Understanding Age Ratio Problems

This question involves finding the present age of an individual given the ratio of ages at two different points in time. We are given the present age ratio of Ram and Ramesh and their age ratio after 7 years. We need to use this information to set up an equation and solve for their current ages.

Setting up the Age Ratio Problem

Let's denote the present ages of Ram and Ramesh using the given ratio. The ratio of the present ages of Ram and Ramesh is 3 : 5. This means we can represent their present ages as a multiple of some unknown value.

• Present age of Ram = \(3x\) years
• Present age of Ramesh = \(5x\) years

Here, \(x\) is a common multiplier that we need to find.

Ages After 7 Years

The question states that after 7 years, the ratio of their ages will be 4 : 5. We need to calculate their ages after adding 7 years to their present ages.

• Ram's age after 7 years = Present age of Ram + 7 = \(3x + 7\) years
• Ramesh's age after 7 years = Present age of Ramesh + 7 = \(5x + 7\) years

The ratio of these future ages is given as 4 : 5. So, we can write the equation:

\(\frac{3x + 7}{5x + 7} = \frac{4}{5}\)

Solving the Age Ratio Equation

To find the value of \(x\), we need to solve the equation. We can do this by cross-multiplying:

\(5 \times (3x + 7) = 4 \times (5x + 7)\)

Now, distribute the numbers on both sides:

\(15x + 35 = 20x + 28\)

Next, collect the \(x\) terms on one side and the constant terms on the other side:

\(35 - 28 = 20x - 15x\)

\(7 = 5x\)

Now, isolate \(x\) by dividing both sides by 5:

\(x = \frac{7}{5}\)

Calculating the Present Age of Ramesh

We are asked to find the present age of Ramesh. We represented Ramesh's present age as \(5x\). Now that we have the value of \(x\), we can substitute it back into the expression for Ramesh's present age:

Present age of Ramesh = \(5x = 5 \times \frac{7}{5}\)

\(= 7\)

So, the present age of Ramesh is 7 years.

Verification

Let's quickly verify our answer. If Ramesh's present age is 7 years and Ram's present age is \(3x = 3 \times \frac{7}{5} = \frac{21}{5} = 4.2\) years, their ratio is \(4.2 : 7\), which simplifies to \(42 : 70\), or \(3 : 5\). This matches the given present ratio.

After 7 years:

• Ram's age = \(4.2 + 7 = 11.2\) years
• Ramesh's age = \(7 + 7 = 14\) years

The ratio of their ages after 7 years is \(11.2 : 14\). Let's check if this simplifies to 4 : 5.

\(\frac{11.2}{14} = \frac{112/10}{14} = \frac{112}{10 \times 14} = \frac{112}{140}\)

Dividing both numerator and denominator by 28:

\(\frac{112 \div 28}{140 \div 28} = \frac{4}{5}\)

The ratio after 7 years is indeed 4 : 5, which matches the information given in the problem. Our calculated age for Ramesh is correct.

Final Answer for Ramesh's Present Age

Based on our calculations, the present age of Ramesh is 7 years.

Revision Table: Age Ratio Concepts

Additional Information: Age Problems and Ratios

Age problems often involve setting up and solving linear equations based on ratios or differences in ages at different points in time. Key things to remember when tackling age problems:

• Consistent Variable: Use the same variable (like \(x\)) to represent the common part of the ratio for all individuals involved.
• Time Affects All: When considering a future or past time, add or subtract the same number of years from everyone's age.
• Ratio as a Fraction: A ratio like \(a:b\) can be written as a fraction \(\frac{a}{b}\) when setting up equations.
• Cross-Multiplication: This is a common technique to solve equations where a fraction equals another fraction (\(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)).
• Check Your Answer: Always substitute the calculated value back into the original problem statement (both present and future/past conditions) to ensure it satisfies all conditions. This helps catch errors.

Understanding how to translate word problems involving ages and ratios into algebraic equations is a crucial skill for solving these types of questions.