What is the maximum percentage by which the company could decrease its

Question

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%

Kudos for a correct solution.

(A) 40% 
(B) 48% 
(C) 49% 
(D) 50% 
(E) 58%

littlewarthog · Accepted Answer

Let's first calculate the total number of peanuts:
The bag contains max. 75 nuts of which the ratio peanuts:cashews:almonds is 10:4:1. So of every 15 nuts, 10 are peanuts, or max 50 in one bag.

Now we can start decreasing the number of peanuts:

\frac{50-x}{75-x}>50%

x must be a positive integer, as it represents the number of peanuts taken out. Therefore, we can solve for x without flipping the greater than-symbol.

50-x>\frac{75-x}{2}

x<25

So we can take out 24 peanuts max. 24 out of 50 peanuts are 48%. Answer B.

EMPOWERgmatRichC · Answer

Hi All,

This is ultimately a question about percentages. The key element to solving this question is to remember that when you remove a peanut, you CHANGE the total number of nuts in the mixture...

To maximize the percentage decrease, I want to maximize the number of nuts that I'm working with. In this way, I'm TESTing VALUES, so I'll use the numbers that would occur if the total number of nuts was 75...

Peanuts = 50 
Cashews = 20 
Almonds = 5

We want to remove as many peanuts as possible while still having peanuts represent MORE than half of the mixture...

The number of cashews and almonds will stay the same though, so we have 20 + 5 = 25 of those non-peanuts in total.

If we had 25 peanuts and 25 non-peanuts, then that would be 50% EXACTLY. We want MORE than 50% though, so we need to add in 1 more peanut. This gives us...

Peanuts = 26 
Cashews = 20 
Almonds = 5

The question asked for the decrease in the number of peanuts as a percentage. We started with 50 peanuts and removed 24 = 24/50 = 48%

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

Bunuel · Answer

The correct answer is B.

PareshGmat · Answer

peanuts ................. cashews ................ almonds ................... Total

10x ............................ 4x ........................... 1x .......................... 15x = 75 ; x = 5(Let x be the multiplication factor)

50 ............................. 20 ............................. 5 ............................ 75

50-a ............................. 20 ............................ 5 .............................. 75-a (Let "a" is the quantity of peanuts removed)

For exact 50% peanuts, equation setup would be

\frac{1}{2}(75-a) = 25

a = 25

For more than 50% peanuts, removal should be less than 25. Say 24. New setup would be as follows

26 ................................ 20 .............................. 5 .............................. 51 (Here 26 is more than 50% of 51)

Decrease per cent  = \frac{24}{50} * 100 = 48%

Answer = B

sinhap07 · Answer

Hi Bunuel

Although I do get the logic in this question, the stem says upto 75 nuts. So why dont I get the answer when I use 15 nuts total when I have 10 peanuts, 4 cashews and 1 almond? I get stuck between 5 and 6 range and then dont know how to proceed.

EMPOWERgmatRichC · Answer

Hi sinhap07,

The question asks for the MAXIMUM percentage decrease, so you have to focus on whatever example would maximize the percentage.

If you work with 15 total nuts, you have....

Peanuts = 10 
Cashews = 4 
Almonds = 1

So you would need to have 6 peanuts (to go along with the 4 cashews and 1 almond) to make sure that peanuts represented MORE than half of the nuts.

In this example, you end up removing 4 of the 10 peanuts, which is a 40% reduction in the number of peanuts. Unfortunately, that is NOT the MAXIMUM PERCENTAGE - that only occurs when you MAXIMIZE the number of starting nuts (75 total).

GMAT assassins aren't born, they're made,
Rich

ShashankDave · Answer

I had to try each option to see which one gave the possibility of the maximum percentage decrease. I have a question regarding your statement "To maximize the percentage decrease, I want to maximize the number of nuts that I'm working with."
 A larger number does not necessarily generate a larger percentage in my opinion. Although, your logic works here, please explain in detail so I can really understand what's going on.

EMPOWERgmatRichC · Answer

Hi ShashankDave,

There's a 'high-concept' Number Property rule built into this question - and most people won't even realize that it's there. The idea is all about how numerators and denominators relate to one another (especially when you increase or decrease both by the same absolute number.

For example....

4/5 = 80%

If we subtract 1 from both the numerator and denominator, we get a DIFFERENT NUMBER....

3/4 = 75%

The percentage drop in the numerator was greater than the percentage drop in the denominator, which is why the fraction decreased in value relative to 4/5. When we add 1 to both the numerator and denominator, the opposite occurs....

5/6 = 83 1/3%

In this question, the number of peanuts - relative to the TOTAL number of nuts- is important, since each individual peanut represents a smaller PERCENTAGE of the total when we have more and more total nuts to work with.

For example....
When there are 10 peanuts and 15 total nuts, each peanut represents 6 2/3% of the nuts
When there are 20 peanuts and 30 total nuts, each peanut represents 3 1/3% of the nuts

We're asked to remove the maximum percentage of peanuts so that the number of peanuts is as close to 50% as possible (while still being GREATER than 50%). As the each peanut becomes a smaller and smaller percent of the total, we can get closer and closer to the 50% threshold (while still being GREATER than it). The question limits us to up to 75 total nuts, so we have to try to maximize the number of nuts to minimize the relative percent that each peanut represents - and to maximize the possible percent that we can reduce the number of peanuts.

GMAT assassins aren't born, they're made, 
Rich

BillyZ · Answer

Official solution from Veritas Prep.

B—if your goal is to maximize the percentage decrease in peanuts, you should check the minimum and maximum situations given the ratio to see which one gives you the largest possible percentage decrease. Because the initial ratio means that the total number of pieces is a multiple of 15, you can try 15, and because the maximum number allowable is 75, you should also try 75.

If there are 15 nuts, then there are 10 peanuts, 4 cashews, and 1 almond. That means that you can remove 4 peanuts (40%) and still have more than 50% peanuts.

On the higher side, if there are 75 nuts, that means 50 peanuts, 20 cashews, and 5 almonds. Here, with 25 other nuts, you can remove 24 peanuts (48%) and still have more than 50% peanuts. Accordingly, the correct answer is B.

gracie · Answer

multiplying 10:4:1 by 5,
each bag of 75 nuts has
50 peanuts and
25 non-peanuts (20 cashews+5 almonds)
25+1=minimum of 26 peanuts needed to maintain majority
50-26=24 maximum number of peanuts decreased
24/50=48% maximum percentage of peanuts decreased

grassmonkey · Answer

Hi Rich,

Quick clarification if you don't mind.

In both the "high" and "low" cases, peanuts are 66.7% of the total. We need to decrease the nuts in each case so that they represent slightly MORE than 50%. I can clearly see why we know that we will need to decrease the NUMBER of nuts more in the "high" case to achieve that decrease in the proportion of peanuts from 66.7% to 50%, but I am having a little bit of trouble understanding why that means that the percentage decrease in the number of nuts is also higher in the "high" case?

I can convince myself by doing the calculations, but intuitively how do you know that the percentage decrease is going to be greater in the "high" case.

Thanks!

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