The present ages of Sweet and Suman are in the ratio $4 : 5$. Six years back, the ratio of their ages was $2 : 3$. What is the present age of Sweet?

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**Solution:**

1. Let Sweet's present age be $4x$ and Suman's present age be $5x$.

2. Six years ago, their ages were:
   - Sweet: $4x - 6$
   - Suman: $5x - 6$

3. According to the problem, the ratio of their ages six years ago was $2 : 3$, so:
   $$\frac{4x - 6}{5x - 6} = \frac{2}{3}$$

4. Cross-multiply to solve for $x$:
   $$3(4x - 6) = 2(5x - 6)$$
   $$12x - 18 = 10x - 12$$

5. Rearranging terms:
   $$12x - 10x = -12 + 18$$
   $$2x = 6$$
   $$x = 3$$

6. Calculate Sweet's present age:
   $$4x = 4 	imes 3 = 12 	ext{ years}$$

**Answer:** The present age of Sweet is $12$ years.

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