9.1. Several consecutive natural numbers are written on the board. Exactly $52 \%$ of them are even. How many even numbers are written on the board
# Answer: 13.

Solution. Since the natural numbers written down are consecutive, even and odd numbers alternate. According to the condition, there are more even numbers, which means the sequence starts and ends with even numbers.

First method. Let $n$ be the number of even numbers, then the number of odd numbers is $(n-1)$. Thus, the even numbers constitute $\frac{n}{2 n-1} \cdot 100 \%$ of all the numbers written on the board. We get the equation $100 n=52(2 n-1)$, from which $n=13$.

Second method. Let a total of $x$ numbers be written. Then among them, $\frac{13}{25} x$ are even and $\frac{12}{25} x$ are odd, and there are exactly one more even numbers. Therefore, $\frac{13}{25} x-\frac{12}{25} x=1$, from which $x=25$. Thus, $\frac{13}{25} x=13$.

Third method. There is one more even number, which means one number constitutes ( $52-48$ ) \% of their total quantity. Therefore, the number of even numbers is $\frac{52}{52-48}=13$.

## Grading criteria:

+ the correct answer and a complete solution (by any method) are provided

$\pm$ correct, overall, reasoning is provided, but the answer is given to the wrong question (for example, only the total number of written numbers is found)

Ғ correct reasoning is provided, but a computational error is made

- only the answer is provided