Understanding the Age Ratio Problem

This problem involves finding the current ages of two people, Sudhir and Ashish, based on a given ratio of their current ages and another ratio of their ages from the past. We are then asked to calculate Sudhir's age in the future.

Let's break down the information given:

• The current age ratio of Sudhir and Ashish is 5:7.
• Twelve years ago, the ratio of their ages was 1:2.
• We need to find Sudhir's age five years from now.

Setting up Equations for Ages

We can represent the current ages using a variable based on the given ratio. Since the ratio is 5:7, let:

• Sudhir's current age $= 5x$ years
• Ashish's current age $= 7x$ years

Now, consider their ages 12 years ago:

• Sudhir's age 12 years ago $= (5x - 12)$ years
• Ashish's age 12 years ago $= (7x - 12)$ years

The problem states that 12 years ago, the ratio of their ages was 1:2. We can write this as an equation:

\begin{equation*} \frac{5x - 12}{7x - 12} = \frac{1}{2} \end{equation*}

Solving for the Unknown Variable 'x'

To find the value of $x$, we need to solve the equation we set up. We can do this by cross-multiplication:

\begin{align*} 2 \times (5x - 12) &= 1 \times (7x - 12) \\ 10x - 24 &= 7x - 12 \end{align*}

Now, we gather the terms with $x$ on one side and the constant terms on the other side:

\begin{align*} 10x - 7x &= -12 + 24 \\ 3x &= 12 \\ x &= \frac{12}{3} \\ x &= 4 \end{align*}

So, the value of $x$ is 4.

Calculating the Current Ages

Now that we have the value of $x$, we can find the current ages of Sudhir and Ashish:

• Sudhir's current age $= 5x = 5 \times 4 = 20$ years
• Ashish's current age $= 7x = 7 \times 4 = 28$ years

Finding Sudhir's Age After Five Years

The question asks for Sudhir's age after five years from now. To find this, we add 5 years to Sudhir's current age:

Sudhir's age after 5 years = Sudhir's current age + 5 years

Sudhir's age after 5 years $= 20 + 5 = 25$ years.

Let's quickly verify the ages 12 years ago based on our calculated current ages:

• Sudhir's age 12 years ago = $20 - 12 = 8$ years
• Ashish's age 12 years ago = $28 - 12 = 16$ years

The ratio of their ages 12 years ago would be $8:16$, which simplifies to $1:2$. This matches the condition given in the problem, confirming our calculations are correct.

Therefore, the age of Sudhir after five years from now will be 25 years.

Revision Table: Key Information

Additional Information on Age Problems

Age-related word problems are common in mathematics and competitive exams. They typically involve setting up linear equations based on relationships between ages at different points in time. Here are some tips for solving age problems:

• Always define variables clearly, usually representing the current age.
• Express ages in the past or future in terms of the current age variable(s) and the given number of years.
• Translate the given ratios or sums/differences of ages into algebraic equations.
• Solve the equations systematically to find the value of the variable(s).
• Once the variable is found, calculate the required age(s) as asked in the question.
• Always check your answer by substituting the calculated ages back into the original problem conditions to ensure they are satisfied.

Understanding how to represent ages at different time points (past, present, future) and translating ratios into equations are key skills for mastering age problems.