The problem states that the average age of a group of people increases by 10 when a person aged 19 years is replaced by someone aged 59 years. We need to find the number of people in the group.

Let's assume there are n people in the group.

1. When the 19-year-old is replaced by a 59-year-old, the total increase in the sum of ages is: $59 - 19 = 40$ So, the total age of the group increases by 40 years.

2. Since the average age of the group increases by 10 years, the total increase in the sum of ages for the whole group (when the average increases by 10) is: $10 \times n$

3. Setting these two expressions equal gives: $40 = 10 \times n$

4. Solving for n: $n = \frac{40}{10} = 4$

So, there are 4 people in the group.

Would you like further details or have any other questions? Here are some related questions to deepen your understanding:

1. What happens to the total sum of ages if a person is replaced by someone younger?
2. How does the average age change when the group size increases?
3. How do you calculate the average of a group of numbers?
4. How would the answer change if the new person's age was higher or lower than 59?
5. How can we use algebra to solve similar age-related word problems?

Tip: When dealing with averages and changes in group data, setting up equations based on the total sum can simplify the problem.