If the average age of three persons is $56$ years and their ages are in the ratio $2 : 5 : 7$, then find the age of the youngest person.

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**Solution:**

1. Let the ages of the three persons be $2x$, $5x$, and $7x$ respectively.

2. The average age is given as $56$ years. Therefore,
   
   $$\frac{2x + 5x + 7x}{3} = 56$$

3. Simplify the numerator:
   
   $$\frac{14x}{3} = 56$$

4. Multiply both sides by $3$:
   
   $$14x = 56 	imes 3 = 168$$

5. Solve for $x$:
   
   $$x = \frac{168}{14} = 12$$

6. The youngest person's age is $2x$:
   
   $$2 	imes 12 = 24 	ext{ years}$$

**Answer:** 24 years

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