Solving the Age Word Problem: Rohan and Nikhil

This question asks us to find Rohan's age in the future when a specific condition about his age relative to his younger brother, Nikhil, is met. We are given their current age difference and the future relationship between their ages.

Understanding the Problem

We know two main things:

• Nikhil is 8 years younger than Rohan. This means Rohan is 8 years older than Nikhil. The age difference between them will always remain 8 years.
• We want to find their ages at a future point in time when Rohan is exactly twice as old as Nikhil.

Setting up the Equations

Let's use variables to represent their ages at the future time we are interested in.

• Let Rohan's age in the future be \(R_{future}\).
• Let Nikhil's age in the future be \(N_{future}\).

From the problem statement, we can write two equations based on the conditions:

1. The age difference is always 8 years. So, at the future time: \(R_{future} - N_{future} = 8\), which can be written as \(R_{future} = N_{future} + 8\).
2. At this future time, Rohan will be twice as old as Nikhil: \(R_{future} = 2 \times N_{future}\).

Solving the Equations

Now we have a system of two simple linear equations:

\(R_{future} = N_{future} + 8\)
\(R_{future} = 2 N_{future}\)

We can solve this system by substituting the second equation into the first one. Since both equations are equal to \(R_{future}\), we can set them equal to each other:

\(N_{future} + 8 = 2 N_{future}\)

Now, we need to solve for \(N_{future}\):

Subtract \(N_{future}\) from both sides of the equation:

\(8 = 2 N_{future} - N_{future}\)
\(8 = N_{future}\)

So, Nikhil's age at that future point in time will be 8 years.

The question asks for Rohan's age at that future point. We can find Rohan's age (\(R_{future}\)) using either of the original equations. Let's use the second one, as it's simpler:

\(R_{future} = 2 \times N_{future}\)
\(R_{future} = 2 \times 8\)
\(R_{future} = 16\)

So, Rohan's age at that future point will be 16 years.

Verification

Let's check if our solution fits the original conditions:

• Rohan's future age is 16, Nikhil's future age is 8.
• Is Rohan 8 years older than Nikhil? \(16 - 8 = 8\). Yes, the age difference is 8 years.
• Is Rohan twice as old as Nikhil? \(16 = 2 \times 8\). Yes, Rohan is twice as old as Nikhil.

Both conditions are satisfied by these ages.

Conclusion

When Rohan is twice as old as Nikhil, Rohan will be 16 years old.

Comparing with Options

Our calculated age for Rohan is 16 years, which matches option 4.

Revision Table

Additional Information

Age word problems are common in mathematics and can often be solved by setting up equations based on the information given. The key is to carefully identify the unknown quantities and the relationships between them at different points in time.

When dealing with age problems, remember that:

• If the age difference between two people is \(d\) now, it will always be \(d\) in the future and was \(d\) in the past.
• If someone is \(x\) years old now, in \(y\) years they will be \(x + y\) years old. In \(z\) years ago, they were \(x - z\) years old.
• Phrases like "twice as old," "half as old," or "a third of the age" translate directly into multiplication or division in your equations.

In this specific problem, setting up the equations based on the future ages was the most direct path to the solution. We defined the future ages and wrote the conditions directly using those variables, avoiding the need to introduce a variable for the number of years from now.