Упр.473 Часть 2 ГДЗ Дорофеев Петерсон 5 класс (Математика)

Задача

а) (1 5/9 : 7 + 1 5/6)/(6 1/6 · 3);
(1 5/9 :7+1 5/6)/(6 1/6•3)=((1•9+5)/9 :7+1 5/6)/((6•6+1)/6•3)=(14/(9•7)+1 5/6)/(37/6•3)=((2•7)/(9•7)+1 5/6)/((37•3)/(2•3))=((2•2)/(9•2)+1 (5•3)/(6•3))/(37/2)=(4/18+1 15/18)/(37/2)=
=(1+(4+15)/18)/(37/2)=(1+19/18)/(37/2)=(1+1 1/18)/(37/2)=(2 1/18)/(37/2)=((2•18+1)/18)/(37/2)=(37/18)/(37/2)=37/18 :37/2=37/18•2/37=
=(37•2)/(18•37)=2/(2•9)=1/9
б) (2 3/7 · 1 5/11 + 2 3/7 · 2 6/11)/(2 3/7 · 1 5/11);
(2 3/7•1 5/11+2 3/7•2 6/11)/(2 3/7•1 5/11)=(2 3/7•(1 5/11+2 6/11))/(2 3/7•1 5/11)=(1+2+(5+6)/11)/((1•11+5)/11)=(3+11/11)/(16/11)=(3+1)/(16/11)=4/(16/11)=
=4:16/11=4•11/16=(4•11)/16=(4•11)/(4•4)=2 3/4
в) (1+1/(1+1/(1+1/2)))/(2+1/(2+1/2)).
(1+1/(1+1/(1+1/2)))/(2+1/(2+1/2))=(1+1/(1+1/(1 1/2)))/(2+1/(2 1/2))=(1+1/(1+1/((1•2+1)/2)))/(2+1/((2•2+1)/2))=(1+1/(1+1/(3/2)))/(2+1/(5/2))=(1+1/(1+1:3/2))/(2+1:5/2)=(1+1/(1+1•2/3))/(2+1•2/5)=(1+1/(1+2/3))/(2+2/5)=
=(1+1/(1 2/3))/(2 2/5)=(1+1:1 2/3)/(2 2/5)=(1+1:(1•3+2)/3)/((2•5+2)/5)=(1+1:5/3)/((10+2)/5)=(1+1•3/5)/(12/5)=(1+3/5)/(12/5)=(1 3/5)/(12/5)=((1•5+3)/5)/(12/5)=
=(8/5)/(12/5)=8/5 :12/5=8/5•5/12=(8•5)/(5•12)=(2•4)/(3•4)=2/3

Подробный ответ

а) Преобразуем смешанные числа в неправильные дроби и выполним действия по порядку:

$$ \frac{1\frac{5}{9}:7+1\frac{5}{6}}{6\frac{1}{6}\cdot 3} = \frac{\frac{14}{9}:7+\frac{11}{6}}{\frac{37}{6}\cdot 3} = \frac{\frac{14}{63}+\frac{11}{6}}{\frac{37}{2}} = \frac{\frac{2}{9}+\frac{11}{6}}{\frac{37}{2}} $$

$$ \frac{2}{9}+\frac{11}{6} = \frac{4}{18}+\frac{33}{18} = \frac{37}{18} $$

$$ \frac{\frac{37}{18}}{\frac{37}{2}} = \frac{37}{18}\cdot\frac{2}{37} = \frac{1}{9} $$

б) Вынесем общий множитель в числителе:

$$ \frac{2\frac{3}{7}\cdot 1\frac{5}{11}+2\frac{3}{7}\cdot 2\frac{6}{11}}{2\frac{3}{7}\cdot 1\frac{5}{11}} = \frac{2\frac{3}{7}\cdot\left(1\frac{5}{11}+2\frac{6}{11}\right)}{2\frac{3}{7}\cdot 1\frac{5}{11}} $$

Сократим на общий множитель $$2\frac{3}{7}$$:

$$ = \frac{1\frac{5}{11}+2\frac{6}{11}}{1\frac{5}{11}} = \frac{3+\frac{11}{11}}{1\frac{5}{11}} = \frac{4}{\frac{16}{11}} = 4\cdot\frac{11}{16} = \frac{11}{4} = 2\frac{3}{4} $$

в) Вычислим значение дроби:

$$ \frac{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}{2+\frac{1}{2+\frac{1}{2}}} $$

Сначала найдём внутренние значения:

$$ 1+\frac{1}{2}=\frac{3}{2}, \qquad \frac{1}{\frac{3}{2}}=\frac{2}{3} $$

$$ 1+\frac{2}{3}=\frac{5}{3}, \qquad \frac{1}{\frac{5}{3}}=\frac{3}{5} $$

$$ 1+\frac{3}{5}=\frac{8}{5} $$

В знаменателе:

$$ 2+\frac{1}{2}=\frac{5}{2}, \qquad \frac{1}{\frac{5}{2}}=\frac{2}{5} $$

$$ 2+\frac{2}{5}=\frac{12}{5} $$

Теперь делим дроби:

$$ \frac{\frac{8}{5}}{\frac{12}{5}} = \frac{8}{5}\cdot\frac{5}{12} = \frac{8}{12} = \frac{2}{3} $$