Write down the given ratios

Identify the given ratios: $A:B = 3:4$ and $B:C = 5:6$.

$A:B = 3:4, \quad B:C = 5:6$

Apply the chain rule for ratios

To find the ratio $A:C$, multiply the fraction forms of the two ratios: $\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}$.

$\frac{A}{C} = \frac{3}{4} \times \frac{5}{6}$

Simplify the expression

Multiply the numerators and denominators, then simplify by canceling common factors: $\frac{3 \times 5}{4 \times 6} = \frac{15}{24} = \frac{5}{8}$.

$\frac{A}{C} = \frac{5}{8}$