M27-16

Question

A fruit stand sold 76 oranges to 19 customers. How many customers bought only one orange?

(1) No customer bought more than 4 oranges.
 
(2) The difference between the number of oranges bought by any two customers is even.

Bunuel · Accepted Answer

Official Solution:

A fruit stand sold 76 oranges to 19 customers. How many customers bought only one orange? 
 
(1) No customer bought more than 4 oranges. 
 
Since no customer bought more than 4 oranges, the total number of oranges sold is  at most   19*4=76 . However, we know that the total number of oranges sold is  exactly  76. Therefore, each customer must have bought exactly 4 oranges, because if any customer bought less than 4, the total number of oranges sold would be less than 76. Consequently, no customer bought less than 4 oranges, which means that no customer bought only one orange. Thus, statement (1) alone is sufficient to answer the question.
 
(2) The difference between the number of oranges bought by any two customers is even. 
 
 For the difference between the number of oranges bought by  any  two customers to be even, either all customers must have bought an odd number of oranges or all customers must have bought an even number of oranges. However, if all customers bought an odd number of oranges, then the sum of the oranges sold would be the sum of 19 odd numbers, which would be odd. This cannot be the case since we know that 76 is even. Therefore, it must be the case that all customers bought an even number of oranges. Consequently, no customer bought only one orange, which is an odd number. Hence, statement (2) alone is sufficient to answer the question.

Answer: D

tronghieu1987 · Answer

The explanation for (2) is great Bunnel. I was frustrated with trying pairs of (1,5), (1,7)....

Icecream87 · Answer

Statement two was a hard one. Great explanation!

notMD · Answer

I thought that 0 is an even number? If it's so then in case 2, two or more people can buy the same odd amount of fruit ( 1, 3, 3 etc or 3, 5, 5 etc.) as long as the sum is 76. Therefore, one may or may not buy exactly 1 orange
--> option 2 is insufficient

Please help

chetan2u · Answer

Hi,
You rae correct that 0 is an even number..
and you are also correct soeonecan buy 1,1,3 or 3,5,5...
but you have to look at two points
1)the total number of oranges..it is an even number
2) since the difference is even.. either all numbers are even or all numbers will be odd,,..
why? say you have someone with 4 and other with 3.. the difference will be odd...

having seen these two points
we have to ask ourselves  " Is it possible to have odd number of people having odd number of oranges and still having total number of oranges as even?" 
The answer is NO... Because   Sum of odd quantity of numbers having odd values will always be odd  
So any odd value gets discarded and along with it goes out the value of '1'..
Thus suff

Senthil7 · Answer