Solving Age Word Problems: Finding Vishal and Armaan's Current Ages

This problem involves finding the current ages of two people, Vishal and Armaan, based on given information about the sum of their current ages and a relationship between their ages in the past. These are classic age word problems often solved using algebraic equations.

Setting Up the Equations for Vishal and Armaan's Ages

Let's use variables to represent the current ages:

• Let \(V\) represent Vishal's current age in years.
• Let \(A\) represent Armaan's current age in years.

We are given two pieces of information:

1. The sum of their current ages is 70 years.
2. 5 years ago, Vishal was twice as old as Armaan.

Let's translate these statements into algebraic equations.

Equation 1: Sum of Current Ages

The first statement says the sum of their current ages is 70 years. So, we have:

\(V + A = 70 \quad \cdots (1)\)

Equation 2: Ages 5 Years Ago

The second statement talks about their ages 5 years ago.

• Vishal's age 5 years ago was \(V - 5\).
• Armaan's age 5 years ago was \(A - 5\).

According to the problem, 5 years ago, Vishal was twice as old as Armaan. This gives us the equation:

\(V - 5 = 2 \times (A - 5)\)

Let's simplify this second equation:

\(V - 5 = 2A - 10\)

\(V = 2A - 10 + 5\)

\(V = 2A - 5 \quad \cdots (2)\)

Solving the System of Equations

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

We can solve this system using the substitution method. Since Equation (2) already gives us an expression for \(V\), we can substitute this expression into Equation (1).

Substitute \(V = 2A - 5\) into Equation (1):

\((2A - 5) + A = 70\)

Now, let's solve for \(A\):

\(3A - 5 = 70\)

\(3A = 70 + 5\)

\(3A = 75\)

\(A = \frac{75}{3}\)

\(A = 25\)

So, Armaan's current age is 25 years.

If needed, we could also find Vishal's current age by substituting the value of \(A\) back into either equation. Using Equation (1):

\(V + 25 = 70\)

\(V = 70 - 25\)

\(V = 45\)

Vishal's current age is 45 years. Let's quickly check this with the second condition: 5 years ago, Vishal was \(45-5=40\) and Armaan was \(25-5=20\). Indeed, \(40 = 2 \times 20\).

Conclusion

Based on the calculations, Armaan's current age is 25 years.

Revision Table: Key Concepts in Age Problems

Additional Information: Solving Simultaneous Equations

Many age word problems, like the one about Vishal and Armaan, lead to a system of simultaneous linear equations. Here are common methods to solve them:

1. Substitution Method:

• Solve one equation for one variable in terms of the other variable.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.
• Substitute the value found back into one of the original equations to find the value of the other variable.

This method was used in the solution for Vishal and Armaan's ages.

2. Elimination Method:

• Multiply one or both equations by constants so that the coefficients of one variable are opposites (e.g., \(+2x\) and \(-2x\)).
• Add the equations together to eliminate one variable.
• Solve the resulting single-variable equation.
• Substitute the value found back into one of the original equations to find the value of the other variable.

Both methods are effective, and the choice often depends on the specific structure of the equations.