Solving the Roshan and Usha Age Problem

This question is an age word problem that requires setting up and solving a system of linear equations. We are given relationships between the current ages of Roshan and Usha, and also a relationship between their ages from 8 years ago. Our goal is to find Roshan's current age.

Defining Variables for Ages

Let's define variables to represent the current ages:

• Let Roshan's current age be \(R\) years.
• Let Usha's current age be \(U\) years.

Setting up Equations from the Given Information

We translate the information provided in the problem into algebraic equations:

1. "Roshan’s current age is 2 years less than 1.6 times that of Usha’s."
   • 1.6 times Usha's age is \(1.6U\).
   • 2 years less than that is \(1.6U - 2\).
   • So, Roshan's current age \(R\) is equal to this expression.
   • Equation 1: \(R = 1.6U - 2\)
2. "8 years ago, Usha’s age was 1 year more than half of Roshan’s age."
   • Roshan's age 8 years ago was \(R - 8\).
   • Usha's age 8 years ago was \(U - 8\).
   • Half of Roshan's age 8 years ago was \(\frac{1}{2}(R - 8)\).
   • 1 year more than half of Roshan's age 8 years ago was \(\frac{1}{2}(R - 8) + 1\).
   • Usha's age 8 years ago \(U - 8\) was equal to this expression.
   • Equation 2: \(U - 8 = \frac{1}{2}(R - 8) + 1\)

Solving the System of Equations for Roshan's Age

We now have a system of two linear equations with two variables, \(R\) and \(U\):

• Equation 1: \(R = 1.6U - 2\)
• Equation 2: \(U - 8 = \frac{1}{2}(R - 8) + 1\)

Let's simplify Equation 2 first:

\(U - 8 = \frac{1}{2}R - \frac{1}{2}(8) + 1\)

\(U - 8 = \frac{1}{2}R - 4 + 1\)

\(U - 8 = \frac{1}{2}R - 3\)

Now, isolate \(U\) in this simplified Equation 2:

\(U = \frac{1}{2}R - 3 + 8\)

\(U = \frac{1}{2}R + 5\)

Now we can substitute this expression for \(U\) into Equation 1 (\(R = 1.6U - 2\)) to solve for \(R\):

\(R = 1.6\left(\frac{1}{2}R + 5\right) - 2\)

Distribute the 1.6 on the right side:

\(R = 1.6 \times \frac{1}{2}R + 1.6 \times 5 - 2\)

\(R = 0.8R + 8 - 2\)

\(R = 0.8R + 6\)

Now, gather the \(R\) terms on one side:

\(R - 0.8R = 6\)

\(0.2R = 6\)

To find \(R\), divide both sides by 0.2:

\(R = \frac{6}{0.2}\)

\(R = \frac{6}{\frac{2}{10}}\)

\(R = 6 \times \frac{10}{2}\)

\(R = 6 \times 5\)

\(R = 30\)

So, Roshan's present age is 30 years.

Verifying the Solution

Let's check if this value of \(R=30\) satisfies the original conditions. First, find Usha's age using \(U = \frac{1}{2}R + 5\):

\(U = \frac{1}{2}(30) + 5 = 15 + 5 = 20\)

Usha's current age is 20 years.

Now check the original statements:

• "Roshan’s current age is 2 years less than 1.6 times that of Usha’s."
  • 1.6 times Usha's age: \(1.6 \times 20 = 32\).
  • 2 years less than that: \(32 - 2 = 30\).
  • Roshan's age is 30. This matches \(R=30\). (Statement 1 is correct)
• "8 years ago, Usha’s age was 1 year more than half of Roshan’s age."
  • Roshan's age 8 years ago: \(30 - 8 = 22\).
  • Usha's age 8 years ago: \(20 - 8 = 12\).
  • Half of Roshan's age 8 years ago: \(\frac{1}{2}(22) = 11\).
  • 1 year more than that: \(11 + 1 = 12\).
  • Usha's age 8 years ago was 12. This matches \(U-8=12\). (Statement 2 is correct)

Both statements are satisfied with Roshan's age being 30 and Usha's age being 20. Therefore, Roshan's present age is 30 years.

Conclusion on Roshan's Present Age

Based on the calculations and verification, Roshan's present age is 30 years.

Revision Table: Key Concepts in Age Word Problems

Additional Information on Solving Word Problems

Age word problems are common in algebra and quantitative aptitude tests. The key to solving them is carefully reading the problem statement and breaking it down into smaller parts.

• Identify the unknowns: Figure out which ages you need to find. Assign variables to these.
• Note different time periods: Pay attention to whether the information is about the present, past, or future. Adjust ages by adding or subtracting years accordingly.
• Look for relationships: Identify phrases like "is twice as old as," "is 5 years older than," "is half the age of," etc. These describe how the ages are related and help you form equations.
• Formulate equations: Write down the algebraic equations that represent the relationships described in the problem. You usually need as many independent equations as you have variables.
• Solve the equations: Use algebraic techniques (substitution, elimination) to find the values of your variables.
• Check your answer: Substitute the values you found back into the original word problem or equations to ensure they make sense and satisfy all the conditions. This step is crucial to catch potential errors.

Practice with various types of age problems, including those involving more than two people, ratios of ages, or relationships spanning multiple time periods, will improve your ability to solve them efficiently.