Aufgaben zur Multiplikation und Division von Dezimalbrüchen

Kommas "wegdenken", schriftlich multiplizieren, Anzahl der Nachkommastellen zählen und im Ergebnis das Komma mit $1+0=11+0=1$ Nachkommastellen setzen.

$\begin{array}{c}\underset{‾}{\mathrm{2,5}\cdot 10}\\ \phantom{200}000\\ \underset{‾}{\phantom{\mathrm{2,0}}250}\\ \phantom{20}\mathrm{25,0}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash underline\{2\{,\}5\backslash cdot10\}\backslash \backslash \backslash hphantom\{200\}000\backslash \backslash \backslash underline\{\backslash hphantom\{2\{,\}0\}250\}\backslash \backslash \backslash hphantom\{20\backslash ,\}25\{,\}0\backslash end\{array\}$

$\mathrm{25,0}=2525\{,\}0=25$

$\mathrm{2,5}\cdot 102\{,\}5\backslash cdot10$

$\begin{array}{c}\underset{‾}{\mathrm{2,5}\cdot \mathrm{0,1}}\\ 25\\ \underset{‾}{\phantom{\mathrm{2,}\cdot ,}000}\\ \mathrm{0,25}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{r\}\backslash underline\{2\{,\}5\backslash cdot0\{,\}1\}\backslash \backslash 25\backslash \backslash \backslash underline\{\backslash hphantom\{2\{,\}\backslash cdot\{,\}\}000\}\backslash \backslash 0\{,\}25\backslash end\{array\}$

$\mathrm{2,5}\cdot \mathrm{0,1}2\{,\}5\backslash cdot\; 0\{,\}1$

Kommas "wegdenken", schriftlich multiplizieren, Anzahl der Nachkommastellen zählen und im Ergebnis das Komma mit $1+1=21+1=2$ Nachkommastellen setzen.

$\begin{array}{c}\underset{‾}{\mathrm{5,1}\cdot \mathrm{2,9}}\\ 459\\ \underset{‾}{\phantom{,\cdot ,}1020}\\ \mathrm{14,79}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{r\}\backslash underline\{5\{,\}1\backslash cdot2\{,\}9\}\backslash \backslash 459\backslash \backslash \backslash underline\{\backslash hphantom\{\{,\}\backslash cdot\{,\}\}1020\}\backslash \backslash 14\{,\}79\backslash end\{array\}$

Kommas "wegdenken", schriftlich multiplizieren, Nachkommastellen zählen und im Ergebnis das Komma mit $2+3=52+3=5$ Nachkommastellen setzen.

$\begin{array}{c}\underset{‾}{\mathrm{2,45}\cdot \mathrm{0,671}}\\ 245\\ 17150\\ \underset{‾}{\phantom{\mathrm{2,}\cdot ,}147000}\\ \mathrm{1,64395}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{r\}\backslash underline\{2\{,\}45\backslash cdot0\{,\}671\}\backslash \backslash 245\backslash \backslash 17150\backslash \backslash \backslash underline\{\backslash hphantom\{2\{,\}\backslash cdot\{,\}\}147000\}\backslash \backslash 1\{,\}64395\backslash end\{array\}$

$\begin{array}{c}\underset{‾}{9\cdot \mathrm{0,686}}\\ 54\\ 720\\ \underset{‾}{\phantom{9\cdot ,}5400}\\ \mathrm{6,174}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{r\}\backslash underline\{9\backslash cdot0\{,\}686\}\backslash \backslash 54\backslash \backslash 720\backslash \backslash \backslash underline\{\backslash hphantom\{9\backslash cdot\{,\}\}5400\}\backslash \backslash 6\{,\}174\backslash end\{array\}$

Kommas "wegdenken", schriftlich multiplizieren, Nachkommastellen zählen und im Ergebnis das Komma mit $1+0=11+0=1$ Nachkommastellen setzen.

$\begin{array}{c}\underset{‾}{\mathrm{2,5}\cdot 100}\\ 00\\ 000\\ \underset{‾}{\phantom{2\cdot ,}2500}\\ \mathrm{250,0}\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{r\}\backslash underline\{2\{,\}5\backslash cdot100\}\backslash \backslash 00\backslash \backslash 000\backslash \backslash \backslash underline\{\backslash hphantom\{2\backslash cdot\{,\}\}2500\}\backslash \backslash 250\{,\}0\backslash end\{array\}$

$\mathrm{2,5}\cdot 100=\mathrm{2,50}\cdot 100=2502\{,\}5\; \backslash cdot\; 100\; =\; 2\{,\}50\; \backslash cdot\; 100\; =\; 250$

Merke: Bei der Multiplikation mit $100100$ wird das Komma um zwei Stellen nach rechts verschoben.

$\mathrm{8,45}:100=\mathrm{0,0845}8\{,\}45:100=0\{,\}0845$

Teilt man einen Dezimalbruch durch 100, so verschiebt sich das Komma um 2 Stellen nach links.

$\mathrm{8,45}:100=845:100008\{,\}45:100=845:10000$

Multipliziert man den Dividenden und den Divisor mit 100, so ändert sich der Wert der Division nicht. Dadurch wird das Komma bei beiden um zwei Stellen nach rechts verschoben.

$\begin{array}{c}\phantom{-}845:10000=\mathrm{0,0845}\\ \underset{‾}{-\phantom{84}0}\\ \phantom{-}8450\\ \underset{‾}{-\phantom{845}0}\\ \phantom{-}84500\\ \underset{‾}{-80000}\\ \phantom{-0}45000\\ \phantom{0}\underset{‾}{-40000}\\ \phantom{-00}50000\\ \phantom{00}\underset{‾}{-50000}\\ \phantom{-000000}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}845:10000=0\{,\}0845\backslash \backslash \backslash underline\{-\backslash hphantom\{84\}0\}\backslash \backslash \backslash hphantom\{-\}8450\backslash \backslash \backslash underline\{-\backslash hphantom\{845\}0\}\backslash \backslash \backslash hphantom\{-\}84500\backslash \backslash \backslash underline\{-80000\}\backslash \backslash \backslash hphantom\{-0\}45000\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-40000\}\backslash \backslash \backslash hphantom\{-00\}50000\backslash \backslash \backslash hphantom\{00\}\backslash underline\{-50000\}\backslash \backslash \backslash hphantom\{-000000\}0\backslash end\{array\}$

Benutze die schriftliche Division.

$16:\mathrm{0,25}=1600:2516:0\{,\}25=1600:25$

Multipliziert man den Divisor und den Dividenden mit 100 (Komma um zwei Stellen nach rechts verschieben), ändert das den Wert der Division nicht.

$\begin{array}{c}\phantom{-}1600:25=64\\ \underset{‾}{-150}\\ \phantom{-1}100\\ \phantom{0}\underset{‾}{-100}\\ \phantom{-000}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}1600:25=64\backslash \backslash \backslash underline\{-150\}\backslash \backslash \backslash hphantom\{-1\}100\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-100\}\backslash \backslash \backslash hphantom\{-000\}0\backslash end\{array\}$

Benutze die schriftliche Division.

$\mathrm{8,5}:\mathrm{0,160}=\mathrm{8,5}:\mathrm{0,16}8\{,\}5\; :\; 0\{,\}160\; =\; 8\{,\}5\; :\; 0\{,\}16$

Schreibe den Divisor als 0,16 statt 0,160. Das Weglassen von Nullen am Ende eines Dezimalbruchs ändert nichts an dessen Wert.

$\mathrm{8,5}:\mathrm{0,16}=850:168\{,\}5:0\{,\}16\; =\; 850\; :\; 16$

Multipliziere sowohl den Dividenden als auch den Divisor jeweils mit 100, d. h., verschiebe das Komma um 2 Stellen nach rechts. Durch Multiplikation sowohl des Divisors als auch des Dividenden mit der gleichen Zahl ändert sich der Wert der Division nicht.

$\begin{array}{c}\phantom{-}850:16=\mathrm{53,125}\\ \underset{‾}{-80}\\ \phantom{-1}50\\ \phantom{0}\underset{‾}{-48}\\ \phantom{-00}20\\ \phantom{00}\underset{‾}{-16}\\ \phantom{-000}40\\ \phantom{000}\underset{‾}{-32}\\ \phantom{-0000}80\\ \phantom{0000}\underset{‾}{-80}\\ \phantom{-00000}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}850:16=53\{,\}125\backslash \backslash \backslash underline\{-80\}\backslash \backslash \backslash hphantom\{-1\}50\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-48\}\backslash \backslash \backslash hphantom\{-00\}20\backslash \backslash \backslash hphantom\{00\}\backslash underline\{-16\}\backslash \backslash \backslash hphantom\{-000\}40\backslash \backslash \backslash hphantom\{000\}\backslash underline\{-32\}\backslash \backslash \backslash hphantom\{-0000\}80\backslash \backslash \backslash hphantom\{0000\}\backslash underline\{-80\}\backslash \backslash \backslash hphantom\{-00000\}0\backslash end\{array\}$

Benutze die schriftliche Division.

$\mathrm{0,125}:\mathrm{0,5}=125:5000\{,\}125:0\{,\}5=125:500$

Multipliziere Dividend und Divisor jeweils mit 1000, d. h., verschiebe beide Kommas um drei Stellen nach rechts. Der Wert der Division ändert sich nicht, wenn Dividend und Divisor mit der gleichen Zahl multipliziert werden.

$\begin{array}{c}\phantom{-}125:500=\mathrm{0,25}\\ \underset{‾}{-\phantom{12}0}\\ \phantom{-}1250\\ \underset{‾}{-1000}\\ \phantom{-0}2500\\ \phantom{0}\underset{‾}{-2500}\\ \phantom{-0000}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}125:500=0\{,\}25\backslash \backslash \backslash underline\{-\backslash hphantom\{12\}0\}\backslash \backslash \backslash hphantom\{-\}1250\backslash \backslash \backslash underline\{-1000\}\backslash \backslash \backslash hphantom\{-0\}2500\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-2500\}\backslash \backslash \backslash hphantom\{-0000\}0\backslash end\{array\}$

Benutze die schriftliche Division.

$\mathrm{8,45}:\mathrm{0,01}=8458\{,\}45\; :\; 0\{,\}01\; =\; 845$

Teilt man einen Dezimalbruch durch 0,01, so verschiebt sich das Komma um 2 Stellen nach rechts.

$\mathrm{8,45}:\mathrm{0,01}=845:18\{,\}45\; :\; 0\{,\}01\; =\; 845\; :\; 1$

Multipliziert man den Dividenden und den Divisor mit 100, so ändert sich das Ergebnis nicht. Dafür verschiebt sich das Komma bei beiden um zwei Stellen nach rechts.

$\begin{array}{c}\phantom{-}\mathrm{8,1}:9=\mathrm{0,9}\\ \underset{‾}{-0}\\ \phantom{-}81\\ \underset{‾}{-81}\\ \phantom{-8}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}8\{,\}1:9=0\{,\}9\backslash \backslash \backslash underline\{-0\}\backslash \backslash \backslash hphantom\{-\}81\backslash \backslash \backslash underline\{-81\}\backslash \backslash \backslash hphantom\{-8\}0\backslash end\{array\}$

$\begin{array}{c}\phantom{-}\mathrm{7,5}:2=\mathrm{3,75}\\ \underset{‾}{-6}\\ \phantom{-}15\\ \underset{‾}{-14}\\ \phantom{-1}10\\ \phantom{0}\underset{‾}{-10}\\ \phantom{-10}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}7\{,\}5:2=3\{,\}75\backslash \backslash \backslash underline\{-6\}\backslash \backslash \backslash hphantom\{-\}15\backslash \backslash \backslash underline\{-14\}\backslash \backslash \backslash hphantom\{-1\}10\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-10\}\backslash \backslash \backslash hphantom\{-10\}0\backslash end\{array\}$

$\begin{array}{c}\phantom{-}\mathrm{29,48}:11=\mathrm{2,68}\\ \underset{‾}{-22}\\ \phantom{-2}74\\ \phantom{0}\underset{‾}{-66}\\ \phantom{-66}88\\ \phantom{88}\underset{‾}{-88}\\ \phantom{-888}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}29\{,\}48:11=2\{,\}68\backslash \backslash \backslash underline\{-22\}\backslash \backslash \backslash hphantom\{-2\}74\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-66\}\backslash \backslash \backslash hphantom\{-66\}88\backslash \backslash \backslash hphantom\{88\}\backslash underline\{-88\}\backslash \backslash \backslash hphantom\{-888\}0\backslash end\{array\}$

Benutze die schriftliche Division.

Sobald du die $88$ hinter dem Komma "runternimmst", musst du ein Komma im Ergebnis setzen.

$\begin{array}{c}\phantom{-}\mathrm{13,811}:7=\mathrm{1,973}\\ \phantom{0}\underset{‾}{-7}\\ \phantom{-0}68\\ \phantom{0}\underset{‾}{-63}\\ \phantom{-00}51\\ \phantom{00}\underset{‾}{-49}\\ \phantom{-000}21\\ \phantom{000}\underset{‾}{-21}\\ \phantom{-0000}0\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash hphantom\{-\}13\{,\}811:7=1\{,\}973\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-7\}\backslash \backslash \backslash hphantom\{-0\}68\backslash \backslash \backslash hphantom\{0\}\backslash underline\{-63\}\backslash \backslash \backslash hphantom\{-00\}51\backslash \backslash \backslash hphantom\{00\}\backslash underline\{-49\}\backslash \backslash \backslash hphantom\{-000\}21\backslash \backslash \backslash hphantom\{000\}\backslash underline\{-21\}\backslash \backslash \backslash hphantom\{-0000\}0\backslash end\{array\}$