Problem Solving

Last month 15 homes were sold in Town X. The average sale price of homes was 150,000 and the median was 130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than 165,000
II. At least one of the homes was sold for more than 130,000 and less than 150,000
III. At least one of the homes was sold for less than 130,000

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

Pls. explain. Thanks.

IMO- C. OA - A

Last month 15 homes were sold in Town X. The average sale price of homes was 150,000 and the median was 130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than 165,000
II. At least one of the homes was sold for more than 130,000 and less than 150,000
III. At least one of the homes was sold for less than 130,000

now avg =150000 m= 130000 n =15

now median is the 8th highest price

now if we maximise values below the median then they can be equal to the median i.e 130000

now 150000 - 130000 = 20000

now since 8 values are equal to 130000 we need the other values to compensate for these i.e by a total of 20000 * 8 = 160000 now to minimise the other 7 vals (worst scebario to minimise max sell price)we need to evenly distribute this amongst them
i.e 160000/7 = 23000 approx

i.e each of the other 7 values higher than median must be 150000 + 23000 =173000

Hnece I has to be true

II cant be said for sure it may happen or not, but it is not necessary here.

III may also happen but it is not necessary

Hnece I is reqd here coz it will be true even for the worst case scenario here.

A

IMO C too.. A does not make sense

Samir! You are right!

arocks and dingo001,

II. At least one of the homes was sold for more than 130,000 and less than 150,000

Take a look at the following Prices:

1. 125,000
2. 125,000
3. 125,000
4. 125,000
5. 125,000
6. 125,000
7. 125,000
8. 130,000
9. 175,000
10. 175,000
11. 175,000
12. 175,000
13. 175,000
14. 175,000
15. 195,000

So II not necessary.
Simillarly,
III. At least one of the homes was sold for less than 130,000

Take a look at the following Prices:
1. 150,000
2. 150,000
3. 150,000
4. 150,000
5. 150,000
6. 150,000
7. 150,000
8. 130,000
9. 150,000
10. 150,000
11. 150,000
12. 150,000
13. 150,000
14. 150,000
15. 170,000

So III not necessary.
whereas, A has to be true.
Ans A.

I agree...Samir great explanation...I got it!!

Suyog,

Don't we arrange the numbers in increasing order before calculating the median. If so, then how does 130K come after 150K?

The reason why i thought III is correct is that - 130K is the median; i thought half of the values would be above 130K and half would be below it. So atleast one home would have been sold for less than 130K. But i understand that this may be true, but not necessarily true, as pointed out by Samir.

Suyog wrote:Samir! You are right!

Take a look at the following Prices:
1. 150,000
2. 150,000
3. 150,000
4. 150,000
5. 150,000
6. 150,000
7. 150,000
8. 130,000
9. 150,000
10. 150,000
11. 150,000
12. 150,000
13. 150,000
14. 150,000
15. 170,000

So III not necessary.
whereas, A has to be true.
Ans A.

yup, we do...
Thanks!