You can only find the median.
Work Step by Step
The median is the central line in the box and whisker plot. The mode (the most frequently occurring number) might be suggested by the shape of the distribution, but there is no way of knowing what it is by reading a box and whisker plot. Similarly, you would need the data set in order to calculate the mean. The median, quartiles, and range are the only values given, and these are not enough to calculate the mean or the mode.