Let the three parts be $A$, $B$, and $C$.

Step 1: Set Up Relationships Between Parts

1. Given that $A$ gets $\frac{5}{4}$ of $B$, we can write: $A = \frac{5}{4}B$

2. The ratio between $B$ and $C$ is given as $3:4$. So, we can express $C$ in terms of $B$: $\frac{B}{C} = \frac{3}{4} \implies C = \frac{4}{3}B$

Step 2: Express Total Sum

The total sum of the parts is given as $430$: $A + B + C = 430$

Substituting the expressions for $A$ and $C$ in terms of $B$: $\frac{5}{4}B + B + \frac{4}{3}B = 430$

Step 3: Simplify the Equation

Combine the terms by finding a common denominator. The common denominator of 4 and 3 is 12.

1. Convert each term: $\frac{5}{4}B = \frac{15}{12}B, \quad B = \frac{12}{12}B, \quad \frac{4}{3}B = \frac{16}{12}B$

2. Adding them: $\frac{15}{12}B + \frac{12}{12}B + \frac{16}{12}B = \frac{43}{12}B$

Thus, the equation becomes: $\frac{43}{12}B = 430$

Step 4: Solve for $B$

Multiply both sides by