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Теория: Десятичная дробь как частное при делении натуральных чисел

Задание

Найдите частное:

\(\displaystyle 3:16=\),

Решение

Поделим \(\displaystyle 3\) на \(\displaystyle 16\) в столбик:

 

\(\displaystyle 3\)\(\displaystyle 1\)\(\displaystyle 6\)
 \(\displaystyle ?\) 

 

Первое действие.

Так как \(\displaystyle 3<16\) и больше разрядов у числа нет, то пишем нуль первой цифрой в частном и ставим запятую после него:

 

\(\displaystyle 3\)\(\displaystyle 1\)\(\displaystyle 6\) 
 \(\displaystyle \color{blue}{0}\)\(\displaystyle \color{red}{,}\)\(\displaystyle ?\)

 

и добавляем нуль к делимому справа (то есть к \(\displaystyle 3\) справа):

 

\(\displaystyle 3\)\(\displaystyle \color{red}{0}\)\(\displaystyle 1\)\(\displaystyle 6\) 
  \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle ?\)

 

Второе действие.

Делим \(\displaystyle 30\) на \(\displaystyle 16\):

\(\displaystyle 30= \color{green}{1}·16+14=\color{blue}{16}+14\).

 

пишем \(\displaystyle \color{green}{1}\) в частном  и вычитаем \(\displaystyle 30-\color{blue}{16}=\bf14\):

 

 \(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 1\)\(\displaystyle 6\)  
\(\displaystyle -\)  \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle \color{green}{1}\)\(\displaystyle ?\)
 \(\displaystyle \color{blue}{1}\)\(\displaystyle \color{blue}{6}\)    
 \(\displaystyle \bf1\)\(\displaystyle \bf4\)    

 

Третье действие.

Добавляем нуль к разности справа:

 

 \(\displaystyle 3\)\(\displaystyle 0\) \(\displaystyle 1\)\(\displaystyle 6\)  
\(\displaystyle -\)   \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle ?\)
 \(\displaystyle 1\)\(\displaystyle 6\)     
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle \color{red}{0}\)    

 

Делим \(\displaystyle 140\) на \(\displaystyle 32\):

 

\(\displaystyle 140= \color{green}{8}·16+12=\color{blue}{128}+12\).

 

Пишем \(\displaystyle \color{green}{8}\) в частном и вычитаем \(\displaystyle 140-\color{blue}{128}=\bf12\):

 

 \(\displaystyle 3\)\(\displaystyle 0\) \(\displaystyle 1\)\(\displaystyle 6\)   
\(\displaystyle -\)   \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle \color{green}{8}\)\(\displaystyle ?\)
 \(\displaystyle 1\)\(\displaystyle 6\)      
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle 0\)     
\(\displaystyle -\)        
 \(\displaystyle \color{blue}{1}\)\(\displaystyle \color{blue}{2}\)\(\displaystyle \color{blue}{8}\)     
  \(\displaystyle \bf1\)\(\displaystyle \bf2\)     

 

Четвертое действие.

Добавляем нуль к разности справа:

 

 \(\displaystyle 3\)\(\displaystyle 0\)  \(\displaystyle 1\)\(\displaystyle 6\)   
\(\displaystyle -\)    \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle ?\)
 \(\displaystyle 1\)\(\displaystyle 6\)       
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle 0\)      
\(\displaystyle -\)         
 \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 8\)      
  \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle \color{red}{0}\)    

 

Делим \(\displaystyle 120\) на \(\displaystyle 16\):

 

\(\displaystyle 120= \color{green}{7}·16+16=\color{blue}{112}+8\).

 

Пишем \(\displaystyle \color{green}{7}\) в частном и вычитаем \(\displaystyle 120-\color{blue}{112}=\bf8\):

 

 \(\displaystyle 3\)\(\displaystyle 0\)  \(\displaystyle 1\)\(\displaystyle 6\)    
\(\displaystyle -\)    \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle \color{green}{7}\)\(\displaystyle ?\)
 \(\displaystyle 1\)\(\displaystyle 6\)        
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle 0\)       
\(\displaystyle -\)          
 \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 8\)       
  \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 0\)      
 \(\displaystyle -\)         
  \(\displaystyle \color{blue}{1}\)\(\displaystyle \color{blue}{1}\)\(\displaystyle \color{blue}{2}\)      
    \(\displaystyle \bf8\)      

 

Пятое действие.

Добавляем нуль к разности справа:

 

 \(\displaystyle 3\)\(\displaystyle 0\)   \(\displaystyle 1\)\(\displaystyle 6\)    
\(\displaystyle -\)     \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle 7\)\(\displaystyle ?\)
 \(\displaystyle 1\)\(\displaystyle 6\)         
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle 0\)        
\(\displaystyle -\)           
 \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 8\)        
  \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 0\)       
 \(\displaystyle -\)          
  \(\displaystyle 1\)\(\displaystyle 1\)\(\displaystyle 2\)       
    \(\displaystyle 8\)\(\displaystyle \color{red}{0}\)      

 

Делим \(\displaystyle 80\) на \(\displaystyle 16\):

 

\(\displaystyle 80= \color{green}{5}·16=\color{blue}{80}\).

 

Пишем \(\displaystyle \color{green}{5}\) в частном и вычитаем \(\displaystyle 80-\color{blue}{80}=\bf0\):

 

 \(\displaystyle 3\)\(\displaystyle 0\)   \(\displaystyle 1\)\(\displaystyle 6\)    
\(\displaystyle -\)     \(\displaystyle 0\)\(\displaystyle ,\)\(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle 7\)\(\displaystyle \color{green}{5}\)
 \(\displaystyle 1\)\(\displaystyle 6\)         
 \(\displaystyle 1\)\(\displaystyle 4\)\(\displaystyle 0\)        
\(\displaystyle -\)           
 \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 8\)        
  \(\displaystyle 1\)\(\displaystyle 2\)\(\displaystyle 0\)       
 \(\displaystyle -\)          
  \(\displaystyle 1\)\(\displaystyle 1\)\(\displaystyle 2\)       
    \(\displaystyle 8\)\(\displaystyle 0\)      
   \(\displaystyle -\)        
    \(\displaystyle \color{blue}{8}\)\(\displaystyle \color{blue}{0}\)      
     \(\displaystyle \bf0\)      

 

Таким образом, \(\displaystyle 3:16=0,1875\).

Ответ: \(\displaystyle 0,1875\).