Shanta is three times of the age of her son. She has a grandson who is half of the age of her son and her granddaughter is the difference of one-third of the age of her son and 3. compare the age of her grandchildren and write who is older in the form of an inequation.

Answer

To solve the problem step by step, we will define the ages of Shanta, her son, grandson, and granddaughter using algebraic expressions and then compare the ages of the grandchildren.

### Step 1: Define the variables
Let the age of Shanta's son be \( x \).

### Step 2: Express Shanta's age
Since Shanta is three times the age of her son, we can express her age as:
\[ \text{Age of Shanta} = 3x \]

### Step 3: Express the age of the grandson
The grandson is half the age of her son. Therefore, we can express the age of the grandson as:
\[ \text{Age of Grandson} = \frac{x}{2} \]

### Step 4: Express the age of the granddaughter
The granddaughter's age is given as the difference of one-third of the age of her son and 3. Thus, we can express her age as:
\[ \text{Age of Granddaughter} = \frac{x}{3} - 3 \]

### Step 5: Set up the inequality
We need to compare the ages of the grandson and granddaughter. We will set up the inequality:
\[ \text{Age of Grandson} > \text{Age of Granddaughter} \]
Substituting the expressions we found:
\[ \frac{x}{2} > \left(\frac{x}{3} - 3\right) \]

### Step 6: Solve the inequality
To solve the inequality, we first eliminate the fractions by finding a common denominator. The common denominator for 2 and 3 is 6. We will multiply the entire inequality by 6:
\[ 6 \cdot \frac{x}{2} > 6 \cdot \left(\frac{x}{3} - 3\right) \]
This simplifies to:
\[ 3x > 2x - 18 \]

### Step 7: Isolate \( x \)
Now, we will isolate \( x \) by subtracting \( 2x \) from both sides:
\[ 3x - 2x > -18 \]
This simplifies to:
\[ x > -18 \]

### Step 8: Conclusion
Since \( x \) represents the age of Shanta's son, which must be a positive value, we can conclude that the grandson is older than the granddaughter.

### Final Inequation
Thus, the required inequation comparing the ages of the grandchildren is:
\[ \frac{x}{2} > \frac{x}{3} - 3 \]