Question

If $n \cdot 1 \cdot \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{6} \cdot \frac{1}{8} \cdot \frac{1}{10}$, what is the value of $n$? Express your answer as a common fraction.

Answer 1

$n\cdot1\cdot\dfrac12\cdot\dfrac13\cdot\dfrac14\cdot\dfrac15=\dfrac n{5!}$

$\dfrac12\cdot\dfrac14\cdot\dfrac16\cdot\dfrac18\cdot\dfrac1{10}=\dfrac{3\cdot5\cdot7\cdot9}{10!}$

So we have

$\dfrac n{5!}=\dfrac{3\cdot5\cdot7\cdot9}{10!}$

$n=\dfrac{3\cdot5\cdot7\cdot9}{6\cdot7\cdot8\cdot9\cdot10}$

$n=\dfrac{3\cdot5}{6\cdot8\cdot10}$

$n=\dfrac1{2\cdot8\cdot2}$

$n=\dfrac1{32}$