Age problems: one model, five versions

Salvatore Mosaico · 2026

Contents

• 1. The starting problem
• 2. Deriving the model (4k, 3k)
• 3. Three moments and the first solution
• 4. Version 1: the present sum is known
• 5. Version 2: the future sum is known
• 6. Version 3: the older person’s age is known
• 7. Version 4: the younger person’s age is known
• 8. Version 5: a known difference and the reversed problem
• 9. One model, five shortcuts
• 10. What collaboration with AI adds

An age problem can look like a maze of “when”, “were” and “will be”. Once its structure is understood, many versions become applications of one model: (4k, 3k). This article adapts the conversation “Risolvi il problema d’età”. Salvatore Mosaico had already obtained the ratio 3/4 through a parametric equation; here we develop it and apply it to different conditions.

1. The starting problem

“I am twice as old as you were when I was as old as you are. When you are as old as I am now, our combined age will be 90. How old are we today?”

The speaker is older. Let x be the older person’s age and y the younger person’s, with x > y > 0. We treat ages as exact durations, not rounded counts of birthdays. The difference x − y stays constant: moving backwards or forwards adds or subtracts the same time for both.

The first statement determines a ratio; the second selects one pair from all compatible pairs. Instead of starting with 40 and 30 and checking them, we want to understand why those ages follow.

2. Deriving the model (4k, 3k)

When the older person was y, it was x − y years ago. The younger person was then y − (x − y) = 2y − x. “I am twice as old as you were” becomes:

x = 2(2y − x)
x = 4y − 2x
3x = 4y
y/x = 3/4

Thus x:y = 4:3. Every compatible pair can be written:

(x, y) = (4k, 3k), k > 0
x − y = 4k − 3k = k

The positive parameter k is exactly the age difference. We have deduced the ratio rather than guessed one that works. Conversely, starting with 4k and 3k and going back k years gives 3k and 2k; 4k is indeed twice 2k. The model is therefore both necessary and sufficient for the first condition.

3. Three moments and the first solution

Always keep the same order: older person, younger person. “When you are as old as I am” means reaching my age today, not catching up while I continue ageing.

The future sum is 5k + 4k = 9k. Following the approach proposed in the conversation, start directly from 90:

k = 90/9 = 10
(4k, 3k) = (4 × 10, 3 × 10) = (40, 30)

The present ages are 40 and 30. Ten years ago they were 30 and 20: 40 is twice 20. Ten years from now they will be 50 and 40, summing to 90. Both conditions are checked.

4. Version 1: the present sum is known

The base statement “I am twice as old as you were when I was as old as you are” holds. Our ages now add up to 63. How old are we?

Use the same model. The present sum contains seven parts, not nine:

7k = 63
k = 63/7 = 9
(4k, 3k) = (36, 27)

The ages are 36 and 27. Nine years ago they were 27 and 18, and 36 is twice 18. Today 36 + 27 = 63. Generally, for present sum S, k = S/7 and the ages are (4S/7, 3S/7).

5. Version 2: the future sum is known

The same base statement holds. When the younger person reaches the older person’s present age, their combined age will be S. Find their present ages.

At that future moment their ages are (5k, 4k), so k = S/9. However, the required ages are the present ones: (4k, 3k). Do not mix the two moments.

9k = S
k = S/9
(4k, 3k) = (4S/9, 3S/9)

For S = 90 we recover (40, 30). For S = 54, k = 6 gives (24, 18); the future check is 30 + 24 = 54. S = 63 gives (28, 21), while S = 27 gives (12, 9). The data change, not the method.

6. Version 3: the older person’s age is known

“I am 24, and twice as old as you were when I was as old as you are.” How old is the other person?

The known age corresponds to four parts:

4k = 24
k = 24/4 = 6
(4k, 3k) = (24, 18)

The ages are 24 and 18. Six years ago they were 18 and 12, and 24 = 2 × 12. Generally, if A is the older person’s age, k = A/4 and the pair is (A, 3A/4). With A = 40 we immediately get (40, 30).

7. Version 4: the younger person’s age is known

The younger person is 18. The older person says the same base statement. How old is the speaker?

Now the known age corresponds to three parts. This numerical example applies the fourth formula from the conversation:

3k = 18
k = 18/3 = 6
(4k, 3k) = (24, 18)

Again the ages are 24 and 18, but the given information differs. Generally, if B is the younger person’s age, k = B/3 and the pair is (4B/3, B). Dividing by four here would assign the data to the wrong person.

8. Version 5: a known difference and the reversed problem

If the base statement holds at the moment considered and the difference is D, the parameter is already known: k = D. The ages are (4D, 3D); with D = 4 we obtain (16, 12).

Salvatore Mosaico’s proposed variant reverses the question:

“Today I am 69 and you are 65. One day I told you: ‘I am twice as old as you were when I was as old as you are.’ How old were we that day?”

The model (4k, 3k) must hold on the day of the statement, not today. The age difference, however, has remained unchanged:

D = 69 − 65 = 4
k = D = 4
(4k, 3k) = (16, 12)
69 − 16 = 65 − 12 = 53

They were 16 and 12, 53 years ago. To check the statement, go back another four years from that day: they were 12 and 8. The sixteen-year-old could correctly say they were twice the other person’s age of 8 at that earlier time.

For present ages A > B, set D = A − B. At the required moment the ages are (4D, 3D); elapsed years are A − 4D = B − 3D. A positive value means the past, zero the present and a negative value the future: the moment is not always in the past.

9. One model, five shortcuts

The ratio 4:3 does not remain constant throughout life; the difference does. The model describes the moment when the base statement holds. Ten years after (40, 30), the ages are (50, 40), no longer in the ratio 4:3.

The parameter k need not be an integer if fractional ages are allowed. If both ages must be whole numbers of years, k must be an integer: the present sum must then be divisible by 7 and the specified future sum by 9. Always read a formula together with its assumptions.

10. What collaboration with AI adds

The initial AI response presented 40 and 30 and showed that 4:3 works. Checking one pair, however, is not the same as deriving all solutions. The author’s reference to his earlier ratio 3/4 and parametric equation helped make the general structure explicit.

The next step was not to solve each variant from scratch, but to use (4k, 3k) consistently and obtain k directly from the given information. The 69–65 variant highlights the distinction between today and the moment when the statement was true. We credit it as the author’s proposal in the conversation, without claiming unverified historical priority.

Understanding a model means recognising the same structure beneath different wording. AI helps develop and check the steps; choosing what to generalise and which question to ask gives the work its direction.

AI-generated photographic cover: an illustrative scene with fictional people, not a photograph of the conversation’s participants.