In a bag, there are three types of colored balls: red, white, and blue. The probability of selecting a red ball from the total balls is $\frac{1}{2}$, and the probability of selecting a blue ball is $\frac{2}{7}$. The number of white balls in the bag is 6. What is the total number of balls in the bag?

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**Solution:**

1. Let the total number of balls be $x$.

2. Given the probability of selecting a red ball is $\frac{1}{2}$, so the number of red balls is:
   $$\frac{1}{2} 	imes x = \frac{x}{2}$$

3. Given the probability of selecting a blue ball is $\frac{2}{7}$, so the number of blue balls is:
   $$\frac{2}{7} 	imes x = \frac{2x}{7}$$

4. The number of white balls is given as 6.

5. Total balls are the sum of red, white, and blue balls:
   $$\frac{x}{2} + 6 + \frac{2x}{7} = x$$

6. Multiply the entire equation by 14 (LCM of 2 and 7) to clear denominators:
   $$14 	imes \frac{x}{2} + 14 	imes 6 + 14 	imes \frac{2x}{7} = 14x$$

Simplifying:
   $$7x + 84 + 4x = 14x$$

7. Combine like terms:
   $$11x + 84 = 14x$$

8. Rearranging:
   $$14x - 11x = 84$$
   $$3x = 84$$

9. Solve for $x$:
   $$x = \frac{84}{3} = 28$$

**Answer:** The total number of balls in the bag is $28$.

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