Joseph had half a cookie. ellen had a quarter of a different cookie. the peace joseph had was smaller. how would ellen's peace be larger?

It is possible in some cases.

Let us assume that the cookie is circular. And Joseph has a cookie that has a radius of 2.5 cm and Ellen has  a cookie that has a radius of 5 cm. 

Now,
lets find the area of both cookies.

So,


As you can see, half of Joseph's cookie is less than a quarter of Ellen's cookie.