a x a/b Trick

When multiplying an integer $a$ by a fraction of the form $\frac{a}{b}$, you can use the formula:

$a \times \frac{a}{b} = [a + (a - b)] + \frac{(a - b)^2}{b}$

If the fraction is less than 1, you subtract the difference $(a-b)$ from $a$. If the fraction is greater than 1, you add it.

For $11 \times \frac{11}{13}$: $a=11, b=13, a-b=-2$. The result is $[11 + (-2)] + \frac{(-2)^2}{13} = 9\frac{4}{13}$.

For $13 \times \frac{13}{12}$: $a=13, b=12, a-b=1$. The result is $[13 + 1] + \frac{1^2}{12} = 14\frac{1}{12}$.

Sometimes it is simpler to just convert to an improper fraction and solve directly.