Solving the Age Ratio Problem: Find Satheesh's Present Age

This problem involves finding the present ages of two individuals, Sai and Satheesh, given their current age ratio and their age ratio after a certain number of years. We can use algebraic methods to solve this type of age problem.

Understanding the Given Information

• Present age ratio of Sai and Satheesh is 5 : 4.
• After three years, the age ratio will be 11 : 9.
• We need to find Satheesh's present age.

Setting up the Equations

Let's represent the present ages of Sai and Satheesh using a variable based on their current ratio.

• Let Sai's present age be $5x$ years.
• Let Satheesh's present age be $4x$ years.

Now, let's consider their ages after three years:

• Sai's age after 3 years will be $5x + 3$ years.
• Satheesh's age after 3 years will be $4x + 3$ years.

The problem states that the ratio of their ages after three years will be 11 : 9. We can write this as an equation:

$\frac{\text{Sai's age after 3 years}}{\text{Satheesh's age after 3 years}} = \frac{11}{9}$

Substituting the expressions for their ages:

$\frac{5x + 3}{4x + 3} = \frac{11}{9}$

Solving for the Variable

Now we need to solve this equation for $x$. We can do this by cross-multiplication:

$9 \times (5x + 3) = 11 \times (4x + 3)$

Distribute the numbers on both sides:

$9 \times 5x + 9 \times 3 = 11 \times 4x + 11 \times 3$

$45x + 27 = 44x + 33$

Now, we need to isolate the term with $x$. Subtract $44x$ from both sides:

$45x - 44x + 27 = 33$

$x + 27 = 33$

Subtract 27 from both sides to find the value of $x$:

$x = 33 - 27$

$x = 6$

Finding Satheesh's Present Age

We defined Satheesh's present age as $4x$ years. Now that we have the value of $x$, we can calculate Satheesh's age:

Satheesh's present age $= 4x = 4 \times 6 = 24$ years.

Let's also check Sai's present age: $5x = 5 \times 6 = 30$ years.

After 3 years, Sai's age would be $30 + 3 = 33$ years, and Satheesh's age would be $24 + 3 = 27$ years. The ratio would be $33 : 27$. Dividing both numbers by 3, we get $11 : 9$, which matches the information given in the problem. This confirms our value for $x$ is correct.

Summary of Present Ages

Therefore, Satheesh's present age is 24 years.

Revision Table: Age Ratio Problem Solution

Additional Information: Understanding Ratio Problems

Ratio problems often involve comparing quantities. When dealing with ages, ratios change over time. If the ratio of two ages is $a:b$, we can represent the ages as $ax$ and $bx$, where $x$ is a common factor. When time passes (say, $t$ years), their ages become $ax + t$ and $bx + t$. The new ratio can then be used to form an equation to solve for $x$. This common factor $x$ is crucial because it allows us to scale the ratio to the actual values of the ages.

Key steps in solving age ratio problems:

• Represent initial ages using variables and the given ratio (e.g., $ax$, $bx$).
• Express ages after the specified time period (e.g., $ax+t$, $bx+t$).
• Form an equation using the new ratio of these future ages.
• Solve the equation to find the value of the variable $x$.
• Substitute the value of $x$ back into the expressions for the ages to find the actual ages.

Always double-check your answer by plugging the calculated ages back into the conditions given in the problem.