Understanding Age Ratio Problems

This problem involves finding the present age of Rajni based on the given ratios of present ages and future ages of two individuals, Neelam and Rajni.

Setting Up the Age Variables

We are given that the ratio of the present ages of Neelam and Rajni is 6 : 7. We can represent their present ages using a common variable, say x.

• Let Neelam's present age be \( 6x \) years.
• Let Rajni's present age be \( 7x \) years.

Calculating Future Ages

The problem states that after four years, their ages will be in the ratio of 8 : 9. We need to calculate their ages after four years.

• Neelam's age after 4 years = \( 6x + 4 \) years.
• Rajni's age after 4 years = \( 7x + 4 \) years.

Forming the Equation

According to the problem, the ratio of their ages after four years is 8 : 9. We can set up an equation using this information:

$$ \frac{\text{Neelam's age after 4 years}}{\text{Rajni's age after 4 years}} = \frac{8}{9} $$

Substituting the expressions for their future ages:

$$ \frac{6x + 4}{7x + 4} = \frac{8}{9} $$

Solving the Equation for x

To find the value of x, we will cross-multiply:

$$ 9(6x + 4) = 8(7x + 4) $$

Distribute the numbers on both sides:

$$ 54x + 36 = 56x + 32 $$

Now, rearrange the terms to solve for x. Subtract \( 54x \) from both sides:

$$ 36 = 2x + 32 $$

Subtract 32 from both sides:

$$ 36 - 32 = 2x $$

$$ 4 = 2x $$

Divide by 2:

$$ x = \frac{4}{2} $$

$$ x = 2 $$

Determining Rajni's Present Age

We found that \( x = 2 \). The question asks for the present age of Rajni. Rajni's present age was represented as \( 7x \).

Rajni's present age = \( 7x = 7 \times 2 \)

Rajni's present age = 14 years.

Final Answer

Rajni's present age is 14 years.