Solving the Virat and Mohinder Age Word Problem

This problem involves finding the current ages of two people, Virat and Mohinder, based on conditions given about their ages at different points in time. We can solve this by setting up a system of linear equations.

Setting Up Variables for Present Ages

Let's use variables to represent their current ages:

• Virat's present age = \(V\) years
• Mohinder's present age = \(M\) years

Condition 1: Age Relationship in the Future

The problem states that seven years from now, Virat will be twice as old as Mohinder.

• Virat's age 7 years from now: \(V + 7\)
• Mohinder's age 7 years from now: \(M + 7\)

According to the condition, we can write the equation:

\(V + 7 = 2(M + 7)\)

Let's simplify this equation to get it into a standard linear form:

Condition 2: Age Relationship in the Past

The problem also states that five years ago, Mohinder's age was one year less than \(\frac{2}{5}\) of Virat's age.

• Virat's age 5 years ago: \(V - 5\)
• Mohinder's age 5 years ago: \(M - 5\)

According to the condition, we can write the equation:

\(M - 5 = \frac{2}{5}(V - 5) - 1\)

Let's simplify this equation. To remove the fraction, we can multiply the entire equation by 5:

Now, let's rearrange the terms to get the variables on one side and the constant on the other:

Solving the System of Linear Equations

Now we have a system of two linear equations with two variables:

• Equation 1: \(V - 2M = 7\)
• Equation 2: \(-2V + 5M = 10\)

We can solve this system using the elimination method. Our goal is to eliminate one variable by making its coefficients equal and opposite in the two equations, and then adding the equations.

Let's multiply Equation 1 by 2:

Now, add Equation 3 and Equation 2:

Combine like terms:

So, Mohinder's present age is 24 years.

Now that we have the value of \(M\), we can substitute \(M = 24\) into Equation 1 (or Equation 2) to find Virat's present age \(V\). Using Equation 1:

Add 48 to both sides of the equation:

So, Virat's present age is 55 years.

To verify, check the conditions with \(V=55\) and \(M=24\):

• 7 years from now: Virat = 55 + 7 = 62, Mohinder = 24 + 7 = 31. Is 62 twice 31? \(2 \times 31 = 62\). Yes.
• 5 years ago: Virat = 55 - 5 = 50, Mohinder = 24 - 5 = 19. Is 19 one year less than 2/5 of 50? \(\frac{2}{5} \times 50 = 2 \times 10 = 20\). Is 19 one year less than 20? \(20 - 1 = 19\). Yes.

Both conditions are satisfied, confirming the calculated ages are correct.

Conclusion: Virat's Present Age

Based on the conditions given in the problem, Virat's present age is 55 years.

Revision Table: Key Concepts in Age Word Problems

Additional Information on Solving Age Problems

Age problems are a classic type of word problem often found in algebra. They require you to set up mathematical equations based on the relationships between people's ages at different points in time (past, present, or future). The key is to carefully read the problem and correctly translate the given information into algebraic expressions.

Here are some general steps and tips for tackling age problems:

• Identify the Unknowns: Figure out whose ages you need to find. Usually, it's best to let a variable represent their present age.
• Express Ages at Different Times: If the problem mentions ages in the past or future, write expressions for the ages at those specific times based on the present age variable. For example, if the present age is \(x\), the age 5 years ago is \(x-5\), and the age in 10 years is \(x+10\).
• Formulate Equations: Use the relationships given in the problem statement (e.g., "A is twice as old as B", "A's age is 5 more than B's age") to create equations involving the age expressions.
• Solve the Equations: You will typically end up with one or two linear equations. Solve these equations using standard algebraic methods like substitution, elimination, or graphing (though substitution/elimination are most common for two variable systems).
• Check Your Answer: Once you find the values for the variables, plug them back into the original word problem to make sure all the conditions are met. This helps catch any calculation errors.

Paying close attention to the wording, especially phrases involving "years ago", "years from now", "times as old", "less than", and "more than", is crucial for setting up the equations correctly.