At present, the sum of the ages of a father and his son is 33 years. Determine the smallest number of years until the father's age is 4 times the son's age.

f+s=33

f+x = 4(s+x)

clearly, f > 4x now, since f/s decreases as both grow
So, f > 24, meaning s < 7

You want f to be close to 4s now, so try

s=6, f=27
In one year, s=7, f=28
Looks like 1 is the smallest value.

Thank you, Steve

Let the son's age be represented by 'x' years.

According to the given information, the father's age can be represented as (33 - x) years.

We need to find the smallest number of years until the father's age is 4 times the son's age. Mathematically, this can be represented as:

33 - x + n = 4x

where 'n' represents the number of years to be added to the father's age to make it 4 times the son's age.

Simplifying the equation:

33 + n = 5x

To determine the smallest number of years, we need to find the smallest positive integer value of 'n' such that '33 + n' is divisible by 5 (since '5x' should be an integer).

The remainder when dividing '33 + n' by 5 should be 0.

Let's find the smallest positive value of 'n' that satisfies this condition:

n = 5 - (33 modulo 5)
n = 5 - 3
n = 2

Therefore, the smallest number of years until the father's age is 4 times the son's age is 2 years.

To solve this problem, let's assume the son's age is x years.

We are given that the sum of the ages of the father and the son is 33. So, if the son's age is x, then the father's age would be (33 - x).

According to the problem, we need to find the smallest number of years until the father's age is 4 times the son's age.

Mathematically, this can be represented as: (33 - x) + y = 4x, where y is the number of years until the father's age is 4 times the son's age.

Simplifying the equation, we have: 33 + y = 5x

To find the smallest value of y, we need to find the smallest value of x that satisfies this equation.

Now, let's consider the possible values of x starting from 1:
- If x = 1, then 33 + y = 5. However, this equation doesn't have an integer solution for y.
- If x = 2, then 33 + y = 10. Solving for y, we get y = 10 - 33 = -23.
- If x = 3, then 33 + y = 15. Solving for y, we get y = 15 - 33 = -18.
- If x = 4, then 33 + y = 20. Solving for y, we get y = 20 - 33 = -13.
- If x = 5, then 33 + y = 25. Solving for y, we get y = 25 - 33 = -8.
- If x = 6, then 33 + y = 30. Solving for y, we get y = 30 - 33 = -3.
- If x = 7, then 33 + y = 35. Solving for y, we get y = 35 - 33 = 2.

From these values, we can see that when x = 7, y is positive (indicating the number of years until the father's age is 4 times the son's age). Therefore, the smallest number of years until the father's age is 4 times the son's age is 2 years.