If the ratio of three numbers are $3:4:5$ and their LCM is $1200$, then the smallest number is:

Let the three numbers be $3x$, $4x$, and $5x$, where $x$ is a common positive multiplier.
Since $3$, $4$, and $5$ are mutually co-prime integers, their relative $\text{LCM}$ component is simply their product:

$$\text{LCM}(3x,4x,5x)=3\times 4\times 5\times x=60x$$

We are given that the $\text{LCM}$ is $1200$:

$$60x=1200⟹x=\frac{1200}{60}=20$$

The smallest number among the three is $3x$:

$$\text{Smallest Number}=3\times 20=60$$