https://file.aiquickdraw.com/mathbot/file/1725377407664jcg2d1by.jpg

Let's analyze the problem given in the image. 

The text says:
15. "Un hombre tiene 30 años más que su hijo y 31 menos que su padre. La suma de las edades de estas tres personas es 100 años. ¿Cuál es la edad de cada una?"

### Translation:
A man is 30 years older than his son and 31 years younger than his father. The sum of the ages of these three people is 100 years. What is the age of each one?

### Let’s define variables for each person’s age:

- Let \(x\) be the age of the son.
- Let \(y\) be the age of the man.
- Let \(z\) be the age of the father.

### From the problem statement, we have the following relationships:

1. \(y = x + 30\) (The man is 30 years older than his son)
2. \(z = y + 31\) (The man is 31 years younger than his father)
3. \(x + y + z = 100\) (The sum of their ages is 100)

### Substituting equations 1 and 2 into equation 3:

Let's substitute \(y = x + 30\) and \(z = (x + 30) + 31 = x + 61\) into the third equation:

\[
x + (x + 30) + (x + 61) = 100
\]

Combine like terms:

\[
3x + 91 = 100
\]

Solve for \(x\):

\[
3x = 100 - 91 = 9
\]
\[
x = 3
\]

Now, substitute \(x = 3\) back into the equations for \(y\) and \(z\):

\[
y = x + 30 = 3 + 30 = 33
\]
\[
z = x + 61 = 3 + 61 = 64
\]

### Conclusion:

- The son is **3 years** old.
- The man is **33 years** old.
- The father is **64 years** old.

Would you like more details, or do you have any questions?

Here are 5 related questions that expand on this topic:
1. How would the solution change if the man was 40 years older than his son instead of 30?
2. If the total age sum was 120 instead of 100, what would their ages be?
3. What if the father was 20 years younger than given, how would that affect the ages?
4. How would you set up and solve this problem using a different method?
5. How can we check if our calculated ages are correct?

**Tip:** Always clearly define your variables and relationships when solving word problems to avoid confusion and ensure accuracy.

Un hombre tiene 30 años más que su hijo y 31 menos que su padre. La suma de las edades de estas tres personas es 100 años. ¿Cuál es la edad de cada una?

Learn how to solve a word problem involving the ages of a man, his son, and his father. The man is 30 years older than his son and 31 years younger than his father, with a combined age sum of 100 years. Ideal for students in Grades 6-8.

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