Bahía Furnishing has several sofas for sale. The median pric

Question

Bahia Furnishing has several sofas for sale, ranging in price from $400 to $2,990. If median price and the average price of the sofas are $900 and $750, respectively, what is the smallest possible number of sofas for sale?

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26

Show Answer

Official Answer

D

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26

paragw · Answer

Here's my solution for this question

LibrarianTurtle · Answer

Average: 750 Median: 900 Highest Value: 2990 Lowest Value: 400 Either a set has odd or even number of terms Odd Terms: 2N+1 Even Terms: 2N In order to to minimize the number of terms we have to minimize (Number of terms)*(Average)= Total Sum Since "2990" is the highest value if we minimize the total sum (Imaginative), number of terms will be smaller Number of terms is odd (2N+1): Median will be the middle number leaving "2N" numbers for the rest of the set. Half of 2N=(N) is less than or equal to median and the other half (N) is greater than or equal to median. Set lower part of N all 400 to minimize the sum and all (N-1) as 900 Lowest Value (400) ------- N ------- Median (900) --------N -1 ------- Highest Value (2990) To minimize the sum suppose that all the N less than median are 400 and all the N greater than median are 900(Equal) Sum(min)= 400*N+900+900*(N-1)+2990=1300N +2990 1300N+2990<=(2N+1)*750 1300N+2990<=1500N+750 2240<=200N N>=11.2 N has to be integer min N=12 Number of terms = 2*12+1=25 Number of terms is even (2N) Median will be the average of middle two numbers leaving (N-1) for less than median and (N-1) greater than median. Sum of middle two numbers divided by 2 has to equal 900 (Median). Therefore, lower half of the set means less than 900 (One of the two middle terms) not including 900 and it becomes (N-1) since we include lowest Value and exclude 900. Higher half of the set means greater than 900 (one of the two middle terms) and it becomes (N-2) because we don't include 900 and 2990. Lowest Value (400) ------- N-1 ------- (900) -- (900) -------- N-2 ------- Highest Value (2990) Sum(min) 400*(N-1)+1800+900*(N-2)+2990=1300N+2590 1300N+2590<=(2N)*750 1300N+2590<=1500N 2590<=200N N>=12.95 N has to be integer so N=13 Number of terms= 2*13=26 When N=12 number of terms is 25 (smaller) Ans. (D) IMO

kevincan · Answer

Note that the minimum will occur when n is odd. Why?

Kinshook · Answer

Bahia Furnishing has several sofas for sale, ranging in price from $400 to $2,990.

If median price and the average price of the sofas are $900 and $750, respectively, what is the smallest possible number of sofas for sale?

Low price = 400
High price = 2990
Median price = 900
Average price = 750

Let the smallest possible number of sofas = x

Case 1: x = 2k + 1 ; odd
Min sum = 400k + 900+ 900(k-1) + 2990 <= 750 (2k+1)
1300k + 2990 <= 1500k + 750
200k >= 2240
k >= 2240/200 = 11.2
Min k = 12
Min n = 2k + 1 = 25

Case 2: x = 2k; even
Min sum = 400(k-1) + 900 + 900(k-1) + 2990 <= 750*2k
1300k - 400 + 2990 <= 1500k
200k >= 2590
k >= 2590/200 = 12.95
Min k = 13
Min n = 26

The smallest possible number of sofas for sale = min (25,26) = 25

IMO D

LibrarianTurtle · Answer

Do you mean when the number of terms is odd ?

egmat · Answer

Hi LibrarianTurtle,

Yes, exactly. When kevincan writes "n is odd,"  n is the total number of sofas (the number of terms in your set) . He's pointing out that the smallest possible count - the  25  you found - comes from the  odd  case, not the even one. So your reading is correct.

Now the interesting part is  why  odd wins, and it comes straight out of the two setups you already built.

Where the difference lives: the median

-  Odd set (2k+1):  only  one  sofa sits at the median of  900 . Every other sofa in the lower half can be pushed all the way down to the cheapest allowed price,  400 . That drives the sum as low as possible.
-  Even set (2k):  the median is the  average  of the  two  middle sofas, so those two must together add up to  1800 . You can't shove both of them down to  400  - the middle of the set is forced to carry more value.

That extra weight in the middle is the whole story. Look at your own minimum sums:

- Odd: min sum =  1300k + 2990 
- Even: min sum =  1300k + 2590

The even case  looks  cheaper per k, but it holds  2k  sofas versus  2k+1 , so each sofa has to average higher to hit  750  - and that constraint bites harder. Working it out, odd needs  k = 12 → 25 sofas , while even needs  k = 13 → 26 sofas .

So the minimum lands on the odd case because a single median value lets you stack the maximum number of cheap  400  sofas without disturbing the median.

Feel it with a tiny set

Compare medians directly:

- Odd:  {400, 900, 2990}  - the  900  is one real sofa; the  400  is free to stay at rock bottom.
- Even:  {400, x, y, 2990}  with median  900  - now  x + y = 1800 , so the middle can't both be  400 .

Same concept, just shrunk: the odd set spends less on its middle, which is exactly why it minimizes the count.

Answer: D

LibrarianTurtle · Answer

Oh I see, I made a silly mistake as I was writing my explanation (It was a long one)

. Much appreciate for the help. I was just thinking what kevincan ment, now I got it.

Bahía Furnishing has several sofas for sale. The median pric

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