Solving Age Word Problems with Equations

This problem asks us to find Kartik's current age given the sum of his age and his parents' ages, and the relationships between their ages. We can solve this by setting up a system of equations based on the information provided.

Setting Up the Age Equations

Let's define variables for the ages:

• Let Kartik's current age be \( K \) years.
• Let his mother's current age be \( M \) years.
• Let his father's current age be \( F \) years.

Now, we translate the given statements into mathematical equations:

1. The sum of their ages is 122 years: \( K + M + F = 122 \)
2. Kartik’s age is half of his mother’s age: \( K = \frac{1}{2} M \)
3. His mother is 7 years younger than his father: \( M = F - 7 \)

Solving the System of Equations

We have a system of three equations with three variables. Our goal is to find the value of \( K \).

From equation (2), we can express \( M \) in terms of \( K \):

\( M = 2K \)

From equation (3), we can express \( F \) in terms of \( M \):

\( F = M + 7 \)

Now, we can substitute the expression for \( M \) from equation (2) into the equation for \( F \):

\( F = (2K) + 7 \)

So, we have \( M \) and \( F \) expressed in terms of \( K \). Let's substitute these expressions into equation (1):

\( K + (2K) + (2K + 7) = 122 \)

Now, we simplify and solve for \( K \):

\( K + 2K + 2K + 7 = 122 \)

Combine the terms with \( K \):

\( 5K + 7 = 122 \)

Subtract 7 from both sides of the equation:

\( 5K = 122 - 7 \)

\( 5K = 115 \)

Divide by 5 to find \( K \):

\( K = \frac{115}{5} \)

\( K = 23 \)

So, Kartik's current age is 23 years.

We can verify this answer by finding the parents' ages:

• Mother's age \( M = 2K = 2 \times 23 = 46 \) years.
• Father's age \( F = M + 7 = 46 + 7 = 53 \) years.
• Sum of ages \( K + M + F = 23 + 46 + 53 = 122 \) years. This matches the given information.

Conclusion: Kartik's Age

Based on the calculations, Kartik's age is 23 years.

Revision Table: Age Problem Summary

Additional Information: Tackling Age Word Problems

Age word problems are common in mathematics and can be solved effectively using algebraic methods. Here are some tips:

• Read Carefully: Understand the relationships between the ages given in the problem.
• Assign Variables: Use different variables (like \( K, M, F \)) to represent the unknown ages.
• Formulate Equations: Translate each sentence or piece of information into a mathematical equation using your variables.
• Solve the System: Use substitution or elimination methods to solve the system of equations you have created. Substitution is often effective when one variable is easily expressed in terms of another.
• Check Your Answer: Once you find the values of the variables, plug them back into the original equations or the problem statement to ensure they satisfy all conditions.

Practice with different types of age problems will help you become more comfortable setting up and solving the equations.