The ratio of the ages of Ram to Shyam 6 years ago was $1:2$. The present age of Ram is 5 years more than the present age of Prem. The ratio of the present ages of Prem and Shyam is $5:14$. What will be the age of Ram after 10 years (in years)?

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**Solution:**

1. Let the present ages of Ram, Shyam, and Prem be $R$, $S$, and $P$ respectively.

2. Given that 6 years ago, the ratio of Ram's age to Shyam's age was $1:2$. Therefore,
   $$\frac{R - 6}{S - 6} = \frac{1}{2} \implies 2(R - 6) = S - 6 \implies 2R - 12 = S - 6 \implies S = 2R - 6$$

3. The present age of Ram is 5 years more than the present age of Prem:
   $$R = P + 5$$

4. The ratio of the present ages of Prem and Shyam is $5:14$:
   $$\frac{P}{S} = \frac{5}{14} \implies 14P = 5S$$

5. Substitute $S$ from step 2 into the above:
   $$14P = 5(2R - 6) = 10R - 30$$

6. Substitute $R = P + 5$ from step 3:
   $$14P = 10(P + 5) - 30 = 10P + 50 - 30 = 10P + 20$$

7. Simplify:
   $$14P - 10P = 20 \implies 4P = 20 \implies P = 5$$

8. Find $R$:
   $$R = P + 5 = 5 + 5 = 10$$

9. Find $S$:
   $$S = 2R - 6 = 2 	imes 10 - 6 = 20 - 6 = 14$$

10. The question asks for Ram's age after 10 years:
    $$R + 10 = 10 + 10 = 20$$

**Answer:** $20$ years

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