Vasya wrote down several consecutive integers. Which of the following could not be the
percentage of odd numbers among them?
(A) 40 (B) 45 (C) 48 (D) 50 (E) 60
To determine which percentage of odd numbers among the consecutive integers is not possible, we need to consider the characteristics of consecutive integers.
In a sequence of consecutive integers, we have two possibilities:
1. The sequence contains only even numbers.
2. The sequence contains an equal number of odd and even numbers.
For option (A) 40%, it is possible to have a sequence with only even numbers. For example, the sequence {2, 4, 6, 8, 10} would yield a 0% odd numbers, which is less than 40%.
For option (B) 45%, it is not possible to have exactly 45% odd numbers in a sequence of consecutive integers. The percentage of odd numbers can only be a whole number or 0% due to the nature of consecutive integers. Therefore, option (B) is not possible.
For option (C) 48%, it is not possible to have exactly 48% odd numbers in a sequence of consecutive integers. Again, the percentage of odd numbers can only be a whole number or 0%, making option (C) not possible.
For option (D) 50%, it is possible to have an equal number of odd and even numbers in a sequence of consecutive integers. For example, the sequence {1, 2, 3, 4, 5} would yield a 50% odd numbers.
For option (E) 60%, it is possible to have a sequence with only odd numbers. For example, the sequence {1, 3, 5, 7, 9} would yield a 100% odd numbers, which is greater than 60%.
Therefore, the percentage of odd numbers that could not be possible among the consecutive integers is (B) 45%.
To determine which of the given percentages could not be the percentage of odd numbers among the consecutive integers written by Vasya, we need to analyze the choices.
To begin, let's consider the fact that consecutive integers follow a clear pattern: odd numbers alternate with even numbers.
Suppose Vasya starts with an odd number. In this case, every other number will be even. Thus, the percentage of odd numbers will always be 50%.
Similarly, suppose Vasya starts with an even number. In this case, every other number will be odd. Again, the percentage of odd numbers will be 50%.
Therefore, we can conclude that the percentage of odd numbers among the consecutive integers will always be either 50% or one less than 50%.
Now, let's examine the given choices:
(A) 40%: This percentage could be obtained if the consecutive integers consist of mostly even numbers. It is possible to have a sequence where 4 out of every 10 numbers are odd, resulting in a 40% odd number percentage. Therefore, this choice is possible.
(B) 45%: This percentage could be obtained if the consecutive integers consist of slightly more even numbers than odd numbers. It is possible to have a sequence where 9 out of every 20 numbers are odd, resulting in a 45% odd number percentage. Therefore, this choice is possible.
(C) 48%: This percentage could be obtained if the consecutive integers consist of slightly more even numbers than odd numbers. It is possible to have a sequence where 12 out of every 25 numbers are odd, resulting in a 48% odd number percentage. Therefore, this choice is possible.
(D) 50%: As explained earlier, it is always possible to have a sequence where exactly half of the numbers are odd, resulting in a 50% odd number percentage. Therefore, this choice is possible.
(E) 60%: This percentage cannot be obtained since it is greater than 50%. As we have established, the percentage of odd numbers among the consecutive integers will always be either 50% or one less than 50%. Hence, this choice is not possible.
Therefore, the answer is (E) 60%.
The number of odds and evens cannot differ by more than 1.
40 = 2/5, so 2 odds and 3 evens works
48 = 24/25
50 = 1/2
60 = 3/5
45 = 9/20 which cannot be. If there are 9 odds, there cannot be 11 evens.