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 $x$ represents their Highest Common Factor (HCF).
Since the coefficients 3, 4, and 5 are mutually coprime, their combined LCM is:

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

Given that the LCM is $1200$:

$$60x=1200⟹x=\frac{1200}{60}=20$$

The smallest number among them is $3x$:

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