Gladys wrote down four numbers. The mean of these numbers is 13, the median is 12 and the mode is 10. Find the four numbers. Answer ...

1. Use the mode to identify repeated numbers and set up the ordered list.

Since the mode is $10$, the number $10$ must occur more often than any other number.

For four numbers in ascending order, take the first two numbers as
$10,\ 10,\ a,\ b$
where $a \geq 10$ and $b \geq a$.

2. Use the median of four numbers to find the third number.

For four numbers, the median is the mean of the two middle numbers.

So
$\frac{10+a}{2}=12$
$10+a=24$
$a=14$

3. Use the mean to find the fourth number.

The mean is $13$, so the total of the four numbers is
$4\times 13=52$

Hence
$10+10+14+b=52$
$34+b=52$
$b=18$