Question

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?

Answer 1

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,
$Area\ of\ Joseph's\ cookie = \pi (2.5)^2\\\\ Area\ of\ Joseph's\ cookie = 19.63cm^2\\\\Now,\\\\Area\ of\ Ellen's\ cookie = \pi (5)^2\\\\ Area\ of\ Joseph's\ cookie =78.54cm^2\\\\Now,\\\\ \frac{1}{2}\ of \ Joseph's \ cookie = \frac{1}{2} *19.63=9.82cm^2\\\\ \frac{1}{4} \ of\ Ellen's\ cookie= \frac{1}{4} *78.54=19.64cm^2$

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