2.276 Выполните действия: *) (+).18 12 31 в) (4-3).. б) 5 (-) 2015 г) 5-4 12 -) e) ( 15 11\5 31 4\1 18 27

а) $$\left(\frac{4}{9} + \frac{5}{12}\right) \cdot \frac{18}{31}$$

Приведем дроби в скобках к общему знаменателю 36:

$$\frac{4}{9} = \frac{4 \cdot 4}{9 \cdot 4} = \frac{16}{36}$$, $$\frac{5}{12} = \frac{5 \cdot 3}{12 \cdot 3} = \frac{15}{36}$$

Тогда:

$$\left(\frac{16}{36} + \frac{15}{36}\right) \cdot \frac{18}{31} = \frac{31}{36} \cdot \frac{18}{31} = \frac{1}{2}$$

б) $$\frac{6}{25} \cdot \left(\frac{11}{15} - \frac{9}{20}\right)$$

Приведем дроби в скобках к общему знаменателю 60:

$$\frac{11}{15} = \frac{11 \cdot 4}{15 \cdot 4} = \frac{44}{60}$$, $$\frac{9}{20} = \frac{9 \cdot 3}{20 \cdot 3} = \frac{27}{60}$$

Тогда:

$$\frac{6}{25} \cdot \left(\frac{44}{60} - \frac{27}{60}\right) = \frac{6}{25} \cdot \frac{17}{60} = \frac{1}{25} \cdot \frac{17}{10} = \frac{17}{250}$$

г) $$\left(5-4\frac{4}{7}\right) \cdot \left(7\frac{1}{6} - 6\frac{5}{12}\right)$$ Переведем смешанные числа в неправильные дроби:

$$5-4\frac{4}{7} = \frac{35}{7} - \frac{32}{7} = \frac{3}{7}$$ $$7\frac{1}{6} = \frac{43}{6} = \frac{86}{12}$$ $$6\frac{5}{12} = \frac{77}{12}$$ Тогда:

$$\frac{3}{7} \cdot \left(\frac{86}{12} - \frac{77}{12}\right) = \frac{3}{7} \cdot \frac{9}{12} = \frac{3}{7} \cdot \frac{3}{4} = \frac{9}{28}$$

д) $$\left(\frac{1}{24} - \frac{5}{12}\right) \cdot \left(\frac{4}{8} - 3\frac{5}{24}\right)$$ $$ \frac{5}{12} = \frac{10}{24} $$ $$ \frac{4}{8} = \frac{1}{2} = \frac{12}{24} $$ $$ \frac{12}{24} - \frac{77}{24} = \frac{12-77}{24} = -\frac{65}{24}$$ $$\left(\frac{1}{24} - \frac{10}{24}\right) \cdot \left(\frac{12}{24} - \frac{77}{24}\right) = -\frac{9}{24} \cdot -\frac{65}{24} = \frac{3 \cdot 3 \cdot 5 \cdot 13}{8 \cdot 3 \cdot 8 \cdot 3} = \frac{195}{192} = \frac{65}{64}$$

e) $$\left(1\frac{2}{15} - \frac{11}{15}\right) \cdot \left(5\frac{3}{18} - 4\frac{1}{27}\right)$$ $$ \frac{17}{15} - \frac{11}{15} = \frac{6}{15} = \frac{2}{5} $$ $$ 5\frac{3}{18} = 5\frac{1}{6} = \frac{31}{6} $$ $$ 4\frac{1}{27} = \frac{109}{27} $$ $$\frac{31}{6} - \frac{109}{27} = \frac{31 \cdot 9}{6 \cdot 9} - \frac{109 \cdot 2}{27 \cdot 2} = \frac{279}{54} - \frac{218}{54} = \frac{61}{54}$$ $$\frac{2}{5} \cdot \frac{61}{54} = \frac{1}{5} \cdot \frac{61}{27} = \frac{61}{135}$$

Ответ:

• а) $$\frac{1}{2}$$
• б) $$\frac{17}{250}$$
• г) $$\frac{9}{28}$$
• д) $$\frac{65}{64}$$
• e) $$\frac{61}{135}$$