medium difficulty8 min

Age and number based linear models

Age and number models are word problems where unknown ages or numbers are represented by variables and converted into linear equations.

In one line: Age changes with time, but the relationship must stay true.

Practice This Concept

Understand it

These problems often involve present age, past age, future age, digits, or two numbers with a given relation. The key is to translate the statement carefully, because the algebra is usually simple once the equations are written correctly. These questions are common in exams because they test understanding of both language and equations.

Example

If a father is 3 times as old as his son and their total age is 48, we can form linear equations and solve for each age.

Formula and rule

Present age, past age, and future age are related by simple addition or subtraction of years.

Digits and numbers can also be written as x and y with given conditions.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

A father is 3 times as old as his son.

1After 12 years, he will be twice as old.
2Let the son’s present age be x and the father’s be 3x.
3Then 3x + 12 = 2(x + 12), so x = 12 and the father is 36.

Method you can reuse

1Choose variables for the unknown ages or numbers.
2Write the present relation.
3Translate future or past conditions carefully.
4Solve the pair.
5Check whether the answer matches the story.

Visual tip

A small time-line table with 'present', 'after 5 years', and '5 years ago' helps avoid sign mistakes.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In age problems, what must be written carefully when the question mentions years before or after today?

Think first, then show answer

You must change each age by the given number of years before forming the equation. Past age means subtracting years, and future age means adding years to the present value.

Common wrong answer

A common wrong answer is to add years to only one person or to both at the wrong time. The timing must match the statement exactly.

Don't mix this up

Students often use the same variable for both people or forget to express future and past ages correctly.

Exam tip

Write the present quantity first, then add or subtract the time change exactly as the question says.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

The exam may ask for present ages, digits of a number, or two numbers using age-related language and time change.

1
Enough for 1 mark
Age and number models are word problems where unknown ages or numbers are represented by variables and converted into linear equations.
2
Add for 2 marks
Example: If a father is 3 times as old as his son and their total age is 48, we can form linear equations and solve for each age.
3
Add for 3 marks
These problems often involve present age, past age, future age, digits, or two numbers with a given relation.

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