My grandfather was 8 times older than me 16 years ago. He would be 3 times of my age, 8 years from now. Eight years ago, what was the ratio of my age to that of my grandfather?

Options

A. `3:8`
B. `1:5`
C. `1:2`
D. None

Answer

To solve the problem step by step, we will define variables for our current ages and then use the information given in the problem to set up equations.

### Step 1: Define Variables
Let:
- \( x \) = my current age
- \( y \) = my grandfather's current age

### Step 2: Set Up the First Equation
According to the problem, 16 years ago, my grandfather was 8 times older than me. This can be expressed as:
\[ y - 16 = 8(x - 16) \]

### Step 3: Simplify the First Equation
Expanding the equation:
\[ y - 16 = 8x - 128 \]
Rearranging gives:
\[ y = 8x - 112 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
The problem also states that 8 years from now, my grandfather will be 3 times my age. This can be expressed as:
\[ y + 8 = 3(x + 8) \]

### Step 5: Simplify the Second Equation
Expanding this equation:
\[ y + 8 = 3x + 24 \]
Rearranging gives:
\[ y = 3x + 16 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
Now we have two equations:
1. \( y = 8x - 112 \)
2. \( y = 3x + 16 \)

Setting them equal to each other:
\[ 8x - 112 = 3x + 16 \]

### Step 7: Isolate \( x \)
Subtract \( 3x \) from both sides:
\[ 5x - 112 = 16 \]
Add 112 to both sides:
\[ 5x = 128 \]
Divide by 5:
\[ x = \frac{128}{5} = 25.6 \]

### Step 8: Find \( y \)
Substituting \( x \) back into Equation 1:
\[ y = 8\left(\frac{128}{5}\right) - 112 \]
Calculating:
\[ y = \frac{1024}{5} - 112 = \frac{1024}{5} - \frac{560}{5} = \frac{464}{5} = 92.8 \]

### Step 9: Calculate Ages 8 Years Ago
Now, we need to find the ages 8 years ago:
- My age 8 years ago: \( x - 8 = 25.6 - 8 = 17.6 \)
- My grandfather's age 8 years ago: \( y - 8 = 92.8 - 8 = 84.8 \)

### Step 10: Find the Ratio
The ratio of my age to my grandfather's age 8 years ago is:
\[ \text{Ratio} = \frac{17.6}{84.8} \]
To simplify this, we can divide both by 17.6:
\[ \text{Ratio} = \frac{1}{4.8} = \frac{5}{24} \]

### Final Answer
The ratio of my age to that of my grandfather's age 8 years ago is \( 5:24 \).

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