Step 1: Express the numbers using a common multiplier

Since the two numbers are given in the ratio $2: 3$, we can let the numbers be $2x$ and $3x$, where $x$ is the HCF of the numbers.

$2x \text{ and } 3x$

Step 2: Relate LCM with the algebraic numbers

The LCM of two numbers $2x$ and $3x$ is given by the product of their ratio parts and the common factor $x$: $\text{LCM} = 2 \times 3 \times x = 6x$

$6x = 84$

Step 3: Solve for the common multiplier $x$

Equating $6x$ to the given LCM value of 84: $x = \frac{84}{6} = 14$

$x = 14$

Step 4: Find the sum of the numbers

The sum of the two numbers is $(2x + 3x) = 5x$. Substitute $x = 14$: $\text{Sum} = 5 \times 14 = 70$

$\text{Sum} = 70$