• Rechnen mit Zinsen
  • Deliah Herbstritt
  • 21.07.2020
  • Mathematik
  • Zinsen
  • E (Expertenstandard)
  • 8
  • Einzelarbeit
  • Arbeitsblatt
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1
Ein Kapital fest angelegt. Berechne das Kapital am Ende des angegebenen Zeitraums:
  • K = 500,00€; p = 2%; n = 4 Jahre
  • K = 2 500,00€; p = 1,5%; n = 3 Jahre
  • K = 15 000,00€; p = 0,75%; n = 5 Jahre
Lösung a1
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+2%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 2\%
q=1,02\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}02

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n
K4=500,001,024\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_4=500{,}00€\cdot1{,}02^4
K4541,22\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_4\approx541{,}22€
Lösung b
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+1,5%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 1{,}5\%
q=1,015\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}015

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n
K3=2500,001,0153\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_3=2\,500{,}00€\cdot 1{,}015^3
K32614,20\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_3\approx2\,614{,}20€
Lösung c
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+0,75%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 0{,}75\%
q=1,0075\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}0075

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n
K5=15000,001,00755\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_5=15\,000{,}00€\cdot 1{,}0075^5
K515571,00\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_5\approx15\,571{,}00€
2

Anfangskapital

Zinssatz

Laufzeit

Endkapital

12 000,00€

1,25%

10 Jahre

13 587,25€

5 500,00€

0,5%

4 Jahre

5 610,83€

750,00€

1,2%

2 Jahre

768,11€

3
Wie hoch war das Anfangskapital?
K4 = 3 090,68€; p = 2,5%
Lösung3
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+2,5%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 2{,}5\%
q=1,025\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}025

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n |:qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q^n
Kn:qn=K0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n:q^n=K_0
3090,58:1,0254=K0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 3\,090{,}58€ : 1{,}025^4 = K_0
2799,91=K0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2\,799{,}91€ = K_0

Antwort: Das Anfangskapital betrug 2 799,91€.
4
Um wie viel Prozent vergrößert sich ein Kapital von 2 500,00€ bei einem Zinssatz von 1,5% in 3 Jahren?
Lösung4
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+1,5%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 1{,}5\%
q=1,015\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}015

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n
K3=2500,001,0153\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_3=2\,500{,}00€\cdot 1{,}015^3
K32614,20\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_3\approx2\,614{,}20€

Dreisatz:
2 500,00€ - 100%
1,00€ - 0,04%
2 614,20€ - 104,57%

104,57% - 100% = 4,57%

Antwort: Prozentual vergrößert sich das Kapital um 4,57%.
5
Wie viel Zinsen erhält man bei 0,25% für 5 500,00€ in 5 Jahren?
Lösung5
q=1+p%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + p\%
q=1+0,25%\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1 + 0{,}25\%
q=1,0025\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} q = 1{,}0025

Kn=K0qn\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_n=K_0\cdot q^n
K5=5500,001,00255\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_5 = 5\,500{,}00€\cdot 1{,}0025^5
K55569,09\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} K_5 \approx 5\,569{,}09

Z=K5K0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} Z=K_5-K_0
Z=5569,095500,00\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} Z = 5\,569{,}09€-5\,500{,}00
Z=69,09\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} Z=69{,}09€

Antwort: In 5 Jahren erhält man 69,09€ Zinsen.