## Task 3 - 020623

If you swap the units and tens digits of a two-digit number, you get a new number that is $4 \frac{1}{2}$ times as large as the original number.

a) What is the number?

b) How did you find it?

Show that there is only one such number!
The sought number is $10 a+b$, where $a$ is a digit from 1 to 9 and $b$ is a digit from 0 to 9. This results in the equation

$$
10 b+a=4 \frac{1}{2} \cdot(10 a+b)=45 a+4 \frac{1}{2} b=45 a+4.5 b \quad \rightarrow \quad 44 a=\frac{11}{2} b
$$

and thus $8 a=b$.

Since $b$ is less than 10 and $a$ must be at least 1, it follows uniquely that $a=1$ and $b=8$. The sought number is 18.