At a home decor shop, 12 lamps were on display on Sunday evening, and

Question

At a home decor shop, 12 lamps were on display on Sunday evening, and each lamp had a different price. If, on Monday morning, the price of each of the 12 lamps was changed, did the average (arithmetic mean) price of the 12 lamps increase?

(1) The 6 most expensive lamps each increased in price by 8%.
(2) The 6 least expensive lamps each decreased in price by 8%.

Bunuel · Accepted Answer

GMAT Club Official Solution:

At a home decor shop, 12 lamps were on display on Sunday evening, and each lamp had a different price. If, on Monday morning, the price of each of the 12 lamps was changed, did the average (arithmetic mean) price of the 12 lamps increase?

The question asks whether the total price of the 12 lamps increased after the price changes.

(1) The 6 most expensive lamps each increased in price by 8%.

This is not sufficient, since we do not know what happened to the 6 least expensive lamps.

Not sufficient.

(2) The 6 least expensive lamps each decreased in price by 8%.

This is not sufficient, since we do not know what happened to the 6 most expensive lamps.

Not sufficient.

(1)+(2) Since all 12 prices are different, the total price of the 6 most expensive lamps must be greater than the total price of the 6 least expensive lamps. So 8% of the total price of the 6 most expensive lamps is greater than 8% of the total price of the 6 least expensive lamps. Therefore, the total increase is greater than the total decrease, so the overall total price increased. Hence, the average price also increased. Sufficient.

Answer: C.

AdarshSambare · Answer

We need to determine whether the average price increased.
Let total original price = T.
 Statement (1): 
The 6 most expensive lamps increased by 8%.
But we do not know what happened to the other 6 lamps.
 
 They could stay same → average increases. 
 They could decrease a lot → average decreases.  
 Not sufficient. 
 
Statement (2): 
The 6 least expensive lamps decreased by 8%.
But we do not know what happened to the other 6 lamps.
 
 They could stay same → average decreases. 
 They could increase a lot → average increases.  
 Not sufficient.

Together:
Top 6 increase by 8%.
Bottom 6 decrease by 8%.
Let:
Sum of top 6 = H
Sum of bottom 6 = L
Since top 6 are each more expensive than bottom 6,
H > L
Net change:
+0.08H − 0.08L
= 0.08(H − L)
 Since H > L, net change is positive. 
Therefore total price, and hence average price, increased.
Both together are sufficient.
 Answer: C

BongBideshini · Answer

(1) 
The 6 most expensive lamps each increased by 8%.
No information about the 6 cheaper lamps. Their prices may decrease enough to offset the increase.
Insufficient

(2)

The 6 least expensive lamps each decreased by 8%.
No information about the 6 expensive lamps. Their increases may or may not outweigh the decrease.
 Insufficient

(1) + (2) 
Since the expensive lamps have a larger total value than the cheaper lamps, an 8% increase on the larger group is greater than an 8% decrease on the smaller group.
So:
increase amount > decrease amount
=>total sum increases
=>average increases

Sufficient

Answer: C

At a home decor shop, 12 lamps were on display on Sunday evening, and

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