Problem Solving

Hi Gurus,
Here is a simple question (i may sound stupid) but i not able to comprehend properly.Especially the statement in bold.Would appreciate if any one can explain the solution in simple language.
A trail mix company keeps costs down by employing the
peanuts:cashews:almonds ratio of 10:4:1 in each bag of up to 75 total nuts.
What is the maximum percentage by which the company could decrease its
number of peanuts per bag and still have peanuts constitute more than half
the total amount of nuts?
(A) 40%
(B) 48%
(C) 49%
(D) 50%
(E) 58%
Answer is:B
The way i tried is and got stuck:( :
Peanut:Cashew:Almond
original mix :10x+4x+x=75>>>50+20+5=75
More than half the number nuts~ 38(75/2)
and dead..

dhiren8182 wrote: A trail mix company keeps costs down by employing the
peanuts:cashews:almonds ratio of 10:4:1 in each bag of up to 75 total nuts.
What is the maximum percentage by which the company could decrease its
number of peanuts per bag and still have peanuts constitute more than half
the total amount of nuts?
(A) 40%
(B) 48%
(C) 49%
(D) 50%
(E) 58%

Goal: To decrease the number of peanuts as much as possible.
Constraint: The remaining peanuts must constitute more than 1/2 of the remaining nuts.

We can PLUG IN THE ANSWERS, which represent the greatest percentage by which the number of peanuts can be decreased.
Since the total number of nuts cannot be greater than 75 -- and P : C : A = 10:4:1 = 50:20:5 -- let P=50, C=20, and A=5, for a total of 75 nuts.
Since the question stem asks for the greatest possible percentage, start with the greatest answer choice.

Answer choice E:
If the number of peanuts is reduced by 58%, we get:
New P = 50 - (58/100)(50) = 50 - 29 = 21.
New total = New P + C + A = 21+20+5 = 46.
New P/New total = 21/46, which is less than 1/2.
Since the new number of peanuts is not more than 1/2 of the new total, eliminate E.

Answer choice D:
If the number of peanuts is reduced by 50%, we get:
New P = 50 - (50/100)(50) = 50 - 25 = 25.
New total = New P + C + A = 25+20+5 = 50.
New P/New total = 25/50, which is equal to 1/2.
Since the new number of peanuts is not more than 1/2 of the new total, eliminate D.

Answer choice C.
If the number of peanuts is reduced by 49%, we get:
New P = 50 - (49/100)(50) = 50 - 49/2 = 25.5.
Since the new number of peanuts must be an integer, eliminate C.

Answer choice B:
If the number of peanuts is reduced by 48%, we get:
New P = 50 - (48/100)(50) = 50 - 24 = 26.
New total = New P + C + A = 26+20+5 = 51.
New P/New total = 26/51, which is greater than 1/2.
Success!

The correct answer is B.

A trail mix company keeps costs down by employing the peanuts:cashews:almonds ratio of 10:4:1 in each bag of up to 75 total nuts.
What is the maximum percentage by which the company could decrease its
number of peanuts per bag and still have peanuts constitute more than half
the total amount of nuts?
(A) 40%
(B) 48%
(C) 49%
(D) 50%
(E) 58%

If Peanuts = 10x
then cashews = 4x
and Almonds = 1x

Only Peanuts have to be decreased
therefore Cashews and Almonds will remain 4x+1x = 5x
i.e. Peanuts should be more than 5 in order to remain more than 50% i.e. Min. Peanuts required = 5x+1
i.e. 5x-1 peanuts to be taken off the bag


%Decrease in peanuts = {10x-(5x+1)}*100/10x = (5x-1)*10/x

But 15x = 75 i.e. x=5 [Substitute back]

%Decrease in peanuts = (5x-1)*10/x = (5*5-1)*10/5 = 48%

Answer: Option B

Here's a quicker way. Since the ratio has 15 parts (10+4+1), we know that the number of nuts must divide by 15. So we have the following possibilities:

(1) Exactly 15 nuts in the mix
Here, we could reduce the mix to at least 6 peanuts, 4 cashews, and 1 almond. Going from 10 peanuts to 6 peanuts is a 40% reduction, so A is possible.

(2) Exactly 30 nuts in the mix
We could reduce the mix to at least 11 peanuts, 8 cashews, and 2 almonds. Going from 20 peanuts to 11 peanuts is a 45% reduction, so we know the answer is at least 45%.

At this point, we should notice that the % will become greater as the total number of nuts in the mix increases. This leads us to consider the final case

(5) Exactly 75 nuts in the mix
Here we'd start with 50 peanuts, 20 cashews, and 5 almonds. We could drop the peanuts to 26 and still have more than half of the mix be peanuts, so we went from 50 peanuts to 26 peanuts, a 48% reduction. Success!