If a principal P amounts to A in two years when compounded half yearly with r% interest. The same principal P amounts to A in two years when compounded annually with R% interest, then which of the following relationship is true?

In the first case, we are told that at a rate r, it is compounding semi-annually. 
$$P\left(1+\frac{r}{200}\right)^4=A$$

In the second case, we are told that a rate R is compounded annually. 
$$P\left(1+\frac{R}{100}\right)^2=A$$

Equating them, we get, 
$$\left(1+\frac{R}{100}\right)^2=\left(1+\frac{r}{200}\right)^4$$

$$1+\frac{R}{100}=\left(1+\frac{r}{200}\right)^2$$

Taking R/100=R and r/100 as r for ease of calculation,  

$$\left(1+R\right)=\left(1+\frac{r}{2}\right)^2$$

$$1+R=1+\frac{r^2}{4}+r$$

$$R=r+\frac{r^2}{4}$$

So, R will naturally be greater than r.