The ratio of the present ages of Ram and Shyam is ` 3 : 2`. Which of the following cannot be the ratio of their ages 20 years ago?

Options

A. ` 8 : 5`
B. ` 17 : 10`
C. ` 9 :5`
D. ` 7 : 5`

Explanation

To solve the problem step by step, we will analyze the ages of Ram and Shyam and how they change over time.

### Step 1: Define the Present Ages
Let the present ages of Ram and Shyam be represented in terms of a variable \( x \).
- Present age of Ram = \( 3x \)
- Present age of Shyam = \( 2x \)

### Step 2: Calculate Their Ages 20 Years Ago
To find their ages 20 years ago, we subtract 20 from their present ages.
- Age of Ram 20 years ago = \( 3x - 20 \)
- Age of Shyam 20 years ago = \( 2x - 20 \)

### Step 3: Set Up the Ratio of Their Ages 20 Years Ago
The ratio of their ages 20 years ago can be expressed as:
\[ \text{Ratio} = \frac{3x - 20}{2x - 20} \]

### Step 4: Determine the Condition for the Ratio
We know that the present ratio of their ages is \( \frac{3}{2} \) or \( 1.5 \). For the ratio of their ages 20 years ago to be valid, it must be greater than \( \frac{3}{2} \) because both ages are decreasing by the same amount (20 years).

### Step 5: Analyze the Inequality
We need to ensure:
\[ \frac{3x - 20}{2x - 20} > \frac{3}{2} \]

### Step 6: Cross-Multiply to Solve the Inequality
Cross-multiplying gives:
\[ 2(3x - 20) > 3(2x - 20) \]
Expanding both sides:
\[ 6x - 40 > 6x - 60 \]

### Step 7: Simplify the Inequality
Subtract \( 6x \) from both sides:
\[ -40 > -60 \]
This inequality is always true, indicating that the ratio of their ages 20 years ago will always be greater than \( \frac{3}{2} \).

### Step 8: Check the Given Options
Now, we need to check which of the given options cannot be the ratio of their ages 20 years ago. The options are:
1. \( \frac{8}{5} \) = 1.6 (greater than 1.5)
2. \( \frac{17}{10} \) = 1.7 (greater than 1.5)
3. \( \frac{9}{5} \) = 1.8 (greater than 1.5)
4. \( \frac{7}{5} \) = 1.4 (less than 1.5)

### Step 9: Identify the Incorrect Option
From the options, \( \frac{7}{5} \) = 1.4 is less than 1.5, which means it cannot be the ratio of their ages 20 years ago.

### Conclusion
Thus, the answer is:
**The ratio that cannot be the ratio of their ages 20 years ago is \( \frac{7}{5} \).**

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