Auf dem Planeten Schnell gibt es die Längeneinheiten LANG und KURZ sowie die Zeiteinheiten VIEL und WENIG.

Dabei gilt $1\text{ LANG}=65\text{ KURZ }1\backslash \; \backslash text\{LANG\}=65\backslash \; \backslash text\{KURZ\}\backslash$ und $1\text{ VIEL}=170\text{ WENIG }1\backslash text\{\; VIEL\}=170\backslash text\{\; WENIG\}\backslash$.

Rechne die Geschwindigkeit $234\frac{\mathrm{K}\mathrm{U}\mathrm{R}\mathrm{Z}}{\mathrm{W}\mathrm{E}\mathrm{N}\mathrm{I}\mathrm{G}}234\backslash frac\{\backslash mathrm\{KURZ\}\}\{\backslash mathrm\{WENIG\}\}$ in die Einheit $\frac{\mathrm{L}\mathrm{A}\mathrm{N}\mathrm{G}}{\mathrm{V}\mathrm{I}\mathrm{E}\mathrm{L}}\backslash frac\{\backslash mathrm\{LANG\}\}\{\backslash mathrm\{VIEL\}\}$ um.

$1\text{ LANG}=65\text{ KURZ}\Rightarrow 1\text{ KURZ }={\displaystyle \frac{\text{LANG}}{65}}1\backslash text\{\; LANG\}=65\backslash text\{\; KURZ\}\backslash quad\backslash Rightarrow\backslash \; \backslash \; 1\backslash text\{\; KURZ\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{LANG\}\}\{65\}$

$1\text{ VIEL}=170\text{ WENIG }\Rightarrow 1\text{ WENIG }={\displaystyle \frac{\text{VIEL}}{170}}1\backslash text\{\; VIEL\}\; =\; 170\backslash text\{\; WENIG\}\backslash \; \backslash Rightarrow\backslash \; 1\backslash text\{\; WENIG\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{VIEL\}\}\{170\}$

Umrechnen von $234\frac{\mathrm{K}\mathrm{U}\mathrm{R}\mathrm{Z}}{\mathrm{W}\mathrm{E}\mathrm{N}\mathrm{I}\mathrm{G}}234\backslash frac\{\backslash mathrm\{KURZ\}\}\{\backslash mathrm\{WENIG\}\}$ in $\frac{\mathrm{L}\mathrm{A}\mathrm{N}\mathrm{G}}{\mathrm{V}\mathrm{I}\mathrm{E}\mathrm{L}}\backslash frac\{\backslash mathrm\{LANG\}\}\{\backslash mathrm\{VIEL\}\}$; stetze für $\text{KURZ }={\displaystyle \frac{\text{LANG}}{65}}\backslash text\{KURZ\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{LANG\}\}\{65\}$ und für $\text{WENIG }={\displaystyle \frac{\text{VIEL}}{170}}\backslash text\{WENIG\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{VIEL\}\}\{170\}$

$234{\displaystyle \frac{\frac{\text{LANG}}{65}}{\frac{\text{VIEL}}{170}}}=234{\displaystyle \frac{170\text{ LANG}}{65\text{ VIEL}}}234\backslash dfrac\{\backslash frac\{\backslash text\{LANG\}\}\{65\}\}\{\backslash frac\{\backslash text\{VIEL\}\}\{170\}\}\backslash \; =\backslash \; 234\backslash dfrac\{170\backslash text\{\; LANG\}\}\{65\backslash text\{\; VIEL\}\}$

Damit du kürzen kannst, zerlege $234234$, $170170$ und $6565$:

$234=2\cdot 117=2\cdot 3\cdot 39=2\cdot 3\cdot 3\cdot 13234=2\backslash cdot\; 117=2\backslash cdot3\backslash cdot39=2\backslash cdot3\backslash cdot3\backslash cdot13$

$170=2\cdot 5\cdot 17170=2\backslash cdot5\backslash cdot17$

$65=5\cdot 1365=5\backslash cdot13$

$234\frac{\mathrm{K}\mathrm{U}\mathrm{R}\mathrm{Z}}{\mathrm{W}\mathrm{E}\mathrm{N}\mathrm{I}\mathrm{G}}=\frac{2\cdot 3\cdot 3\cdot 13\cdot 2\cdot 5\cdot 17}{5\cdot 13}\frac{\mathrm{L}\mathrm{A}\mathrm{N}\mathrm{G}}{\mathrm{V}\mathrm{I}\mathrm{E}\mathrm{L}}=2\cdot 2\cdot 3\cdot 3\cdot 17\frac{\mathrm{L}\mathrm{A}\mathrm{N}\mathrm{G}}{\mathrm{V}\mathrm{I}\mathrm{E}\mathrm{L}}=612\frac{\mathrm{L}\mathrm{A}\mathrm{N}\mathrm{G}}{\mathrm{V}\mathrm{I}\mathrm{E}\mathrm{L}}234\backslash frac\{\backslash mathrm\{KURZ\}\}\{\backslash mathrm\{WENIG\}\}\backslash \; =\backslash \; \backslash frac\{2\backslash cdot3\backslash cdot3\backslash cdot13\backslash cdot2\backslash cdot5\backslash cdot17\}\{5\backslash cdot13\}\backslash frac\{\backslash mathrm\{LANG\}\}\{\backslash mathrm\{VIEL\}\}\; \backslash \; =\; 2\backslash cdot2\backslash cdot3\backslash cdot3\backslash cdot17\backslash frac\{\backslash mathrm\{LANG\}\}\{\backslash mathrm\{VIEL\}\}\backslash \; =\; \backslash \; 612\backslash frac\{\backslash mathrm\{LANG\}\}\{\backslash mathrm\{VIEL\}\}$