This year, the sum of Mr. Bellini's age and his son's age is 38 years. 4 years ago, Mr. Bellini's was 5 times as old as his son. How old is his son this year? How old is Mr. Bellini this year?

man's age --- x

son's age ---- 38-x

4 yrs ago:
man -- x-4
son ---34-x

at that time: x-4 = 5(34-x)
x-4=170 - 5x
6x = 174
x = 29

man is 29 , son is 9

check:
4 years ago, man was 25, son was 5
was the man 5 times as old ? YEAhh

Let's solve this step by step:

Step 1: Assign variables.
Let's assume Mr. Bellini's age this year is represented by "B" and his son's age this year is represented by "S."

Step 2: Translate the given information into equations.
From the information provided, we can create two equations:
Equation 1: B + S = 38 (the sum of their ages this year is 38)
Equation 2: B - 4 = 5(S - 4) (4 years ago, Mr. Bellini's age was 5 times his son's age)

Step 3: Simplify equation 2.
Distribute 5 to (S - 4): B - 4 = 5S - 20

Step 4: Rearrange equation 2.
Rearrange the equation to solve for B in terms of S: B = 5S - 20 + 4
Simplifying, we get B = 5S - 16.

Step 5: Solve the system of equations.
Substitute the value of B in equation 1 with the expression from equation 4:
(5S - 16) + S = 38
Combine like terms: 6S - 16 = 38
Add 16 to both sides: 6S = 54
Divide by 6: S = 9

Step 6: Calculate Mr. Bellini's age this year.
Substitute the value of S (9) into equation 1:
B + 9 = 38
Subtract 9 from both sides: B = 29

Therefore, Mr. Bellini's son is 9 years old this year, while Mr. Bellini is 29 years old this year.

To solve this problem, we need to set up a system of equations. Let's define Mr. Bellini's age as "B" and his son's age as "S."

Given that the sum of their ages is 38 years, we have the equation:

B + S = 38

We're also given that 4 years ago, Mr. Bellini was 5 times as old as his son. This means that Mr. Bellini's age 4 years ago (B - 4) was equal to 5 times his son's age 4 years ago (5 * (S - 4)). We can express this equation as:

B - 4 = 5 * (S - 4)

Now, we can solve these equations simultaneously to find the values of B (Mr. Bellini's age) and S (his son's age).

Let's rearrange the first equation to isolate one variable. Subtracting S from both sides, we have:

B = 38 - S

Now, substitute this expression for B in the second equation:

(38 - S) - 4 = 5 * (S - 4)

Expand the equation:

34 - S = 5S - 20

Add S to both sides:

34 = 6S - 20

Add 20 to both sides:

54 = 6S

Divide both sides by 6:

9 = S

Therefore, Mr. Bellini's son is 9 years old this year.

Now, substitute the value of S back into the first equation to find Mr. Bellini's age:

B + 9 = 38

Subtract 9 from both sides:

B = 38 - 9

B = 29

Therefore, Mr. Bellini is 29 years old this year.