Finding Present Ages Using Age Difference and Past Relationship

The problem asks us to find the present ages of two individuals, Sunita and Sheela, given two conditions about their ages.

Let's break down the information provided:

• The difference between their present ages is 12 years.
• 9 years ago, the elder's age was 4 times the younger's age.

Setting up Equations for Ages

Let the present age of Sunita be \(S\) years and the present age of Sheela be \(H\) years.

According to the first condition, the difference in their ages is 12 years. This means one is 12 years older than the other. Let's assume Sunita is the elder one for now. So, the first equation is:

\(S - H = 12 \quad (1)\)

From this equation, we can express \(S\) in terms of \(H\):

\(S = H + 12\)

Now, let's consider their ages 9 years ago:

• Sunita's age 9 years ago was \(S - 9\) years.
• Sheela's age 9 years ago was \(H - 9\) years.

The second condition states that 9 years ago, the elder's age was 4 times the younger's age. Since we assumed Sunita is elder, her age 9 years ago (\(S - 9\)) was 4 times Sheela's age 9 years ago (\(H - 9\)). So, the second equation is:

\(S - 9 = 4 \times (H - 9) \quad (2)\)

Solving the System of Equations

We now have a system of two linear equations:

\(S = H + 12\)

\(S - 9 = 4(H - 9)\)

We can substitute the expression for \(S\) from equation (1) into equation (2):

\((H + 12) - 9 = 4(H - 9)\)

Simplify the left side:

\(H + 3 = 4(H - 9)\)

Distribute the 4 on the right side:

\(H + 3 = 4H - 36\)

Now, we need to isolate \(H\). Subtract \(H\) from both sides:

\(3 = 4H - H - 36\)

\(3 = 3H - 36\)

Add 36 to both sides:

\(3 + 36 = 3H\)

\(39 = 3H\)

Divide by 3 to find \(H\):

\(H = \frac{39}{3}\)

\(H = 13\)

So, Sheela's present age is 13 years.

Now substitute the value of \(H\) back into the equation \(S = H + 12\) to find Sunita's present age:

\(S = 13 + 12\)

\(S = 25\)

So, Sunita's present age is 25 years.

The present ages are 25 years and 13 years. Since we assumed Sunita was elder (\(S > H\)), and we got \(25 > 13\), our assumption was consistent.

Verifying the Solution

Let's check if these ages satisfy both original conditions:

1. Difference in present ages: \(25 - 13 = 12\) years. (Satisfied)
2. Ages 9 years ago:
   • Sunita's age 9 years ago: \(25 - 9 = 16\) years.
   • Sheela's age 9 years ago: \(13 - 9 = 4\) years.
   Elder's age (16) is 4 times younger's age (4): \(16 = 4 \times 4\). (Satisfied)

Both conditions are met, so the present ages are 25 years and 13 years.

Comparing with Options

The calculated present ages are 13 years and 25 years. Looking at the options:

• A. 11 and 23 (Difference is 12. 9 years ago: 2 & 14. 14 ≠ 4*2)
• B. 15 and 27 (Difference is 12. 9 years ago: 6 & 18. 18 ≠ 4*6)
• C. 13 and 25 (Difference is 12. 9 years ago: 4 & 16. 16 = 4*4. This matches)
• D. 23 and 35 (Difference is 12. 9 years ago: 14 & 26. 26 ≠ 4*14)

Option C provides the ages 13 and 25, which is consistent with our solution.

Conclusion

Based on the given conditions and our calculations, the present ages of Sunita and Sheela are 25 years and 13 years.

Revision Table: Age Word Problems

Additional Information: Solving Age Problems

Age problems are common in mathematics and often appear in competitive exams. They typically involve finding the present age of one or more people based on conditions about their ages at different points in time (past, present, or future). Here are some tips for solving age problems:

• Always assign variables to the unknown present ages. Using present ages as the base makes it easier to calculate ages in the past or future.
• Carefully read the conditions provided. Identify the different time periods mentioned (e.g., '9 years ago', 'after 5 years').
• For each time period mentioned, express the ages of the individuals in terms of the variables assigned to their present ages. For example, if the present age is \(x\), the age 9 years ago is \(x-9\). If the present age is \(y\), the age after 5 years is \(y+5\).
• Translate the relationships given in the word problem (e.g., 'sum of ages is...', 'one age is twice the other...', 'difference is...') into algebraic equations using the age expressions for the relevant time periods.
• Solve the system of equations formed. You might have one or two equations depending on the number of unknowns and conditions.
• After finding the values of the variables, always double-check your answer by substituting these values back into the original problem statements to ensure all conditions are satisfied.
• Pay attention to whether the question asks for present ages or ages at a different time.

Understanding how to set up and solve linear equations is fundamental to solving age problems effectively.