For a $0\mathrm{.}3$ m$3$ soil sample, moist mass is measured to be equal to $500$ g$.$ The sample

weighs $400$ g when oven dried. For the same soil, water contents at liquid and plastic

limits are known to be equal to $55$ and $15\%,$ respectively. Calculate the plasticity

index $($PI$)$ and liquidity index $($LI$)$ for this soil. For the same sample, the clay fraction

$($by weight$)$ determined from a hydrometer analysis is equal to $20\%.$ Calculate the

activity number.