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Краткие ответы

Блок 1

Задание 1:

1. \(\frac{12}{5}\); 2) \(\frac{15}{128}\); 3) \(-\frac{17}{25}\); 4) \(\frac{7}{20}\).

Задание 2:

1. \(14{,}7\); 2) \(6{,}8\); 3) \(8{,}16\); 4) \(4\).

Задание 3: \(39\).

Задание 4:

1. \(\frac{42}{5}\); 2) \(18\).

Блок 2

Задание 1:

1. \(-\frac72\); 2) \(\frac9{10}\); 3) \(\frac{22}{25}\); 4) \(\frac{22}{5}\).

Задание 2:

1. \(\frac{75}{2}\); 2) \(\frac{83}{4}\); 3) \(\frac{455}{71}\); 4) \(12\).

Задание 3:

1. \(\frac34\); 2) \(7\); 3) \(\frac{16}{5}\); 4) \(\frac{49}{10}\).

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Решение и принцип

При сложении и вычитании дробей приводим их к общему знаменателю. При делении на дробь умножаем на обратную дробь. Смешанное число сначала переводим в неправильную дробь.

Блок 1

Задание 1

1. \[ \frac27\cdot\frac{42}{5} =\frac{2\cdot6}{5} =\frac{12}{5}. \]

2. \[ \frac3{16}\cdot\frac58=\frac{15}{128}. \]

3. \[ \frac1{10}-\frac{39}{50} =\frac5{50}-\frac{39}{50} =-\frac{34}{50} =-\frac{17}{25}. \]

4. \[ \frac15+\frac3{20} =\frac4{20}+\frac3{20} =\frac7{20}. \]

Задание 2

\[ 8{,}8+5{,}9=14{,}7; \]

\[ 9{,}2-2{,}4=6{,}8; \]

\[ 1{,}6\cdot5{,}1=8{,}16; \]

\[ 6{,}8:1{,}7=4. \]

Задание 3

Знаменатель \(90\):

\[ \frac56=\frac{75}{90},\qquad \frac25=\frac{36}{90}. \]

Тогда

\[ \frac56-\frac25 =\frac{75}{90}-\frac{36}{90} =\frac{39}{90}. \]

Числитель полученной дроби: \(\boxed{39}\).

Задание 4

1. \[ \frac1{\frac1{28}+\frac1{12}} = \frac1{\frac3{84}+\frac7{84}} = \frac1{\frac{10}{84}} = \frac1{\frac5{42}} = \frac{42}{5}. \]

2. \[ \frac1{\frac1{14}-\frac1{63}} = \frac1{\frac9{126}-\frac2{126}} = \frac1{\frac7{126}} = \frac1{\frac1{18}} =18. \]

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Блок 2

Задание 1

1. \[ \left(\frac1{12}-\frac{17}{15}\right)\cdot\frac{10}{3} = \left(\frac5{60}-\frac{68}{60}\right)\cdot\frac{10}{3} = -\frac{21}{20}\cdot\frac{10}{3} =-\frac72. \]

2. \[ \left(\frac7{15}+\frac{19}{30}\right)\cdot\frac9{11} = \left(\frac{14}{30}+\frac{19}{30}\right)\cdot\frac9{11} = \frac{11}{10}\cdot\frac9{11} =\frac9{10}. \]

3. \[ \left(\frac3{25}-\frac2{35}\right)\cdot14 = \left(\frac{21}{175}-\frac{10}{175}\right)\cdot14 = \frac{11}{175}\cdot14 =\frac{22}{25}. \]

4. \[ \left(\frac1{30}+\frac7{10}\right)\cdot6 = \left(\frac1{30}+\frac{21}{30}\right)\cdot6 = \frac{11}{15}\cdot6 =\frac{22}{5}. \]

Задание 2

1. \[ \left(2\frac3{17}+\frac1{34}\right)\cdot17 = \left(\frac{37}{17}+\frac1{34}\right)\cdot17 = \frac{75}{34}\cdot17 =\frac{75}{2}. \]

2. \[ \left(1\frac14-\frac3{19}\right)\cdot19 = \left(\frac54-\frac3{19}\right)\cdot19 = \frac{83}{76}\cdot19 =\frac{83}{4}. \]

3. \[ 4\frac38:\left(\frac38+\frac4{13}\right) = \frac{35}{8}:\left(\frac{39}{104}+\frac{32}{104}\right) \]

\[ = \frac{35}{8}:\frac{71}{104} = \frac{35}{8}\cdot\frac{104}{71} =\frac{455}{71}. \]

4. \[ 5\frac25:\left(3\frac14-2\frac45\right) = \frac{27}{5}:\left(\frac{13}{4}-\frac{14}{5}\right) \]

\[ = \frac{27}{5}:\frac9{20} = \frac{27}{5}\cdot\frac{20}{9} =12. \]

Задание 3

1. \[ \frac{3{,}6}{5{,}9-1{,}1} = \frac{3{,}6}{4{,}8} =\frac{36}{48} =\frac34. \]

2. \[ \frac{1{,}3+9{,}2}{1{,}5} = \frac{10{,}5}{1{,}5} =7. \]

3. \[ \frac{28}{17{,}5\cdot0{,}5} = \frac{28}{8{,}75} = \frac{16}{5}. \]

4. \[ \frac{5{,}6\cdot0{,}7}{0{,}8} = \frac{3{,}92}{0{,}8} = \frac{49}{10}. \]

Проверка: \(1{,}7\cdot4=6{,}8\), \(1{,}5\cdot7=10{,}5\), а \(\frac5{42}\cdot\frac{42}{5}=1\).