If $5A=13B=7C$, find the value of $A:B:C$.

Given:
$$5A=13B=7C=k$$
Therefore,
$$A=\frac{k}{5},B=\frac{k}{13},C=\frac{k}{7}$$
The ratio $A:B:C$ is:
$$A:B:C=\frac{1}{5}:\frac{1}{13}:\frac{1}{7}$$
To convert to integers, multiply each term by the LCM of $5,13,\text{ and }7$, which is $5\times 13\times 7=455$:
$$A=\frac{455}{5}=91$$
$$B=\frac{455}{13}=35$$
$$C=\frac{455}{7}=65$$
Thus, $A:B:C=91:35:65$.