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gesamt:541 Möglichkeiten
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0x (1cent) + 0x (2cents) + 20x (5cents)
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45x (1cent) + 10x (2cents) + 7x (5cents)
45x (1cent) + 15x (2cents) + 5x (5cents)
45x (1cent) + 20x (2cents) + 3x (5cents)
45x (1cent) + 25x (2cents) + 1x (5cents)
46x (1cent) + 2x (2cents) + 10x (5cents)
46x (1cent) + 7x (2cents) + 8x (5cents)
46x (1cent) + 12x (2cents) + 6x (5cents)
46x (1cent) + 17x (2cents) + 4x (5cents)
46x (1cent) + 22x (2cents) + 2x (5cents)
46x (1cent) + 27x (2cents) + 0x (5cents)
47x (1cent) + 4x (2cents) + 9x (5cents)
47x (1cent) + 9x (2cents) + 7x (5cents)
47x (1cent) + 14x (2cents) + 5x (5cents)
47x (1cent) + 19x (2cents) + 3x (5cents)
47x (1cent) + 24x (2cents) + 1x (5cents)
48x (1cent) + 1x (2cents) + 10x (5cents)
48x (1cent) + 6x (2cents) + 8x (5cents)
48x (1cent) + 11x (2cents) + 6x (5cents)
48x (1cent) + 16x (2cents) + 4x (5cents)
48x (1cent) + 21x (2cents) + 2x (5cents)
48x (1cent) + 26x (2cents) + 0x (5cents)
49x (1cent) + 3x (2cents) + 9x (5cents)
49x (1cent) + 8x (2cents) + 7x (5cents)
49x (1cent) + 13x (2cents) + 5x (5cents)
49x (1cent) + 18x (2cents) + 3x (5cents)
49x (1cent) + 23x (2cents) + 1x (5cents)
50x (1cent) + 0x (2cents) + 10x (5cents)
50x (1cent) + 5x (2cents) + 8x (5cents)
50x (1cent) + 10x (2cents) + 6x (5cents)
50x (1cent) + 15x (2cents) + 4x (5cents)
50x (1cent) + 20x (2cents) + 2x (5cents)
50x (1cent) + 25x (2cents) + 0x (5cents)
51x (1cent) + 2x (2cents) + 9x (5cents)
51x (1cent) + 7x (2cents) + 7x (5cents)
51x (1cent) + 12x (2cents) + 5x (5cents)
51x (1cent) + 17x (2cents) + 3x (5cents)
51x (1cent) + 22x (2cents) + 1x (5cents)
52x (1cent) + 4x (2cents) + 8x (5cents)
52x (1cent) + 9x (2cents) + 6x (5cents)
52x (1cent) + 14x (2cents) + 4x (5cents)
52x (1cent) + 19x (2cents) + 2x (5cents)
52x (1cent) + 24x (2cents) + 0x (5cents)
53x (1cent) + 1x (2cents) + 9x (5cents)
53x (1cent) + 6x (2cents) + 7x (5cents)
53x (1cent) + 11x (2cents) + 5x (5cents)
53x (1cent) + 16x (2cents) + 3x (5cents)
53x (1cent) + 21x (2cents) + 1x (5cents)
54x (1cent) + 3x (2cents) + 8x (5cents)
54x (1cent) + 8x (2cents) + 6x (5cents)
54x (1cent) + 13x (2cents) + 4x (5cents)
54x (1cent) + 18x (2cents) + 2x (5cents)
54x (1cent) + 23x (2cents) + 0x (5cents)
55x (1cent) + 0x (2cents) + 9x (5cents)
55x (1cent) + 5x (2cents) + 7x (5cents)
55x (1cent) + 10x (2cents) + 5x (5cents)
55x (1cent) + 15x (2cents) + 3x (5cents)
55x (1cent) + 20x (2cents) + 1x (5cents)
56x (1cent) + 2x (2cents) + 8x (5cents)
56x (1cent) + 7x (2cents) + 6x (5cents)
56x (1cent) + 12x (2cents) + 4x (5cents)
56x (1cent) + 17x (2cents) + 2x (5cents)
56x (1cent) + 22x (2cents) + 0x (5cents)
57x (1cent) + 4x (2cents) + 7x (5cents)
57x (1cent) + 9x (2cents) + 5x (5cents)
57x (1cent) + 14x (2cents) + 3x (5cents)
57x (1cent) + 19x (2cents) + 1x (5cents)
58x (1cent) + 1x (2cents) + 8x (5cents)
58x (1cent) + 6x (2cents) + 6x (5cents)
58x (1cent) + 11x (2cents) + 4x (5cents)
58x (1cent) + 16x (2cents) + 2x (5cents)
58x (1cent) + 21x (2cents) + 0x (5cents)
59x (1cent) + 3x (2cents) + 7x (5cents)
59x (1cent) + 8x (2cents) + 5x (5cents)
59x (1cent) + 13x (2cents) + 3x (5cents)
59x (1cent) + 18x (2cents) + 1x (5cents)
60x (1cent) + 0x (2cents) + 8x (5cents)
60x (1cent) + 5x (2cents) + 6x (5cents)
60x (1cent) + 10x (2cents) + 4x (5cents)
60x (1cent) + 15x (2cents) + 2x (5cents)
60x (1cent) + 20x (2cents) + 0x (5cents)
61x (1cent) + 2x (2cents) + 7x (5cents)
61x (1cent) + 7x (2cents) + 5x (5cents)
61x (1cent) + 12x (2cents) + 3x (5cents)
61x (1cent) + 17x (2cents) + 1x (5cents)
62x (1cent) + 4x (2cents) + 6x (5cents)
62x (1cent) + 9x (2cents) + 4x (5cents)
62x (1cent) + 14x (2cents) + 2x (5cents)
62x (1cent) + 19x (2cents) + 0x (5cents)
63x (1cent) + 1x (2cents) + 7x (5cents)
63x (1cent) + 6x (2cents) + 5x (5cents)
63x (1cent) + 11x (2cents) + 3x (5cents)
63x (1cent) + 16x (2cents) + 1x (5cents)
64x (1cent) + 3x (2cents) + 6x (5cents)
64x (1cent) + 8x (2cents) + 4x (5cents)
64x (1cent) + 13x (2cents) + 2x (5cents)
64x (1cent) + 18x (2cents) + 0x (5cents)
65x (1cent) + 0x (2cents) + 7x (5cents)
65x (1cent) + 5x (2cents) + 5x (5cents)
65x (1cent) + 10x (2cents) + 3x (5cents)
65x (1cent) + 15x (2cents) + 1x (5cents)
66x (1cent) + 2x (2cents) + 6x (5cents)
66x (1cent) + 7x (2cents) + 4x (5cents)
66x (1cent) + 12x (2cents) + 2x (5cents)
66x (1cent) + 17x (2cents) + 0x (5cents)
67x (1cent) + 4x (2cents) + 5x (5cents)
67x (1cent) + 9x (2cents) + 3x (5cents)
67x (1cent) + 14x (2cents) + 1x (5cents)
68x (1cent) + 1x (2cents) + 6x (5cents)
68x (1cent) + 6x (2cents) + 4x (5cents)
68x (1cent) + 11x (2cents) + 2x (5cents)
68x (1cent) + 16x (2cents) + 0x (5cents)
69x (1cent) + 3x (2cents) + 5x (5cents)
69x (1cent) + 8x (2cents) + 3x (5cents)
69x (1cent) + 13x (2cents) + 1x (5cents)
70x (1cent) + 0x (2cents) + 6x (5cents)
70x (1cent) + 5x (2cents) + 4x (5cents)
70x (1cent) + 10x (2cents) + 2x (5cents)
70x (1cent) + 15x (2cents) + 0x (5cents)
71x (1cent) + 2x (2cents) + 5x (5cents)
71x (1cent) + 7x (2cents) + 3x (5cents)
71x (1cent) + 12x (2cents) + 1x (5cents)
72x (1cent) + 4x (2cents) + 4x (5cents)
72x (1cent) + 9x (2cents) + 2x (5cents)
72x (1cent) + 14x (2cents) + 0x (5cents)
73x (1cent) + 1x (2cents) + 5x (5cents)
73x (1cent) + 6x (2cents) + 3x (5cents)
73x (1cent) + 11x (2cents) + 1x (5cents)
74x (1cent) + 3x (2cents) + 4x (5cents)
74x (1cent) + 8x (2cents) + 2x (5cents)
74x (1cent) + 13x (2cents) + 0x (5cents)
75x (1cent) + 0x (2cents) + 5x (5cents)
75x (1cent) + 5x (2cents) + 3x (5cents)
75x (1cent) + 10x (2cents) + 1x (5cents)
76x (1cent) + 2x (2cents) + 4x (5cents)
76x (1cent) + 7x (2cents) + 2x (5cents)
76x (1cent) + 12x (2cents) + 0x (5cents)
77x (1cent) + 4x (2cents) + 3x (5cents)
77x (1cent) + 9x (2cents) + 1x (5cents)
78x (1cent) + 1x (2cents) + 4x (5cents)
78x (1cent) + 6x (2cents) + 2x (5cents)
78x (1cent) + 11x (2cents) + 0x (5cents)
79x (1cent) + 3x (2cents) + 3x (5cents)
79x (1cent) + 8x (2cents) + 1x (5cents)
80x (1cent) + 0x (2cents) + 4x (5cents)
80x (1cent) + 5x (2cents) + 2x (5cents)
80x (1cent) + 10x (2cents) + 0x (5cents)
81x (1cent) + 2x (2cents) + 3x (5cents)
81x (1cent) + 7x (2cents) + 1x (5cents)
82x (1cent) + 4x (2cents) + 2x (5cents)
82x (1cent) + 9x (2cents) + 0x (5cents)
83x (1cent) + 1x (2cents) + 3x (5cents)
83x (1cent) + 6x (2cents) + 1x (5cents)
84x (1cent) + 3x (2cents) + 2x (5cents)
84x (1cent) + 8x (2cents) + 0x (5cents)
85x (1cent) + 0x (2cents) + 3x (5cents)
85x (1cent) + 5x (2cents) + 1x (5cents)
86x (1cent) + 2x (2cents) + 2x (5cents)
86x (1cent) + 7x (2cents) + 0x (5cents)
87x (1cent) + 4x (2cents) + 1x (5cents)
88x (1cent) + 1x (2cents) + 2x (5cents)
88x (1cent) + 6x (2cents) + 0x (5cents)
89x (1cent) + 3x (2cents) + 1x (5cents)
90x (1cent) + 0x (2cents) + 2x (5cents)
90x (1cent) + 5x (2cents) + 0x (5cents)
91x (1cent) + 2x (2cents) + 1x (5cents)
92x (1cent) + 4x (2cents) + 0x (5cents)
93x (1cent) + 1x (2cents) + 1x (5cents)
94x (1cent) + 3x (2cents) + 0x (5cents)
95x (1cent) + 0x (2cents) + 1x (5cents)
96x (1cent) + 2x (2cents) + 0x (5cents)
98x (1cent) + 1x (2cents) + 0x (5cents)
100x (1cent) + 0x (2cents) + 0x (5cents)

Frage:  $$0.9999\overline9 \stackrel{?}{=} 1$$

Beiweis Idee:  $$1 = \frac{1}{3} + \frac{2}{3}$$

Also:

$$\frac{1}{3} = 0.33333\overline3$$   und   $$\frac{2}{3} = 0.666666\overline6$$

sowie:

$$\begin{matrix} & 0.33333333\overline3 \\ + & 0.66666666\overline6 \\ & ======= \\ & 0.99999999\overline9 \end{matrix}$$

folgt:

$$0.9999\overline9 = 1$$  
q.e.d.

example:

$${\mathtt{2}}{i}{\mathtt{\,\times\,}}{\mathtt{2}}{i} = -{\mathtt{4}}$$

examples:

101     103     107     109
113     127     131     137     139     149     151     157     163     167
173     179     181     191     193     197     199     211     223     227
229     233     239     241     251     257     263     269     271     277
281     283     293     307     311     313     317     331     337     347
349     353     359     367     373     379     383     389     397     401
409     419     421     431     433     439     443     449     457     461
463     467     479     487     491     499     503     509     521     523
541     547     557     563     569     571     577     587     593     599
601     607     613     617     619     631     641     643     647     653
659     661     673     677     683     691     701     709     719     727
733     739     743     751     757     761     769     773     787     797
809     811     821     823     827     829     839     853     857     859
863     877     881     883     887     907     911     919     929     937
941     947     953     967     971     977     983     991     997    

(1+x)^8=1+2.15

1+x=(1+2.15)^(1/8)

x=3.15^(1/8)-1

x=0.1542205949321005

x≈15.4%

$$\underset{\,\,\,\,{\textcolor[rgb]{0.66,0.66,0.66}{\rightarrow {\mathtt{u1, r, u2, x}}}}}{{solve}}{\left(\begin{array}{l}{\mathtt{u1}}={\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{\pi}}{\mathtt{\,\times\,}}{\mathtt{r}}\\ {\mathtt{u2}}={\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{\pi}}{\mathtt{\,\times\,}}\left({\mathtt{r}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}\right)\\ {\mathtt{u2}}={\mathtt{u1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{u1}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{r2}}{\mathtt{\,\times\,}}{\mathtt{\pi}}\\ {\mathtt{r}} = {\mathtt{r2}}\\ {\mathtt{u2}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{r2}}{\mathtt{\,\times\,}}{\mathtt{\pi}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\\ {\mathtt{x}} = {\frac{{\mathtt{1}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{\pi}}\right)}}\\ \end{array} \right\}$$

x=1/(2*pi) = ca. 16cm

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$${expand}{\left(\left({\mathtt{x}}{\mathtt{\,\times\,}}{\mathtt{5}}\right){\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}\right)\right)} = {\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{6}}$$

$${expand}{\left({\frac{\left(\left({\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{6}}\right){\mathtt{\,\small\textbf+\,}}\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,-\,}}{\mathtt{6}}\right)\right)}{{\mathtt{2}}}}\right)} = {\mathtt{a}}{\mathtt{\,\small\textbf+\,}}{\mathtt{27}}$$

a) Subtrahiere vom Produkt x und 5 das doppelte der summe x und 3.

(x*5)-2*(x+3)