{"id":36,"date":"2022-12-20T14:42:17","date_gmt":"2022-12-20T14:42:17","guid":{"rendered":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/?page_id=36"},"modified":"2022-12-23T16:02:44","modified_gmt":"2022-12-23T16:02:44","slug":"zinseszinsen-laufzeit","status":"publish","type":"page","link":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/zinseszinsen-laufzeit\/","title":{"rendered":"Zinseszinsen &#8211; Laufzeit"},"content":{"rendered":"\n<p>Die Laufzeit bei Zinseszinsen geben wir immer in Jahren an und k\u00fcrzen diese mit n ab.<\/p>\n\n\n\n<p><strong>Beispielaufgabe: <\/strong>Frau Wunderle m\u00f6chte sich eine Wohnung kaufen, welche 250.000,00 \u20ac kostet. Momentan verf\u00fcgt sie \u00fcber 200.000,00 \u20ac auf ihrem Konto. Wie viele volle Jahre m\u00fcsste Frau Wunderle ihr Kapital auf dem Konto liegen lassen, um sich die Wohnung kaufen zu k\u00f6nnen? Ihre Bank gew\u00e4hrt 4 % Zinsen pro Jahr. Berechnen Sie.<\/p>\n\n\n\n<p>Schau dir zun\u00e4chst das Lernvideo an und l\u00f6se anschlie\u00dfend die Beispielaufgabe.<\/p>\n\n\n<style>.kt-accordion-id_6b0b40-cf .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_6b0b40-cf .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_6b0b40-cf > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_6b0b40-cf > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_6b0b40-cf .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_6b0b40-cf > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_6b0b40-cf:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_6b0b40-cf .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_6b0b40-cf .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_6b0b40-cf kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_0293fd-33\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Lernvideo: Berechnung der Laufzeit n bei Zinseszinsen<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" hcb-fetch-image-from=\"https:\/\/youtu.be\/oUjYapq1AX8\" title=\"Finanzmathe: Zinseszinsen Teil 4 - Berechnung der Laufzeit (n)\" width=\"800\" height=\"450\" src=\"https:\/\/www.youtube.com\/embed\/oUjYapq1AX8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_b6c843-98\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">L\u00f6sung Beispielaufgabe<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>Das Startkapital (200 000,00 \u20ac), der Zinssatz (4%) und das Endkapital (250 000,00 \u20ac) sind gegeben. Wir setzte zun\u00e4chst alle gegebenen Gr\u00f6\u00dfen die Formel der Zinseszinsrechnung ein.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>K_n=K_0*q^n <\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>250 000 = 200 000 * 1,04^n<\/pre><\/div>\n\n\n\n<p>Wir erhalten eine Gleichung, bei der wir den Exponenten (<strong>n<\/strong>) suchen. Diese l\u00f6sen wir mit Hilfe von \u00c4quivalenzumformungen und dem Logarithmus nach n auf.<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>250 000 = 200 000*1,04^n \\ |:200 000 <\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\\frac{250000}{200000} = 1,04^n \\ |\\ log<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>n = log_{1,04}\\  \\frac{250000}{200000} = 5,67<\/pre><\/div>\n\n\n\n<p>oder (je nach Taschenrechner) alternative Berechnung:<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>n= \\frac{log\\ \\frac{250000}{200000}}{log\\ 1,04} = 5,67<\/pre><\/div>\n\n\n\n<p>Wir runden das Ergebnis auf <strong>6 volle Jahre<\/strong>. Frau Wunderle kann sich die Wohnung also sp\u00e4testens nach <strong>6 Jahren<\/strong> leisten.<\/p>\n\n\n\n<p>Allgemein lautet die Formel zur Berechnung des Exponenten\/der Laufzeit n bei Zinseszinsen also:<\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>n =log_q\\ \\frac{K_n}{K_0}<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_ebc86d-1f\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">PDF: \u00dcbungsaufgaben Berechnung der Laufzeit n<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-content\/uploads\/2022\/12\/Uebungsaufgaben-Laufzeit-Zinseszinsen-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Einbettung von Uebungsaufgaben-Laufzeit-Zinseszinsen-1.\"><\/object><a id=\"wp-block-file--media-2ddd5066-7dec-4f34-94b9-4e226d786437\" href=\"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-content\/uploads\/2022\/12\/Uebungsaufgaben-Laufzeit-Zinseszinsen-1.pdf\">Uebungsaufgaben-Laufzeit-Zinseszinsen-1<\/a><a href=\"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-content\/uploads\/2022\/12\/Uebungsaufgaben-Laufzeit-Zinseszinsen-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-2ddd5066-7dec-4f34-94b9-4e226d786437\">Herunterladen<\/a><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Die Laufzeit bei Zinseszinsen geben wir immer in Jahren an und k\u00fcrzen diese mit n ab. Beispielaufgabe: Frau Wunderle m\u00f6chte<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"colormag_page_container_layout":"default_layout","colormag_page_sidebar_layout":"default_layout","footnotes":""},"class_list":["post-36","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/pages\/36","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/comments?post=36"}],"version-history":[{"count":9,"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/pages\/36\/revisions"}],"predecessor-version":[{"id":343,"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/pages\/36\/revisions\/343"}],"wp:attachment":[{"href":"https:\/\/nachhilfe-wirtschaftsschule.de\/wordpress\/wp-json\/wp\/v2\/media?parent=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}