Understanding the Age Problem

This question involves solving a word problem about the ages of three people: Ramesh, his wife Sangeeta, and their son Mayank. We are given relationships between their current ages and their ages in the future. The goal is to find the present age of Sangeeta.

Key Information from the Question

• Ramesh's present age is 7 years more than Sangeeta's present age.
• 6 years from now, Ramesh's age will be three times Mayank's age.
• Mayank's present age is 10 years.

Setting up the Equations for Ages

Let's use variables to represent the present ages:

• Let \(R\) be the present age of Ramesh.
• Let \(S\) be the present age of Sangeeta.
• Let \(M\) be the present age of Mayank.

From the given information, we can write equations:

1. Ramesh's age vs. Sangeeta's age: \(R = S + 7\)
2. Mayank's present age: \(M = 10\)
3. Ages 6 years hence:
   • Ramesh's age in 6 years: \(R + 6\)
   • Mayank's age in 6 years: \(M + 6\)
   Relationship 6 years hence: \(R + 6 = 3 \times (M + 6)\)

Solving for Ramesh's Present Age

We know \(M = 10\). We can substitute this into the equation for ages 6 years hence:

\(R + 6 = 3 \times (M + 6)\)

\(R + 6 = 3 \times (10 + 6)\)

\(R + 6 = 3 \times 16\)

\(R + 6 = 48\)

Now, we can solve for \(R\):

\(R = 48 - 6\)

\(R = 42\)

So, Ramesh's present age is 42 years.

Calculating Sangeeta's Present Age

We know that Ramesh's present age (\(R\)) is 7 years more than Sangeeta's present age (\(S\)). We have the equation:

\(R = S + 7\)

We found Ramesh's present age is 42. Substitute this value into the equation:

\(42 = S + 7\)

Now, solve for \(S\):

\(S = 42 - 7\)

\(S = 35\)

Therefore, Sangeeta's present age is 35 years.

Verification of Ages

Let's check if these ages satisfy all conditions:

• Sangeeta's present age = 35 years
• Ramesh's present age = 42 years (\(35 + 7 = 42\), correct)
• Mayank's present age = 10 years (given)

Ages 6 years hence:

• Ramesh's age in 6 years = \(42 + 6 = 48\) years
• Mayank's age in 6 years = \(10 + 6 = 16\) years

Is Ramesh's age in 6 years three times Mayank's age in 6 years?

\(48 = 3 \times 16\)

\(48 = 48\)

Yes, the condition is satisfied. The calculated ages are consistent with the information given in the problem.

The question asks for the present age of Sangeeta, which we found to be 35 years.

Revision Table: Age Problem Concepts

Understanding how to set up equations from word problems is crucial. Here's a quick summary:

• Identify the unknowns and assign variables (e.g., present age).
• Write down the given information as equations based on the relationships described.
• Pay attention to time references (present, future, past) and how they affect the ages in the equations.
• Solve the system of equations to find the values of the variables.
• Always verify your answer by plugging the calculated ages back into the original problem conditions.

Additional Information: Solving Age Word Problems

Age problems are common in mathematics. They often involve translating sentences about ages into algebraic equations. Key phrases to look for include:

• "is" or "was" or "will be" usually translate to "="
• "more than" or "older than" usually mean addition (+)
• "less than" or "younger than" usually mean subtraction (-)
• "times" or "twice" or "thrice" usually mean multiplication (\(\times\))
• References to time shifts (e.g., "in 5 years", "5 years ago") require adjusting the present age by adding or subtracting the number of years.

Break down complex problems into smaller steps, solving for one variable at a time, until you find the required age.