The times taken, in minutes, by 14 students to complete a Mathematics assignment are recorded. The results are shown in the stem-and-leaf diagram.

1. Find the value with the highest frequency

From the stem-and-leaf diagram, the times include
$23, 23, 23$.

No other time appears more than twice, so the modal time is $23$ minutes.

1. Identify students who took at most 20 minutes and calculate the percentage

Times at most $20$ minutes are $11, 15, 19, 20$, so there are $4$ students.

Total number of students $=14$.

$\frac{4}{14}\times 100\%=28.571\ldots\%=28.6\%$

1. Find the original smallest and greatest times

From the stem-and-leaf diagram, the smallest time is $11$ minutes and the greatest time is $40$ minutes.

Original range $=40-11=29$ minutes.

2. Find the possible new times that make the new range 31 minutes

The new range is $31$ minutes.

If the new time is the new smallest value, then
$40-x=31$
$x=9.$

If the new time is the new greatest value, then
$x-11=31$
$x=42.$