{"id":2406,"date":"2023-05-09T19:36:20","date_gmt":"2023-05-09T19:36:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2406"},"modified":"2024-10-18T20:55:18","modified_gmt":"2024-10-18T20:55:18","slug":"simple-and-compound-interest-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/simple-and-compound-interest-fresh-take\/","title":{"raw":"Simple and Compound Interest: Fresh Take","rendered":"Simple and Compound Interest: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate simple interest and compound interest<\/li>\r\n\t<li>Determine annual percentage yield (APY) based on given interest scenarios<\/li>\r\n\t<li>Solve for time in compound interest calculations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Principal and Simple Interest<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> The <strong>principal <\/strong>is the amount of money that is borrowed or invested. For example, if you take out a student loan for [latex]$10,000[\/latex], the principal is [latex]$10,000[\/latex]. <strong> Simple interest<\/strong> is the interest that is calculated only on the principal amount, without taking into account any interest that has accumulated over time. Simple interest is typically calculated as a percentage of the principal amount and is added to the principal at regular intervals, such as monthly or annually. For example, if you have a savings account with a principal of [latex]$1,000[\/latex] and a simple interest rate of [latex]5\\%[\/latex], you would earn [latex]$50[\/latex] in interest over the course of a year, which would be added to the principal balance of the account. Simple interest is a straightforward way to calculate the interest earned or paid on a loan or investment.<\/div>\r\n<h2>Simple Interest Over Time<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Simple one-time interest<\/strong> refers to the interest that is earned or paid on a principal amount over a single period of time. This means that interest is only calculated once, at the end of the specified period, and it is based on the initial principal amount. Simple one-time interest is typically used in situations where the length of the investment or loan period is short and the interest rate is fixed. It is a straightforward way to calculate interest and can be useful for comparing different investment or loan options.<\/div>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10356045&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ZWCXrbnMN-E&amp;video_target=tpm-plugin-9awv0ur0-ZWCXrbnMN-E\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Simple+Interest+Tutorial.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimple Interest Tutorial\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a [latex]$1,000[\/latex] bond that pays [latex]5\\%[\/latex] interest annually and matures in [latex]5[\/latex] years. How much interest will you earn? [reveal-answer q=\"14596\"]Show Solution[\/reveal-answer] [hidden-answer a=\"14596\"]Each year, you would earn [latex]5\\%[\/latex] interest: [latex]$1000(0.05) = $50[\/latex] in interest. So over the course of five years, you would earn a total of [latex]$250[\/latex] in interest. When the bond matures, you would receive back the [latex]$1,000[\/latex] you originally paid, leaving you with a total of [latex]$1,250[\/latex]. Further explanation about solving this example can be seen <a href=\"https:\/\/youtu.be\/rNOEYPCnGwg\">here<\/a>.\r\n\r\n\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Simple+interest+over+time.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimple interest over time\u201d here (opens in new window).<\/a><\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<h2>Compounding Interest<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Compound interest<\/strong> refers to the interest that is earned on the initial principal amount as well as on the accumulated interest over time. In other words, the interest earned on an investment or the interest charged on a loan is added to the principal amount at regular intervals, and the new balance earns interest in the subsequent period. For example, if you invest [latex]$100[\/latex] in a savings account with a compound interest rate of [latex]5\\%[\/latex] per year, at the end of the first year, you will earn [latex]$5[\/latex] in interest, bringing your balance up to [latex]$105[\/latex]. In the second year, you will earn interest on the new balance of [latex]$105[\/latex], which means you will earn [latex]$5.25[\/latex] in interest. This means that over time, your interest earnings will grow at an increasing rate because you are earning interest on both the initial principal amount and the previously earned interest. Compound interest can lead to significant growth in the value of an investment over time, especially when the interest rate is high and the investment period is long. However, it can also result in higher costs for loans that charge compound interest, as the interest charged on the loan grows over time.<\/div>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10356046&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wf91rEGw88Q&amp;video_target=tpm-plugin-4ssnbmbu-wf91rEGw88Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Investopedia+Video+Compound+Interest+Explained.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInvestopedia Video: Compound Interest Explained\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10356047&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Hn0eLcOSQGw&amp;video_target=tpm-plugin-fvw05pb7-Hn0eLcOSQGw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Compound+Interest+by+The+Organic+Chemistry+Tutor.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCompound Interest\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>You know that you will need [latex]$40,000[\/latex] for your child\u2019s education in [latex]18[\/latex] years. If your account earns [latex]4\\%[\/latex] compounded quarterly, how much would you need to deposit now to reach your goal?[reveal-answer q=\"842460\"]Show Solution[\/reveal-answer] [hidden-answer a=\"842460\"]In this example, we\u2019re looking for [latex]P_0[\/latex].<\/p>\r\n<p><\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]r = 0.04[\/latex]<\/td>\r\n<td>[latex]4\\%[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]n = 4[\/latex]<\/td>\r\n<td>[latex]4[\/latex] quarters in [latex]1[\/latex] year<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t= 18[\/latex]<\/td>\r\n<td>Since we know the balance in [latex]18[\/latex] years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]P_{18} = $40,000[\/latex]<\/td>\r\n<td>The amount we have in [latex]18[\/latex] years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><\/p>\r\n<p><\/p>\r\n<p>In this case, we\u2019re going to have to set up the equation, and solve for [latex]P_0[\/latex].<center>[latex]\\begin{align}&amp;40000={{P}_{0}}{{\\left(1+\\frac{0.04}{4}\\right)}^{18\\times4}}\\\\&amp;40000={{P}_{0}}(2.0471)\\\\&amp;{{P}_{0}}=\\frac{40000}{2.0471}=\\$19539.84 \\\\\\end{align}[\/latex]<\/center>\r\n<p><\/p>\r\n<p>So you would need to deposit [latex]$19,539.84[\/latex] now to have [latex]$40,000[\/latex] in [latex]18[\/latex] years. [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate simple interest and compound interest<\/li>\n<li>Determine annual percentage yield (APY) based on given interest scenarios<\/li>\n<li>Solve for time in compound interest calculations<\/li>\n<\/ul>\n<\/section>\n<h2>Principal and Simple Interest<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> The <strong>principal <\/strong>is the amount of money that is borrowed or invested. For example, if you take out a student loan for [latex]$10,000[\/latex], the principal is [latex]$10,000[\/latex]. <strong> Simple interest<\/strong> is the interest that is calculated only on the principal amount, without taking into account any interest that has accumulated over time. Simple interest is typically calculated as a percentage of the principal amount and is added to the principal at regular intervals, such as monthly or annually. For example, if you have a savings account with a principal of [latex]$1,000[\/latex] and a simple interest rate of [latex]5\\%[\/latex], you would earn [latex]$50[\/latex] in interest over the course of a year, which would be added to the principal balance of the account. Simple interest is a straightforward way to calculate the interest earned or paid on a loan or investment.<\/div>\n<h2>Simple Interest Over Time<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Simple one-time interest<\/strong> refers to the interest that is earned or paid on a principal amount over a single period of time. This means that interest is only calculated once, at the end of the specified period, and it is based on the initial principal amount. Simple one-time interest is typically used in situations where the length of the investment or loan period is short and the interest rate is fixed. It is a straightforward way to calculate interest and can be useful for comparing different investment or loan options.<\/div>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10356045&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ZWCXrbnMN-E&amp;video_target=tpm-plugin-9awv0ur0-ZWCXrbnMN-E\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Simple+Interest+Tutorial.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimple Interest Tutorial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a [latex]$1,000[\/latex] bond that pays [latex]5\\%[\/latex] interest annually and matures in [latex]5[\/latex] years. How much interest will you earn? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q14596\">Show Solution<\/button> <\/p>\n<div id=\"q14596\" class=\"hidden-answer\" style=\"display: none\">Each year, you would earn [latex]5\\%[\/latex] interest: [latex]$1000(0.05) = $50[\/latex] in interest. So over the course of five years, you would earn a total of [latex]$250[\/latex] in interest. When the bond matures, you would receive back the [latex]$1,000[\/latex] you originally paid, leaving you with a total of [latex]$1,250[\/latex]. Further explanation about solving this example can be seen <a href=\"https:\/\/youtu.be\/rNOEYPCnGwg\">here<\/a>.<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Simple+interest+over+time.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimple interest over time\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Compounding Interest<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Compound interest<\/strong> refers to the interest that is earned on the initial principal amount as well as on the accumulated interest over time. In other words, the interest earned on an investment or the interest charged on a loan is added to the principal amount at regular intervals, and the new balance earns interest in the subsequent period. For example, if you invest [latex]$100[\/latex] in a savings account with a compound interest rate of [latex]5\\%[\/latex] per year, at the end of the first year, you will earn [latex]$5[\/latex] in interest, bringing your balance up to [latex]$105[\/latex]. In the second year, you will earn interest on the new balance of [latex]$105[\/latex], which means you will earn [latex]$5.25[\/latex] in interest. This means that over time, your interest earnings will grow at an increasing rate because you are earning interest on both the initial principal amount and the previously earned interest. Compound interest can lead to significant growth in the value of an investment over time, especially when the interest rate is high and the investment period is long. However, it can also result in higher costs for loans that charge compound interest, as the interest charged on the loan grows over time.<\/div>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10356046&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=wf91rEGw88Q&amp;video_target=tpm-plugin-4ssnbmbu-wf91rEGw88Q\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Investopedia+Video+Compound+Interest+Explained.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInvestopedia Video: Compound Interest Explained\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10356047&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Hn0eLcOSQGw&amp;video_target=tpm-plugin-fvw05pb7-Hn0eLcOSQGw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Compound+Interest+by+The+Organic+Chemistry+Tutor.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCompound Interest\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>You know that you will need [latex]$40,000[\/latex] for your child\u2019s education in [latex]18[\/latex] years. If your account earns [latex]4\\%[\/latex] compounded quarterly, how much would you need to deposit now to reach your goal?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q842460\">Show Solution<\/button> <\/p>\n<div id=\"q842460\" class=\"hidden-answer\" style=\"display: none\">In this example, we\u2019re looking for [latex]P_0[\/latex].<\/p>\n<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]r = 0.04[\/latex]<\/td>\n<td>[latex]4\\%[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]n = 4[\/latex]<\/td>\n<td>[latex]4[\/latex] quarters in [latex]1[\/latex] year<\/td>\n<\/tr>\n<tr>\n<td>[latex]t= 18[\/latex]<\/td>\n<td>Since we know the balance in [latex]18[\/latex] years<\/td>\n<\/tr>\n<tr>\n<td>[latex]P_{18} = $40,000[\/latex]<\/td>\n<td>The amount we have in [latex]18[\/latex] years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/p>\n<p>In this case, we\u2019re going to have to set up the equation, and solve for [latex]P_0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&40000={{P}_{0}}{{\\left(1+\\frac{0.04}{4}\\right)}^{18\\times4}}\\\\&40000={{P}_{0}}(2.0471)\\\\&{{P}_{0}}=\\frac{40000}{2.0471}=\\$19539.84 \\\\\\end{align}[\/latex]<\/div>\n<\/p>\n<p>So you would need to deposit [latex]$19,539.84[\/latex] now to have [latex]$40,000[\/latex] in [latex]18[\/latex] years. <\/p><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":89,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2406"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":26,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2406\/revisions"}],"predecessor-version":[{"id":14617,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2406\/revisions\/14617"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/89"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2406\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2406"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2406"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2406"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}