at me if not clear
Cheers
J

jlgdr · Answer

Actually let's take another stab at this just for fun Statement 1 So we have 10%, 10% and 15%, we need to know if weighted average >11% Now then, we have that at least 1200, 1200 and also we have that the third is 550 So basically our weights are 120, 120, and 55 Now 120 is more than twice 55, bet let's just assume ratio 2:2:1 and see what happens 2(10) + 2(10) +15 = 5 -->55/5 = 11. Nice! So we know that 120 is more than twice 55 so this must be higher than 11. Moreover, we are told in question stem that >=1200 (At least 1200) so this makes it even more evident. Statement 1 is sufficient on its own Hope this clarifies Cheers! J

h31337u · Answer

Mrs. K is paid at a reduced rate for contracts completed late, and the contract prices may vary. Her compensation for the first two late contracts in any month is reduced by 10%, and her compensation for any subsequent late contracts in the same month is reduced by 15%. If Mrs. K completed three contracts late, in the same month, was her total compensation for those three contracts reduced by more than 11%?

(1) Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.
(2) Without any reduction, she would have received $1500 for the first of the three late contracts.[/b]

(1) Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.
(2) Without any reduction, she would have received $1500 for the first of the three late contracts.

_________________________________________________________________________________

IMO, the answer is A.

Let the first pay A, the second pay B, and the third pay C. 
The question is asking (A+B+C)*0.11 < 0.1A + 0.15B + 0.15C ?

(1) If A = 1200(the minimum), B = 1200(the minimum), and C = 550.
0.1 x 1200 + 0.15 x 1200 + 550 x 0.15 > 0.11(1200+1200+550)
382.5 > 324.5

so even we chose the minimum value, LHS will always be bigger than RHS. 
sufficient.

(2) do not say anything about B and C. 
insufficient.

TarPhi · Answer

Let's take the numbers as
- 1st 1200
- 2nd 1200
- 3rd 550

The total comes to 322.5/2950 = 10.93

or 0.15*550+0.1*1200*2/(550+1200*2)

So A should be insufficient. :p

WheatyPie · Answer

This reasoning is not accurate. You do not in fact know the non-reduced price for the first 2 - you must check the minimum value.

Crytiocanalyst · Answer

(1) Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.
x1=x2=1200 and x3=500 sufficiently giving us room to calculate the total compensation making us decide whether it should be below 15 %

Clearly sufficient

(2) Without any reduction, she would have received $1500 for the first of the three late contracts.
only x1 is provided which is not enough to solve 
Clearly insufficient

Therefore IMO A

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Mrs. K is paid at a reduced rate for contracts completed

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