Ratio of radii of touching circles

Let the radii of the two circles be $r_1$ and $r_2$ with $r_1 > r_2$. Sum of areas: $\pi r_1^2 + \pi r_2^2 = 153\pi \implies r_1^2 + r_2^2 = 153$

Sum of radii: $r_1 + r_2 = 15$

Squaring both sides of $r_1 + r_2 = 15$: $(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2 = 225$ $153 + 2r_1 r_2 = 225 \implies 2r_1 r_2 = 72 \implies r_1 r_2 = 36$

We need two numbers $r_1, r_2$ whose sum is 15 and product is 36. The numbers are 12 and 3.

Ratio of larger to smaller radius = $\frac{12}{3} = 4$.