The Constant-Difference Model

Age problems scare a lot of students, but they all hide the same friendly secret: the difference between two ages never changes. If your sister is 4 years older than you today, she will be 4 years older when you are both grown up, and she was 4 years older when you were born. Time adds the same number of years to both people, so the gap stays frozen. That one fact is the whole constant-difference model.

Parents: let your child slide the years forward and notice the gap staying fixed before the answer appears. The "Method tip" boxes name what a marker rewards — here, "the age gap stays the same".

By the end you'll be able to spot a constant-difference problem, draw it, and use the fixed gap to answer "in how many years" questions. Let's freeze that gap.

The gap that never changes

Picture two bars, one for each person's age. The longer bar is the older person. The overhang — the difference — is how many years apart they are. Now let a year pass: both bars grow by 1. The overhang is exactly the same as before.

This is what makes age problems solvable. You may not know either age in the future, but you always know their gap, because it is the same as it is today.

Using the fixed gap

Here is a classic. Matthew is 29 and his son is 5. We want to know: in how many years will Matthew be 3 times as old as his son? Slide the years forward. Watch their ages climb together while the gap stays 24 the whole time, and stop when Matthew is exactly 3 times his son.

🔬

When is Matthew 3 times his son's age?

🤔 Predict first: As the years pass, what happens to the GAP between their ages?

It growsIt stays 24It shrinks

At the answer year, Matthew is 3 units and his son is 1 unit, so the gap is 3 − 1 = 2 units. But the gap is always 24, so 2 units = 24 and 1 unit = 12. His son will be 12 — which is 12 − 5 = 7 years from now. Check: in 7 years Matthew is 36 and his son is 12, and 36 = 3 × 12.

Which model is it?

Constant-difference sits next to the other before-and-after models. Tell them apart by what stays the same:

• One gives to another → the total stays the same (constant total).
• The same amount is added to both (like years passing) → the difference stays the same (constant difference).

Age problems are the headline example, but any story where both quantities grow or shrink by the same amount is a constant-difference problem.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. Ben is 8 years older than his sister. In 12 years, how many years older will he be?

   8 years20 years12 years

2. A father is 36 and his son is 9. What is the age difference, and does it ever change?

   27, and it never changes27, but it shrinks each year45, and it grows

3. At the moment one person is 4 times as old as another, the age difference equals how many units?

   4 units3 units5 units

4. A mum is 32 and her child is 8. The gap is 24. When the mum is 3 times as old, the gap is 2 units. What is one unit worth?

   12824

5. In the question above, in how many years will the mum be 3 times as old as her child?

   4 years8 years12 years

6. Which problem is a constant-DIFFERENCE problem rather than a constant-total problem?

   Tom gives Sam $10Two people each save $5 a week, starting at different amountsA jar is shared into two equal parts

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