Problem Analysis: Age Ratio Calculation

This problem involves calculating the ages of Sudha and her grandfather based on information given about their ages in the past and future. We need to find the ratio of their ages 8 years ago.

Here's a breakdown of the information provided:

• Condition 1: 16 years ago, the grandfather was 8 times older than Sudha.
• Condition 2: 8 years from now, the grandfather will be 3 times Sudha's age.
• Goal: Find the ratio of Sudha's age to her grandfather's age 8 years ago.

Step-by-Step Solution: Finding Current Ages

Let's denote Sudha's current age as $S$ and her grandfather's current age as $G$. We can translate the given conditions into mathematical equations.

Condition 1: 16 Years Ago

16 years ago, Sudha's age was $S - 16$ and her grandfather's age was $G - 16$.

According to the problem:

$ G - 16 = 8 \times (S - 16) $

Let's simplify this equation:

$ G - 16 = 8S - 128 $

$ G = 8S - 128 + 16 $

$ G = 8S - 112 \quad \quad \text{(Equation 1)} $

Condition 2: 8 Years From Now

8 years from now, Sudha's age will be $S + 8$ and her grandfather's age will be $G + 8$.

According to the problem:

$ G + 8 = 3 \times (S + 8) $

Let's simplify this equation:

$ G + 8 = 3S + 24 $

$ G = 3S + 24 - 8 $

$ G = 3S + 16 \quad \quad \text{(Equation 2)} $

Solving for Current Ages

Now we have two equations for $G$. We can set them equal to each other to solve for $S$:

$ 8S - 112 = 3S + 16 $

Subtract $3S$ from both sides:

$ 5S - 112 = 16 $

Add $112$ to both sides:

$ 5S = 128 $

Solve for $S$:

$ S = \frac{128}{5} = 25.6 $

So, Sudha's current age is $25.6$ years.

Now, substitute the value of $S$ back into Equation 2 (or Equation 1) to find $G$:

$ G = 3S + 16 $

$ G = 3 \times (25.6) + 16 $

$ G = 76.8 + 16 $

$ G = 92.8 $

So, the grandfather's current age is $92.8$ years.

Calculating Ages 8 Years Ago

The question asks for the ratio of their ages 8 years ago.

• Sudha's age 8 years ago = $S - 8 = 25.6 - 8 = 17.6$ years.
• Grandfather's age 8 years ago = $G - 8 = 92.8 - 8 = 84.8$ years.

Final Ratio Calculation

The ratio of Sudha's age to her grandfather's age 8 years ago is:

$ \frac{\text{Sudha's age 8 years ago}}{\text{Grandfather's age 8 years ago}} = \frac{17.6}{84.8} $

To simplify this ratio, we can remove the decimal by multiplying the numerator and denominator by 10:

$ \frac{17.6 \times 10}{84.8 \times 10} = \frac{176}{848} $

Now, we find the greatest common divisor (GCD) or simplify by dividing by common factors. Let's divide both by 8:

$ \frac{176 \div 8}{848 \div 8} = \frac{22}{106} $

Both numbers are still even, so we divide by 2:

$ \frac{22 \div 2}{106 \div 2} = \frac{11}{53} $

Therefore, the ratio of Sudha's age to her grandfather's age 8 years ago was $11:53$.