Top Times Top

$\frac{2}{5}\cdot \frac{3}{7}$: take a rectangle and cut it into 5 columns and 7 rows, 35 small pieces in all. Shade 2 of the 5 columns, which is $\frac{2}{5}$, and 3 of the 7 rows, which is $\frac{3}{7}$. Where the two shadings overlap is $\frac{2}{5}$ of $\frac{3}{7}$.

2 columns of 5 and 3 rows of 7 overlap in 6 of the 35 pieces.

The overlap is 2 columns by 3 rows, $2\cdot 3$ pieces, which is 6, out of $5\cdot 7$, which is 35. So $\frac{2}{5}\cdot \frac{3}{7}$ is $\frac{6}{35}$: multiply the tops, and multiply the bottoms.

Unlike adding, multiplying needs no common denominator. The bottoms multiply because cutting fifths into sevenths makes pieces 35 times smaller than the whole. And a product of two positive fractions below 1 is smaller than both of them, because you are taking a part of a part.