Understanding the Age Problem

This question asks us to determine how many years from now Rathin's age will be 1.5 times his cousin's age. We are given their current ages.

• Rathin's current age = 16 years
• Cousin's current age = 7 years

Setting Up the Equation for Ages

Let's assume that after 'x' years, Rathin's age will be 1.5 times his cousin's age. We need to find the value of 'x'.

After 'x' years:

• Rathin's age will be \(16 + x\) years.
• Cousin's age will be \(7 + x\) years.

According to the problem statement, after 'x' years, Rathin's age will be 1.5 times his cousin's age. We can write this as an equation:

\( \text{Rathin's age after x years} = 1.5 \times \text{Cousin's age after x years} \)

Substituting the expressions for their ages:

\( 16 + x = 1.5 \times (7 + x) \)

Solving for the Number of Years

Now, we need to solve the equation \(16 + x = 1.5 \times (7 + x)\) for 'x'.

First, distribute the 1.5 on the right side of the equation:

\( 16 + x = (1.5 \times 7) + (1.5 \times x) \)

\( 16 + x = 10.5 + 1.5x \)

Next, we want to get all terms with 'x' on one side of the equation and constant terms on the other side. Let's subtract 'x' from both sides:

\( 16 + x - x = 10.5 + 1.5x - x \)

\( 16 = 10.5 + (1.5 - 1)x \)

\( 16 = 10.5 + 0.5x \)

Now, subtract 10.5 from both sides to isolate the term with 'x':

\( 16 - 10.5 = 10.5 + 0.5x - 10.5 \)

\( 5.5 = 0.5x \)

Finally, divide both sides by 0.5 to find the value of 'x':

\( \frac{5.5}{0.5} = \frac{0.5x}{0.5} \)

\( x = 11 \)

So, after 11 years, Rathin's age will be 1.5 times his cousin's age.

Verifying the Solution

Let's check if our answer is correct. After 11 years:

• Rathin's age will be \(16 + 11 = 27\) years.
• Cousin's age will be \(7 + 11 = 18\) years.

Is Rathin's age 1.5 times his cousin's age?

\( 1.5 \times \text{Cousin's age} = 1.5 \times 18 \)

To calculate \(1.5 \times 18\), we can do \(1 \times 18 + 0.5 \times 18 = 18 + 9 = 27\).

Since \(1.5 \times 18 = 27\), which is Rathin's age after 11 years, our solution is correct.

Ratio of ages after 11 years: \(\frac{\text{Rathin's age}}{\text{Cousin's age}} = \frac{27}{18}\)

Simplifying the fraction \(\frac{27}{18}\): Divide both by 9, which is the greatest common divisor.

\( \frac{27 \div 9}{18 \div 9} = \frac{3}{2} \)

Converting the fraction to a decimal: \(\frac{3}{2} = 1.5\)

This confirms that after 11 years, Rathin's age will be 1.5 times his cousin's age.

Revision Table: Key Values

Additional Information: Solving Age Word Problems

Age word problems are a common type of algebraic problem. They usually involve finding the current age, future age, or past age of one or more people based on given conditions or relationships between their ages.

Here are some tips for solving age word problems:

• Identify Variables: Assign variables (like 'x' or 'y') to the unknown ages or the number of years.
• Express Ages: Write expressions for the ages of the people at different points in time (past, present, future) in terms of the variables. If 'A' is the current age, then 'A + x' is the age after 'x' years, and 'A - y' is the age 'y' years ago.
• Set Up Equations: Translate the conditions given in the problem into algebraic equations using the expressions you created.
• Solve the Equations: Use algebraic techniques to solve the equations for the unknown variable(s).
• Check Your Answer: Plug the value(s) you found back into the original problem statement or equations to ensure they satisfy the conditions.

Understanding how to represent ages at different times is crucial. If the number of years is 'x':

• Age 'x' years from now = Current age + x
• Age 'x' years ago = Current age - x

Fractions or decimals in relationships (like 1.5 times) are handled just like any other number in the algebraic equation.