Solving Age Word Problems: Finding Rohan's Present Age

This problem asks us to find Rohan's present age based on information given about the ages of Rohan and his father at different points in time: 5 years ago and 5 years hence (future).

To solve this type of problem, we use variables to represent the unknown present ages and set up equations based on the given conditions.

Defining Variables for Present Ages

Let's denote:

• Rohan's present age = \(R\) years
• Father's present age = \(F\) years

Setting up Equations from the Given Information

We have two pieces of information, which will give us two equations:

Condition 1: 5 years ago

• 5 years ago, Rohan's age was \(R - 5\) years.
• 5 years ago, Father's age was \(F - 5\) years.

The problem states that 5 years ago, father’s age was 8 times Rohan’s age.

This translates to the equation:

\(F - 5 = 8 \times (R - 5)\)

Let's simplify this equation:

\(F - 5 = 8R - 40\)

Add 5 to both sides:

\(F = 8R - 40 + 5\)

\(F = 8R - 35\) (Equation 1)

Condition 2: 5 years hence (in the future)

• 5 years hence, Rohan's age will be \(R + 5\) years.
• 5 years hence, Father's age will be \(F + 5\) years.

The problem states that 5 years hence, the ratio of father’s age to Rohan’s age will be 10 : 3.

This translates to the equation:

\(\frac{F + 5}{R + 5} = \frac{10}{3}\)

Now, let's cross-multiply to simplify this equation:

\(3 \times (F + 5) = 10 \times (R + 5)\)

\(3F + 15 = 10R + 50\)

Subtract 15 from both sides:

\(3F = 10R + 50 - 15\)

\(3F = 10R + 35\) (Equation 2)

Solving the System of Equations

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

1) \(F = 8R - 35\)

2) \(3F = 10R + 35\)

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

\(3 \times (8R - 35) = 10R + 35\)

Distribute the 3 on the left side:

\(24R - 105 = 10R + 35\)

Now, we need to isolate \(R\). Subtract \(10R\) from both sides:

\(24R - 10R - 105 = 35\)

\(14R - 105 = 35\)

Add 105 to both sides:

\(14R = 35 + 105\)

\(14R = 140\)

Divide by 14 to find \(R\):

\(R = \frac{140}{14}\)

\(R = 10\)

Conclusion: Rohan's Present Age

We found that \(R = 10\). Since \(R\) represents Rohan's present age, Rohan's present age is 10 years.

Verification

Let's check if this answer fits the original conditions. If Rohan's present age is 10, Father's present age can be found using Equation 1:

\(F = 8R - 35 = 8 \times 10 - 35 = 80 - 35 = 45\)

Father's present age is 45 years.

• 5 years ago: Rohan was \(10 - 5 = 5\) years old. Father was \(45 - 5 = 40\) years old. Is 40 = 8 * 5? Yes, 40 = 40. (Condition 1 is satisfied)
• 5 years hence: Rohan will be \(10 + 5 = 15\) years old. Father will be \(45 + 5 = 50\) years old. Is the ratio of Father's age to Rohan's age 10:3? \(\frac{50}{15} = \frac{10 \times 5}{3 \times 5} = \frac{10}{3}\). Yes, the ratio is 10:3. (Condition 2 is satisfied)

Both conditions are met, so our answer is correct.

Revision Table: Key Concepts in Age Problems

Additional Information: Solving Linear Equations

The core of solving age problems often involves solving a system of linear equations. A system of two linear equations with two variables can be solved using several methods:

• Substitution Method: Solve one equation for one variable (e.g., solve for \(F\) in terms of \(R\)) and substitute that expression into the other equation. This reduces the system to a single equation with one variable, which you can then solve. This is the method used in this solution.
• Elimination Method: Multiply one or both equations by a constant so that the coefficients of one variable are opposite. Then, add the equations together to eliminate that variable, leaving a single equation with one variable.
• Graphical Method: Graph both linear equations. The point where the lines intersect is the solution to the system. This method is less precise for non-integer solutions but provides a visual understanding.

Understanding how to set up and solve these equations is crucial for tackling various word problems, including those involving ages, speeds, mixtures, etc.