How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$? I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\frac 35$ must be greater than $\frac 23$.
At this stage she can't calculate the decimal so that she can't realize that $(\frac 23 = 0.66$ and $\frac 35 = 0.6).$
She is $8$ years old.

A $3\times 5$ cake into fifteen $1\times1$ squares in the normal way. Two thirds is taking 10 pieces, three fifths is $9$ pieces.
This generalizes because if you have $p/q$ and $m/n$ (at least with $0<p<q$ and $0<m<n$ you cut a $m\times q$ cake into $mq$ squares. Then either take $p$ of the $q$ rows (for $pn$ pieces,) or $m$ of the $n$ rows, for $mq$ pieces. So $p/q$ is bigger than $m/n$ if and only if $pn$ is bigger than $mq$.
