The phase diagrams for three binary systems, A$-$B$,$ B$-$C and C$-$A$,$ all exhibit no solid solubility. In each binary is a simple eutectic, which extends into the ternary to the invariant ternary eutectic temperature of $400C.$ The liquidus isotherms of the ternary is shown below for this system. Starting with the total composition of $70\%A,10\%B$ and $20\%C$ :

a$.$ Locate the overall composition point $P.$

b$.$ Upon cooling from $1000C,$ at what temperature will solidification begin?

c$.$ What is the composition of the first solid to form?

d$.$ Trace the path in the diagram followed by the liquid during equilibrium solidification.

e$.$ What is the composition of the last material to solidify?

f$.$ If a material that is $20\%$A and $15\%B$ is heated from $300C,$ at what temperature will melting first occur?

g$.$ Consider the microstructural evolution during equilibrium solidification in this ternary system. Identify three different ways where this evolution would be characteristically different.

h$.$ How would the solidification path for Part d change if there is no time for any solid diffusion $($i$.$e$.,$ the Scheil case$).$