Solving the Ram and Sunny Age Problem

This question asks us to find Ram's current age based on information about his and Sunny's ages at two different points in time. We can solve this type of problem by setting up algebraic equations.

Defining Variables for Current Ages

Let's represent the current ages of Ram and Sunny using variables:

• Let \( R \) be Ram's current age in years.
• Let \( S \) be Sunny's current age in years.

Setting Up Equations from the Given Conditions

We are given two conditions:

1. 13 years ago, Ram was twice as old as Sunny.
2. Three years from now, Sunny's age will be 3/5 of Ram's age.

Condition 1: 13 Years Ago

13 years ago:

• Ram's age was \( R - 13 \)
• Sunny's age was \( S - 13 \)

The condition states that Ram's age was twice Sunny's age:

\( R - 13 = 2(S - 13) \)

Let's simplify this equation:

\( R - 13 = 2S - 26 \)

\( R = 2S - 26 + 13 \)

\( R = 2S - 13 \quad \text{(Equation 1)} \)

Condition 2: Three Years From Now

Three years from now:

• Ram's age will be \( R + 3 \)
• Sunny's age will be \( S + 3 \)

The condition states that Sunny's age will be 3/5 of Ram's age:

\( S + 3 = \frac{3}{5}(R + 3) \)

Let's simplify this equation by multiplying both sides by 5 to clear the fraction:

\( 5(S + 3) = 3(R + 3) \)

\( 5S + 15 = 3R + 9 \)

Rearranging the terms to group \( S \) and \( R \):

\( 5S - 3R = 9 - 15 \)

\( 5S - 3R = -6 \quad \text{(Equation 2)} \)

Solving the System of Equations

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

• \( R = 2S - 13 \quad \text{(Equation 1)} \)
• \( 5S - 3R = -6 \quad \text{(Equation 2)} \)

We can use the substitution method. Substitute the expression for \( R \) from Equation 1 into Equation 2:

\( 5S - 3(2S - 13) = -6 \)

Distribute the -3:

\( 5S - 6S + 39 = -6 \)

Combine the \( S \) terms:

\( -S + 39 = -6 \)

Subtract 39 from both sides:

\( -S = -6 - 39 \)

\( -S = -45 \)

Multiply by -1 to solve for \( S \):

\( S = 45 \)

So, Sunny's current age is 45 years.

Now, substitute the value of \( S \) (45) back into Equation 1 to find \( R \):

\( R = 2S - 13 \)

\( R = 2(45) - 13 \)

\( R = 90 - 13 \)

\( R = 77 \)

So, Ram's current age is 77 years.

Verifying the Solution

Let's check if Ram's current age of 77 and Sunny's current age of 45 satisfy the original conditions:

• 13 years ago:

  • Ram's age: \( 77 - 13 = 64 \)
  • Sunny's age: \( 45 - 13 = 32 \)

  Is 64 twice 32? \( 2 \times 32 = 64 \). Yes, the first condition is met.

• 3 years from now:

  • Ram's age: \( 77 + 3 = 80 \)
  • Sunny's age: \( 45 + 3 = 48 \)

  Is 48 equal to 3/5 of 80? \( \frac{3}{5} \times 80 = \frac{3 \times 80}{5} = \frac{240}{5} = 48 \). Yes, the second condition is met.

Both conditions are satisfied, so our calculated current ages for Ram and Sunny are correct.

Ram's current age is 77 years.

The final answer is 77 years.

Revision Table for Age Problems

Additional Information on Age Word Problems and Linear Equations

Age word problems are common types of algebraic problems that involve finding the ages of one or more people based on given information about their ages at different points in time. These problems often require setting up and solving a system of linear equations.

Key Concepts:

• Variables: Represent unknown quantities (usually current ages) with variables like \( x \) or \( y \), or initial letters like \( R \) and \( S \) for clarity.
• Expressions for Past/Future Ages: If the current age is \( A \):
  • Age \( n \) years ago: \( A - n \)
  • Age \( n \) years from now: \( A + n \)
• Forming Equations: Translate phrases involving relationships between ages (like "twice as old", "sum of ages", "ratio of ages") into mathematical equations.
• System of Linear Equations: Most age problems with two or more people at different times will result in a system of two or more linear equations.
• Solving Techniques: Standard methods for solving systems of equations include:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: Multiply equations by constants so that one variable cancels out when the equations are added or subtracted.

Mastering the translation of word problems into mathematical equations is crucial for solving age problems. Careful reading of the problem statement is essential to correctly identify the relationships between the ages at the specified times.