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Great DS question.

by albatross86 » Mon May 24, 2010 1:00 am
Hey guys, I haven't seen this question anywhere else on the forum so thought I would share it, with my explanation.

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.

Original answer:
[spoiler]A[/spoiler]
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by albatross86 » Mon May 24, 2010 1:05 am
Explanation

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 * (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* [0.1(x+y+z) + 0.05z] / (x + y + z)

This further reduces to:

10 + [ 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:

(5 * 550) / (2400 + 550)
= 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 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)
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by liferocks » Mon May 24, 2010 1:18 am
From 1
unreduced price of contract 1=1200+100x
unreduced price of contract 2=1200+100y
unreduced price of contract 3=550

total 2950+100x+100y=T

11% of T=324.50+11x+11y

reduction from contact 1=120+10x
reduction from contact 2=120+10y
reduction from contact 3=82.5

total reduction =322.50+10x+10y=Tr

for any value of x Tr<T..hence reduction from 3 contracts is not more than 11%..sufficient

From 2
no information about cost of contract 2 and 3...not sufficient

Ans option A
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by albatross86 » Mon May 24, 2010 1:25 am
Liferocks - that is an excellent method and I recommend it over mine, thanks!
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by kstv » Mon May 24, 2010 3:16 am
It is a typical DS Q where u do not need to do any calc.

K receives 10% less for the first two late contracts and 15% for the third. There were 3 late contracts.
Was the total reduction less than 11%. To answer this if we know the value of each contracts. Then, it will be possible to work out the total reduction and the %.

Options 2) is inadequate while option 1) clearly tells us the value of each contract. This is enough, calc the exact % or even whther it is < or > 11% is going to cost valuable time.
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by albatross86 » Mon May 24, 2010 3:19 am
KSTV:

Option 1 does not give us the value of each contract, but only a lower limit. I'm not sure how one would jump to the conclusion that that is sufficient to get one side of 11% without first analyzing it.
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by Haaress » Mon May 24, 2010 1:49 pm
I start a fresh on this.

If the 3 contracts are say, X, Y and Z , then the question basically asks whether

10%X + 10%Y +15%Z > 11%X +11%Y + 11%Z ?

This is also the same as 15%Z - 11%Z > 11%X +11%Y - 10%X - 10%Y ?

4%Z > 1%X + 1%Y OR 4Z > X + Y?

From Stmt 1... Z = 550 , and both X and Y are > 1200.
So substituting for x , y and z ( with the minimum values)

4Z > X + Y ...... 4*550 > 1200+1200 . So , Is 2200 > 2400? and thats a resounding NO, meaning that at no time will the reduction of comp be greater than 11%.... and that sufficient to answer the question.

As for Stmt 2. No info is available for contracts 2 and 3 and thus insufficient.

Hope that helps
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by kevincanspain » Mon May 24, 2010 2:07 pm
albatross86 wrote:Hey guys, I haven't seen this question anywhere else on the forum so thought I would share it, with my explanation.

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.

Original answer:
[spoiler]A[/spoiler]
Suppose rates for 3 contracts are x y and z

Is 10x + 10y + 15z > 11(x + y + z) i.e.4z > x + y ?

(1) 4z = 2200 x + y >=2400 SUFF
(2) NOT SUFF
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by kevincanspain » Mon May 24, 2010 2:10 pm
kstv wrote:It is a typical DS Q where u do not need to do any calc.

K receives 10% less for the first two late contracts and 15% for the third. There were 3 late contracts.
Was the total reduction less than 11%. To answer this if we know the value of each contracts. Then, it will be possible to work out the total reduction and the %.

Options 2) is inadequate while option 1) clearly tells us the value of each contract. This is enough, calc the exact % or even whther it is < or > 11% is going to cost valuable time.
Careful! (1) gives the exact amount of one contract and lower bounds for the other two. What would be the answer if (1) said at least $1000 for the others?
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by kstv » Mon May 24, 2010 6:39 pm
1st contract priced $ 550 is reduced by 10%
2nd and 3 contract priced > $1200 reduced by 15%
the resultant reduction will be between 10% to 15%
the 2nd and 3rd contract ($2400) > 4 times the 1st contract (550)
so a 15% reduction of $ 2400 will be definitely > 12.5% of the total
the 2rd & 3rd contract has a > influence than the 3 rd contract on the final outcome

but if the lower limit of 2nd and 3 contract = 1000*2 = 2000
15% reduction = 300& 10% reduction on 550 = 55
this may need a more careful calc.
but 355 is still > 11% of 2550
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by albatross86 » Mon May 24, 2010 8:31 pm
kstv wrote:1st contract priced $ 550 is reduced by 10%
2nd and 3 contract priced > $1200 reduced by 15%
the resultant reduction will be between 10% to 15%
the 2nd and 3rd contract ($2400) > 4 times the 1st contract (550)
so a 15% reduction of $ 2400 will be definitely > 12.5% of the total
the 2rd & 3rd contract has a > influence than the 3 rd contract on the final outcome

but if the lower limit of 2nd and 3 contract = 1000*2 = 2000
15% reduction = 300& 10% reduction on 550 = 55
this may need a more careful calc.
but 355 is still > 11% of 2550
The THIRD contract is 550$, which is reduced by 15%, and the first 2 which were atleast 1200$ were reduced by 10%. So this is incorrect.
Do read the other solutions.
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by albatross86 » Mon May 24, 2010 8:33 pm
kevincanspain wrote:
albatross86 wrote:Hey guys, I haven't seen this question anywhere else on the forum so thought I would share it, with my explanation.

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.

Original answer:
[spoiler]A[/spoiler]
Suppose rates for 3 contracts are x y and z

Is 10x + 10y + 15z > 11(x + y + z) i.e.4z > x + y ?

(1) 4z = 2200 x + y >=2400 SUFF
(2) NOT SUFF
That is just AWESOME. How did I miss that!!!

Thanks a bunch!
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