Frazioni generatrici

((24 - 2)/9 + (42 - 4)/9)·3.6 - (4 - 5/9 + 4/9)/(7/9) - 42/10=

=(22/9 + 38/9)·3.6 - 35/9/(7/9) - 42/10=

=(22/9 + 38/9)·3.6 - 35/9/(7/9) - 21/5=

=20/3·3.6 - 35/9/(7/9) - 21/5=

=20/3·(18/5) - 35/9/(7/9) - 21/5=

=20/3·(18/5) - 5 - 21/5=

=24 - 5 - 21/5=

=(24·5 - 5·5 - 21)/5= 74/5

2,4 (con 4 periodico) = (24 - 2) / 9 = 22 / 9;

4,2 (con 2 periodico) = (42 - 4) / 9 = 38 / 9;

3,6 = 36/10 = 18/5;

0,5 (con 4 periodico) = (5 - 0) / 9 = 5/9;

0,4 (con 4 periodico) = (4 - 0) / 9 = 4/9;

0,7 (con 7 periodico) = (7 - 0) / 9 = 7/9;

4,2 = 42/10 = 21/5;

(22/9 + 38/9) * 18/5 - (4 - 5/9 + 4/9) : 7/9 - 21/5 =

= (60/9)  * (18/5) - (36/9 - 5/9 + 4/9) : 7/9 - 21/5 =

Moltiplicazione [(60 * 18) /(9 * 5);  60 : 5 = 12; 18 / 9 = 2; resta 12 * 2 = 24];

= 24 - (35/9) * 9/7 - 21/5 =

= 24 - (35 * 9 / (9 * 7) - 21/5 =

Moltiplicazione [(35 * 9 / (9 * 7) = 35/7 = 5/1].

= 24 - 5/1 - 21/5 =

= 120/5 - 25/5 - 21/5 =

= 74/5.

Ciao  @sebastiano_mori

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$\small \left(2,\overline4+4,\overline2\right)·3,6-\left(4-0,\overline5+0,\overline4\right) : 0,\overline7-4,2=$

$\small =\left(\dfrac{24-2}{9}+\dfrac{42-4}{9}\right)·\dfrac{\cancel{36}^{18}}{\cancel{10}_5}-\left(4-\dfrac{5-0}{9}+\dfrac{4-0}{9}\right) : \dfrac{7-0}{9}-\dfrac{\cancel{42}^{21}}{\cancel{10}_5}=$

$\small =\left(\dfrac{22}{9}+\dfrac{38}{9}\right)·\dfrac{18}{5}-\left(4-\dfrac{5}{9}+\dfrac{4}{9}\right) : \dfrac{7}{9}-\dfrac{21}{5}=$

$\small =\dfrac{60}{9}·\dfrac{18}{5}-\left(\dfrac{36-5+4}{9}\right)·\dfrac{9}{7}-\dfrac{21}{5}=$

$\small =\dfrac{\cancel{60}^{12}}{\cancel9_1}·\dfrac{\cancel{18}^2}{\cancel5_1}-\dfrac{\cancel{35}^5}{\cancel9_1}·\dfrac{\cancel9^1}{\cancel7_1}-\dfrac{21}{5}=$

$\small =\dfrac{12}{1}·\dfrac{2}{1}-\dfrac{5}{1}·\dfrac{1}{1}-\dfrac{21}{5}=$

$\small =24-5-\dfrac{21}{5}=$

$\small =\dfrac{120-25-21}{5}=$

$\small =\dfrac{74}{5}$