Tips and tricks for speedy mental division with harder fractions

Question

I'm looking for a strategy to solve these kind of questions rapidly. Do you guys have any suggestions?

$$\frac{\frac{15}{25}}{X}=\frac{14}{35}.$$

A. $\frac{3}{5}$

B. $\frac{63}{30}$

C. $\frac{14}{15}$

D. $\frac{9}{6}$

The only thing I can think of, is to look at $$ \frac{14}{35}.$$ and notice that the denominator has increased compared to the numerator, hence I can rule out A and C. Leaving only 2 options.

Answer 1

Strategy? well, I swing the x in the denominator over and flip the coefficient back.

$\frac{\frac{15}{25}}{X}=\frac{14}{35}.$

$\frac{15}{25} =\frac{14}{35}X $

$\frac{15}{25}\frac{35}{14} =X $

$\frac{15}{5}\frac{7}{14} =X $

$\frac{3}{1}\frac{1}{2} =X $

$\frac{3}{2} =X $

Answer 2

Write it as $X=\frac {35}{14}\cdot \frac {15}{25}$ and cancel obvious factors. Do you recognize the factor $7$ in the first one?