The average age of a family (father, mother and children) is $18$. If the father, who is $38$ years old, is not included, then the new average would be $14$. Find the number of children in the family.

The sum of $N$ positive integers (not necessarily distinct) is $20$. The sum of the three smallest of these numbers is $5$. The sum of the three largest of these numbers is $7$. Find $N$.

There is a list of twelve numbers. The first number is $1$, the last number is $12$, and each of the other numbers is one more than the average of its two neighbors. For example, the two neighbors of $b$ are $a$ and $c$. What is the largest number in the list? $$\underset{a}{1}\;,\;\underset{b}{\text{___}}\;,\;\underset{c}{\text{___}}\;,\;\underset{d}{\text{___}}\;,\;\underset{e}{\text{___}}\;,\;\underset{f}{\text{___}}\;,\;\underset{g}{\text{___}}\;,\;\underset{h}{\text{___}}\;,\;\underset{i}{\text{___}}\;,\;\underset{j}{\text{___}}\;,\;\underset{k}{\text{___}}\;,\;\underset{l}{12}\;$$

The median of a set of five positive integers, is one more than the mode, and one less than the mean. What is the largest possible value of the range of the five integers?

There are two boxes, each of which initially contained three jewels.
One day, Jem decided to move her favourite jewel from the small box to the large box. This increased the mean value of the jewels in the small box by $\$100$. To her surprise, this also increased the mean value of the jewels in the large box by $\$100$.
If her favourite jewel is worth $\$500$, what is the total value of all six of Jem’s jewels?

Jane and Mitzi have both done the same number of tests this semester and they both have the same mean mark.

They do one more test.
Jane scores $89$ and her mean score increases to $68$.
Mitzi scores $57$ and her mean score decreases to $64$.
Find the final number of tests taken by each student.