If $7A=5B=2C$, find $A:B:C$.

Given equation:
$$7A=5B=2C$$
Let $7A=5B=2C=k$.From this, we can express $A$, $B$, and $C$ in terms of $k$:
$$A=\frac{k}{7}$$
$$B=\frac{k}{5}$$
$$C=\frac{k}{2}$$
Now, find the ratio $A:B:C$:
$$A:B:C=\frac{k}{7}:\frac{k}{5}:\frac{k}{2}=\frac{1}{7}:\frac{1}{5}:\frac{1}{2}$$
To eliminate the fractions, multiply each term by the Least Common Multiple (LCM) of the denominators (7, 5, and 2).
$$\text{LCM}(7,5,2)=70$$
Multiplying each term by 70:
$$A:B:C=\left(\frac{1}{7}\times 70\right):\left(\frac{1}{5}\times 70\right):\left(\frac{1}{2}\times 70\right)$$
$$A:B:C=10:14:35$$