Given the following weight$-$fraction chain length distribution, calculate i$)$

number$-$fraction chain length distribution, ii$)$ weight$-$average chain length,

iii$)$ number$-$average chain length, and iv$)$ polydispersity. $($please show

your calculations$).$ Recall that $W(r)=\frac{rN(r)}{{\displaystyle \underset{r=1}{\overset{}{}}}rN(r)}$ and $N(r)=\frac{\frac{W(r)}{r}}{{\displaystyle \underset{r=1}{\overset{}{}}}\frac{W(r)}{r}}$