Suresh, Dinesh, and Ramesh became partners in a business by investing money in the ratio of $3 : 6 : 8$. If their investments are increased by $5\%$, $15\%$, and $20\%$, respectively, then what will be the ratio of their profits for one year?

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**Solution:**

1. Initial investment ratio of Suresh, Dinesh, and Ramesh is $3 : 6 : 8$.

2. Increase their investments by the given percentages:
   - Suresh's new investment = $3 	imes 1.05 = 3.15$
   - Dinesh's new investment = $6 	imes 1.15 = 6.90$
   - Ramesh's new investment = $8 	imes 1.20 = 9.60$

3. New investment ratio is $3.15 : 6.90 : 9.60$.

4. To simplify, multiply all by 100 to remove decimals:
   $$315 : 690 : 960$$

5. Find the greatest common divisor (GCD) of 315, 690, and 960, which is 5:
   $$\frac{315}{5} : \frac{690}{5} : \frac{960}{5} = 63 : 138 : 192$$

6. Divide all terms by 3:
   $$\frac{63}{3} : \frac{138}{3} : \frac{192}{3} = 21 : 46 : 64$$

7. Since profit ratio is proportional to investment ratio for one year, the profit ratio is:
   $$21 : 46 : 64$$

**Answer:** $21 : 46 : 64$

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