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16 Sep 2014, 01:27
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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.
(1) No customer bought more than 4 oranges.
(2) The difference between the number of oranges bought by any two customers is even.
16 Sep 2014, 01:27
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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
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
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17 Nov 2015, 16:56
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The explanation for (2) is great Bunnel. I was frustrated with trying pairs of (1,5), (1,7)....
