{"id":392,"date":"2016-10-12T16:44:20","date_gmt":"2016-10-12T16:44:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=392"},"modified":"2021-02-05T23:58:42","modified_gmt":"2021-02-05T23:58:42","slug":"which-equation-to-use","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/which-equation-to-use\/","title":{"raw":"Which Formula to Use?","rendered":"Which Formula to Use?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine which equation to use for a given scenario<\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/03001020\/chimpthink.png\"><img class=\"wp-image-1325 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/03001020\/chimpthink-300x205.png\" alt=\"\" width=\"373\" height=\"254\" \/><\/a>Now that we have surveyed the basic kinds of finance calculations that are used, it may not always be obvious which one to use when you are given a problem to solve.\u00a0Here are some hints on deciding which equation to use, based on the wording of the problem.\r\n<h3>Loans<\/h3>\r\nThe easiest types of problems to identify are loans.\u00a0 Loan problems almost always include words like <strong>loan<\/strong>, <strong>amortize<\/strong> (the fancy word for loans), <strong>finance<\/strong> (i.e. a car), or <strong>mortgage<\/strong> (a home loan). Look for words like monthly or annual payment.\r\n\r\nThe loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.\r\n<div class=\"textbox\">\r\n<h2>Loans Formula<\/h2>\r\n[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{k}\\right)^{-Nk}\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>0<\/sub><\/em> is the balance in the account at the beginning (the principal, or amount of the loan).<\/li>\r\n \t<li><em>d <\/em>is your loan payment (your monthly payment, annual payment, etc)<\/li>\r\n \t<li><em>r<\/em> is the annual interest rate in decimal form.<\/li>\r\n \t<li><em>k<\/em> is the number of compounding periods in one year.<\/li>\r\n \t<li><em>N<\/em> is the length of the loan, in years.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h3>Interest-Bearing Accounts<\/h3>\r\nAccounts that gain interest fall into two main categories. \u00a0The first is on where you put money in an account once and let it sit, the other is where you make regular payments or withdrawals from the account as in a retirement account.\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Interest<\/strong><\/span>\r\n<ul>\r\n \t<li>If you're letting the money sit in the account with nothing but interest changing the balance, then you're looking at a <strong>compound interest<\/strong> problem. Look for words like compounded, or APY.\u00a0Compound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>COMPOUND INTEREST<\/h3>\r\n[latex]P_{N}=P_{0}\\left(1+\\frac{r}{k}\\right)^{Nk}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\r\n \t<li><em>P<sub>0 <\/sub><\/em>is the starting balance of the account (also called initial deposit, or principal)<\/li>\r\n \t<li><em>r<\/em> is the annual interest rate in decimal form<\/li>\r\n \t<li><em>k<\/em> is the number of compounding periods in one year\r\n<ul>\r\n \t<li>If the compounding is done annually (once a year), <em>k<\/em> = 1.<\/li>\r\n \t<li>If the compounding is done quarterly, <em>k<\/em> = 4.<\/li>\r\n \t<li>If the compounding is done monthly, <em>k<\/em> = 12.<\/li>\r\n \t<li>If the compounding is done daily, <em>k<\/em> = 365.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<ul>\r\n \t<li>The exception would be bonds and other investments where the interest is not reinvested; in those cases you\u2019re looking at <strong>simple interest<\/strong>.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>SIMPLE INTEREST OVER TIME<\/h3>\r\n[latex]\\begin{align}&amp;I={{P}_{0}}rt\\\\&amp;A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\\\\end{align}[\/latex]\r\n<ul>\r\n \t<li><em>I<\/em> is the interest<\/li>\r\n \t<li><em>A<\/em> is the end amount: principal plus interest<\/li>\r\n \t<li>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] is the principal (starting amount)<\/li>\r\n \t<li><em>r<\/em> is the interest rate in decimal form<\/li>\r\n \t<li><em>t<\/em> is time<\/li>\r\n<\/ul>\r\nThe units of measurement (years, months, etc.) for the time should match the time period for the interest rate.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong><span style=\"text-decoration: underline;\">Annuities<\/span><\/strong>\r\n<ul>\r\n \t<li>If you're putting money <strong><em>into<\/em><\/strong> the account on a regular basis (monthly\/annually\/quarterly) then you're looking at a <strong>basic annuity<\/strong> problem.\u00a0 Basic annuities are when you are saving money.\u00a0 Usually in an annuity problem, your account starts empty, and has money in the future. Annuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>ANNUITY FORMULA<\/h3>\r\n[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\r\n \t<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\r\n \t<li><em>r <\/em>is the annual interest rate in decimal form.<\/li>\r\n \t<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\r\n<\/ul>\r\nIf the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.\r\n\r\n<\/div>\r\n&nbsp;\r\n<ul>\r\n \t<li>If you're pulling money <em><strong>out<\/strong><\/em> of the account on a regular basis, then you're looking at a <strong>payout annuity<\/strong> problem.\u00a0 Payout annuities are used for things like retirement income, where you start with money in your account, pull money out on a regular basis, and your account ends up empty\u00a0in the future.\u00a0Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>PAYOUT ANNUITY FORMULA<\/h3>\r\n[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{k}\\right)^{-Nk}\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>0<\/sub><\/em> is the balance in the account at the beginning (starting amount, or principal).<\/li>\r\n \t<li><em>d<\/em> is the regular withdrawal (the amount you take out each year, each month, etc.)<\/li>\r\n \t<li><em>r<\/em> is the annual interest rate (in decimal form. Example: 5% = 0.05)<\/li>\r\n \t<li><em>k<\/em> is the number of compounding periods in one year.<\/li>\r\n \t<li><em>N<\/em> is the number of years we plan to take withdrawals<\/li>\r\n<\/ul>\r\n<\/div>\r\nRemember, the most important part of answering any kind of question, money or otherwise, is first to correctly identify what the question is really asking, and then determine what approach will best allow you to solve the problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFor each of the following scenarios, determine if it is a compound interest problem, a savings annuity problem, a payout annuity problem, or a loans problem. Then solve each problem.\r\n<ol>\r\n \t<li>Marcy received an inheritance of $20,000, and invested it at 6% interest. She is going to use it for college, withdrawing money for tuition and expenses each quarter. How much can she take out each quarter if she has 3 years of school left?[reveal-answer q=\"160930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160930\"]This is a payout annuity problem. She can pull out $1833.60 a quarter.[\/hidden-answer]<\/li>\r\n \t<li>Paul wants to buy a new car. Rather than take out a loan, he decides to save $200 a month in an account earning 3% interest compounded monthly. How much will he have saved up after 3 years?[reveal-answer q=\"160931\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160931\"]This is a savings annuity problem. He will have saved up $7,524.11[\/hidden-answer]<\/li>\r\n \t<li>Keisha is managing investments for a non-profit company. \u00a0 \u00a0 \u00a0 They want to invest some money in an account earning 5% interest compounded annually with the goal to have $30,000 in the account in 6 years. How much should Keisha deposit into the account?\u00a0[reveal-answer q=\"160932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160932\"]This is compound interest problem. She would need to deposit $22,386.46.[\/hidden-answer]<\/li>\r\n \t<li>Miao is going to finance new office equipment at a 2% rate over a 4 year term. If she can afford monthly payments of $100, how much new equipment can she buy?\u00a0[reveal-answer q=\"160933\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160933\"]This is a loans problem. She can buy $4,609.33 of new equipment[\/hidden-answer]<\/li>\r\n \t<li>How much would you need to save every month in an account earning 4% interest to have $5,000 saved up in two years?[reveal-answer q=\"160934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160934\"]This is a savings annuity problem. You would need to save $200.46 each month[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the following video, we present more examples of how to use the language in the question to determine which type of equation to use to solve a finance problem.\r\n\r\nhttps:\/\/youtu.be\/V5oG7lLTECs\r\n\r\nIn the next video example, we show how to solve a finance problem that has two stages, the first stage is a savings problem, and the second stage is a withdrawal problem.\r\n\r\nhttps:\/\/youtu.be\/CNkvwMuLuis\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67287&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67306&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67133&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine which equation to use for a given scenario<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/03001020\/chimpthink.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1325 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/03001020\/chimpthink-300x205.png\" alt=\"\" width=\"373\" height=\"254\" \/><\/a>Now that we have surveyed the basic kinds of finance calculations that are used, it may not always be obvious which one to use when you are given a problem to solve.\u00a0Here are some hints on deciding which equation to use, based on the wording of the problem.<\/p>\n<h3>Loans<\/h3>\n<p>The easiest types of problems to identify are loans.\u00a0 Loan problems almost always include words like <strong>loan<\/strong>, <strong>amortize<\/strong> (the fancy word for loans), <strong>finance<\/strong> (i.e. a car), or <strong>mortgage<\/strong> (a home loan). Look for words like monthly or annual payment.<\/p>\n<p>The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.<\/p>\n<div class=\"textbox\">\n<h2>Loans Formula<\/h2>\n<p>[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{k}\\right)^{-Nk}\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>0<\/sub><\/em> is the balance in the account at the beginning (the principal, or amount of the loan).<\/li>\n<li><em>d <\/em>is your loan payment (your monthly payment, annual payment, etc)<\/li>\n<li><em>r<\/em> is the annual interest rate in decimal form.<\/li>\n<li><em>k<\/em> is the number of compounding periods in one year.<\/li>\n<li><em>N<\/em> is the length of the loan, in years.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Interest-Bearing Accounts<\/h3>\n<p>Accounts that gain interest fall into two main categories. \u00a0The first is on where you put money in an account once and let it sit, the other is where you make regular payments or withdrawals from the account as in a retirement account.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Interest<\/strong><\/span><\/p>\n<ul>\n<li>If you&#8217;re letting the money sit in the account with nothing but interest changing the balance, then you&#8217;re looking at a <strong>compound interest<\/strong> problem. Look for words like compounded, or APY.\u00a0Compound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>COMPOUND INTEREST<\/h3>\n<p>[latex]P_{N}=P_{0}\\left(1+\\frac{r}{k}\\right)^{Nk}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\n<li><em>P<sub>0 <\/sub><\/em>is the starting balance of the account (also called initial deposit, or principal)<\/li>\n<li><em>r<\/em> is the annual interest rate in decimal form<\/li>\n<li><em>k<\/em> is the number of compounding periods in one year\n<ul>\n<li>If the compounding is done annually (once a year), <em>k<\/em> = 1.<\/li>\n<li>If the compounding is done quarterly, <em>k<\/em> = 4.<\/li>\n<li>If the compounding is done monthly, <em>k<\/em> = 12.<\/li>\n<li>If the compounding is done daily, <em>k<\/em> = 365.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<ul>\n<li>The exception would be bonds and other investments where the interest is not reinvested; in those cases you\u2019re looking at <strong>simple interest<\/strong>.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>SIMPLE INTEREST OVER TIME<\/h3>\n<p>[latex]\\begin{align}&I={{P}_{0}}rt\\\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\\\\end{align}[\/latex]<\/p>\n<ul>\n<li><em>I<\/em> is the interest<\/li>\n<li><em>A<\/em> is the end amount: principal plus interest<\/li>\n<li>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] is the principal (starting amount)<\/li>\n<li><em>r<\/em> is the interest rate in decimal form<\/li>\n<li><em>t<\/em> is time<\/li>\n<\/ul>\n<p>The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong><span style=\"text-decoration: underline;\">Annuities<\/span><\/strong><\/p>\n<ul>\n<li>If you&#8217;re putting money <strong><em>into<\/em><\/strong> the account on a regular basis (monthly\/annually\/quarterly) then you&#8217;re looking at a <strong>basic annuity<\/strong> problem.\u00a0 Basic annuities are when you are saving money.\u00a0 Usually in an annuity problem, your account starts empty, and has money in the future. Annuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>ANNUITY FORMULA<\/h3>\n<p>[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\n<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\n<li><em>r <\/em>is the annual interest rate in decimal form.<\/li>\n<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\n<\/ul>\n<p>If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<ul>\n<li>If you&#8217;re pulling money <em><strong>out<\/strong><\/em> of the account on a regular basis, then you&#8217;re looking at a <strong>payout annuity<\/strong> problem.\u00a0 Payout annuities are used for things like retirement income, where you start with money in your account, pull money out on a regular basis, and your account ends up empty\u00a0in the future.\u00a0Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>PAYOUT ANNUITY FORMULA<\/h3>\n<p>[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{k}\\right)^{-Nk}\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>0<\/sub><\/em> is the balance in the account at the beginning (starting amount, or principal).<\/li>\n<li><em>d<\/em> is the regular withdrawal (the amount you take out each year, each month, etc.)<\/li>\n<li><em>r<\/em> is the annual interest rate (in decimal form. Example: 5% = 0.05)<\/li>\n<li><em>k<\/em> is the number of compounding periods in one year.<\/li>\n<li><em>N<\/em> is the number of years we plan to take withdrawals<\/li>\n<\/ul>\n<\/div>\n<p>Remember, the most important part of answering any kind of question, money or otherwise, is first to correctly identify what the question is really asking, and then determine what approach will best allow you to solve the problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>For each of the following scenarios, determine if it is a compound interest problem, a savings annuity problem, a payout annuity problem, or a loans problem. Then solve each problem.<\/p>\n<ol>\n<li>Marcy received an inheritance of $20,000, and invested it at 6% interest. She is going to use it for college, withdrawing money for tuition and expenses each quarter. How much can she take out each quarter if she has 3 years of school left?\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160930\">Show Solution<\/span><\/p>\n<div id=\"q160930\" class=\"hidden-answer\" style=\"display: none\">This is a payout annuity problem. She can pull out $1833.60 a quarter.<\/div>\n<\/div>\n<\/li>\n<li>Paul wants to buy a new car. Rather than take out a loan, he decides to save $200 a month in an account earning 3% interest compounded monthly. How much will he have saved up after 3 years?\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160931\">Show Solution<\/span><\/p>\n<div id=\"q160931\" class=\"hidden-answer\" style=\"display: none\">This is a savings annuity problem. He will have saved up $7,524.11<\/div>\n<\/div>\n<\/li>\n<li>Keisha is managing investments for a non-profit company. \u00a0 \u00a0 \u00a0 They want to invest some money in an account earning 5% interest compounded annually with the goal to have $30,000 in the account in 6 years. How much should Keisha deposit into the account?\u00a0\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160932\">Show Solution<\/span><\/p>\n<div id=\"q160932\" class=\"hidden-answer\" style=\"display: none\">This is compound interest problem. She would need to deposit $22,386.46.<\/div>\n<\/div>\n<\/li>\n<li>Miao is going to finance new office equipment at a 2% rate over a 4 year term. If she can afford monthly payments of $100, how much new equipment can she buy?\u00a0\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160933\">Show Solution<\/span><\/p>\n<div id=\"q160933\" class=\"hidden-answer\" style=\"display: none\">This is a loans problem. She can buy $4,609.33 of new equipment<\/div>\n<\/div>\n<\/li>\n<li>How much would you need to save every month in an account earning 4% interest to have $5,000 saved up in two years?\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160934\">Show Solution<\/span><\/p>\n<div id=\"q160934\" class=\"hidden-answer\" style=\"display: none\">This is a savings annuity problem. You would need to save $200.46 each month<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>In the following video, we present more examples of how to use the language in the question to determine which type of equation to use to solve a finance problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Identifying type of finance problem\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/V5oG7lLTECs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next video example, we show how to solve a finance problem that has two stages, the first stage is a savings problem, and the second stage is a withdrawal problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multistage finance problem\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CNkvwMuLuis?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67287&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67306&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=67133&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-392\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Which Equation to Use?. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Identifying type of finance problem. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/V5oG7lLTECs\">https:\/\/youtu.be\/V5oG7lLTECs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Multistage finance problem. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CNkvwMuLuis\">https:\/\/youtu.be\/CNkvwMuLuis<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 67287, 67306, 67133. <strong>Authored by<\/strong>: Abert, Rex. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Which Equation to Use?\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Identifying type of finance problem\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/V5oG7lLTECs\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Multistage finance problem\",\"author\":\"OCLPhase2\\'s 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