If the sum of the diameters of two circles is 30 cm and the difference of their radii is 5 cm, then the ratio of the area of the smaller circle to that of the bigger circle is:

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Let:  
- $$r_1$$ = radius of the bigger circle,  
- $$r_2$$ = radius of the smaller circle.

From the problem:  
1. Sum of diameters: $$2r_1 + 2r_2 = 30 \implies r_1 + r_2 = 15$$.  
2. Difference of radii: $$r_1 - r_2 = 5$$.

**Step 1:** Solve for $$r_1$$ and $$r_2$$:  
Adding equations:  
$$
(r_1 + r_2) + (r_1 - r_2) = 15 + 5 \implies 2r_1 = 20 \implies r_1 = 10.
$$  
Substitute into $$r_1 + r_2 = 15$$:  
$$
10 + r_2 = 15 \implies r_2 = 5.
$$

**Step 2:** Calculate ratio of areas:  
Area ratio = $$\frac{\pi r_2^2}{\pi r_1^2} = \frac{r_2^2}{r_1^2} = \frac{5^2}{10^2} = \frac{25}{100} = \frac{1}{4}.$$

Thus, the ratio is **1 : 4**, matching **option (d)**.

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