M14-26

Question

The value  x  represents the average (arithmetic mean) price of 10 items, where no two items have the same price. If the prices of the 5 cheapest items decrease by 10% and the prices of the 5 most expensive items increase by 10%, which of the following could be true about  x ?
 
I.  x  decreases
 
II.  x  remains unchanged
 
III.  x  increases

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Bunuel · Accepted Answer

Official Solution:

The value  x  represents the average (arithmetic mean) price of 10 items, where no two items have the same price. If the prices of the 5 cheapest items decrease by 10% and the prices of the 5 most expensive items increase by 10%, which of the following could be true about  x ?
 
I.  x  decreases
 
II.  x  remains unchanged
 
III.  x  increases

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Let  m  represent the total original prices of the 5 cheapest items, and let  M  represent the total original prices of the 5 most expensive items. The original cumulative cost of all items is  m + M , while the new cumulative cost will be  0.9m + 1.1M .
 
Is option I possible? We compare  \frac{0.9m + 1.1M}{10}  with  \frac{m + M}{10} . If  \frac{0.9m + 1.1M}{10} \lt \frac{m + M}{10} , it simplifies to  0.9m + 1.1M \lt m + M , then to  0.1M \lt 0.1m , and finally to  M \lt m . This statement is incorrect because  M  represents the sum of the more expensive items and cannot be less than  m . Therefore, option I is not possible.
 
Is option II possible? We analyze whether  \frac{0.9m + 1.1M}{10} = \frac{m + M}{10} . This equation simplifies to  0.9m + 1.1M = m + M , then further simplifies to  0.1M = 0.1m , and finally to  M = m . This cannot be correct, as  M  and  m  represent the sums of different sets of items with distinct prices. Therefore, option II is not possible.
 
Option III is possible because it is accurate that  M \gt m , as  M  totals the prices of the more expensive items.

Answer: A

Barnal · Answer

I think this is a  high-quality  question and I agree with explanation. What do you think of solving this question like this :

Let's say you have Price 1 = 3 (reduce by 50% -> 1.5
Price 2 =5 Increase by 50% ->7,5
(1,5+7.5)/2>(3+5)/2
Boom I go with III only

nealz · Answer

A decrease in 10% can be compensated only by an increase in 11.11%
while a decrease in 9.09% can be compensated by an increase in 10%

Here we have lower value decreased by 10% while the higher value is increased by 10%.
Even for two consecutive numbers, the final average tends to increase(10 decreased by 10% gives 9 while 11 increased by 10% gives 12.1; the average is increased from 10.5 to 10.55), hence the same will follow for 10 numbers giving answer 3.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Josvit · Answer

Well, what I did was the following:

Original Mean =  /10

New Mean =  /10 =  /10 = 0.1(x6+x7+...+x10-x1-x2..-x5)/10 + Original Mean

= Original mean + 1/100 (something positive).

Therefore, it increases.

Only III

Takeaways: Just try rearranging. Include it in your set of weapons.

pinkman27 · Answer

I like the solution - it’s helpful.

polarbaryon · Answer

I don’t quite agree with the solution. I think a clarification that prices must be positive is needed. The "decrease by 10%" of a negatively priced item is not .9x of that item, it's 1.1x; similarly for an item costing 0, a decrease or increase by any percentage is still 0.

bb · Answer

Sure, that argument will work ..... right after you give a link to a store that has negative prices 😂
where I am going to buy 10,000,000 of negatively priced items.

P.S. For things such as people we do not need to specify that the result has to be an integer and for things such as prices and age, we usually do not need to specify that they are positive numbers but perhaps you can find an example and we will correct.

P.P.S. your case for $0 price may hold some water but again, how many stores have products that are priced at zero. At the same time we have a basket of products and no products cost the same price, so even if one of them is indeed zero, the question would still work since others would have to be priced higher.

JCDFHUIASCDUIAS · Answer

I like the solution - it’s helpful.

Sneha_0410 · Answer

Hey Bunuel,

Can we apply this simple logic without any calculations? : since  Highest 5 have a larger total than lowest 5, therefore their 10% increase is larger than the 10% decrease → average increases

Bunuel · Answer

Yes, exactly. Since the 5 most expensive items have a larger total price than the 5 cheapest items,  10% of their total is also larger .

So the amount added to the expensive items is greater than the amount subtracted from the cheap items. Therefore, the total price increases, and so does the average.

M14-26

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