A liquid limit test, conducted on a soil sample in the Casagrande cup

device, gave the following results:

Two determinations for the plastic limit gave water contents of $20\mathrm{.}3\%$ and

$20\mathrm{.}8\%.$ Determine the following:

a$)$ The liquid limit and the plastic limit

b$)$ The plasticity index

c$)$ The liquidity index if the natural water content is $27\mathrm{.}4\%$

d$)$ The void ratio at the liquid limit if Gs$=2\mathrm{.}7$

e$)$ The shrinkage limit

Use the following relationship for the calculation of the shrinkage limit:

$SL=46\mathrm{.}4(\frac{LL+45\mathrm{.}5}{\frac{?}{PI}+46\mathrm{.}4})-43\mathrm{.}5$