{"id":2547,"date":"2023-05-10T21:09:02","date_gmt":"2023-05-10T21:09:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2547"},"modified":"2024-10-18T20:55:20","modified_gmt":"2024-10-18T20:55:20","slug":"annuities-and-loans-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/annuities-and-loans-learn-it-3\/","title":{"raw":"Annuities and Loans: Learn It 3","rendered":"Annuities and Loans: Learn It 3"},"content":{"raw":"<h2>Conventional Loans<\/h2>\r\n<p>You just learned about payout annuities, now, you will learn about <strong>conventional loans<\/strong> (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.<\/p>\r\n<section class=\"textbox proTip\">\r\n<p><strong>New topic, same formula!<\/strong><\/p>\r\n<p>Mathematical formulas sometimes overlap, applying to more than one application. All the exercises and examples in this section use the same formula and techniques that you've already seen.<\/p>\r\n<\/section>\r\n<p>One great thing about loans is that they use exactly the same formula as a payout annuity.<\/p>\r\n<p>To see why, imagine that you had [latex]$10,000[\/latex] invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after [latex]5[\/latex] years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests [latex]$10,000[\/latex] in you. Since you\u2019re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Loans Formula<\/h3>\r\n<center>[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{n}\\right)^{-nt}\\right)}{\\left(\\frac{r}{n}\\right)}[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n\t<li>[latex]P_0[\/latex] is the balance in the account at the beginning (the principal, or amount of the loan).<\/li>\r\n\t<li>[latex]d[\/latex] is your loan payment (your monthly payment, annual payment, etc)<\/li>\r\n\t<li>[latex]r[\/latex] is the annual interest rate in decimal form.<\/li>\r\n\t<li>[latex]n[\/latex] is the number of compounding periods in one year.<\/li>\r\n\t<li>[latex]t[\/latex] is the length of the loan, in years.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<p>Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>When do you use this?<\/strong><\/p>\r\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>\r\n<ul>\r\n\t<li>Compound interest: One deposit<\/li>\r\n\t<li>Annuity: Many deposits<\/li>\r\n\t<li>Payout Annuity: Many withdrawals<\/li>\r\n\t<li>Loans: Many payments<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">You can afford [latex]$200[\/latex] per month as a car payment. If you can get an auto loan at [latex]3\\%[\/latex] interest for [latex]60[\/latex] months ([latex]5[\/latex] years), how expensive a car can you afford? In other words, what loan amount can you pay off with [latex]$200[\/latex] per month?<br \/>\r\n[reveal-answer q=\"129373\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"129373\"]In this example,\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]d = $200[\/latex]<\/td>\r\n<td>the monthly loan payment<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]r= 0.03[\/latex]<\/td>\r\n<td>[latex]3\\%[\/latex] annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]n = 12[\/latex]<\/td>\r\n<td>since we\u2019re doing monthly payments, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t = 5[\/latex]<\/td>\r\n<td>since we\u2019re making monthly payments for [latex]5[\/latex] years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>We\u2019re looking for [latex]P_0[\/latex], the starting amount of the loan.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{{P}_{0}}=\\frac{200\\left(1-{{\\left(1+\\frac{0.03}{12}\\right)}^{-5(12)}}\\right)}{\\left(\\frac{0.03}{12}\\right)}\\\\&amp;{{P}_{0}}=\\frac{200\\left(1-{{\\left(1.0025\\right)}^{-60}}\\right)}{\\left(0.0025\\right)}\\\\&amp;{{P}_{0}}=\\frac{200\\left(1-0.861\\right)}{\\left(0.0025\\right)}=\\$11,120 \\\\\\end{align}[\/latex]<\/p>\r\n<p>You can afford a [latex]$11,120[\/latex] loan.<\/p>\r\n<p>You will pay a total of [latex]$12,000[\/latex] ([latex]$200[\/latex] per month for [latex]60[\/latex] months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, you\u2019re paying [latex]$12,000-$11,120 = $880[\/latex] interest total.<\/p>\r\n<p>Details of this example are examined in this video.<\/p>\r\n<p>[embed]https:\/\/youtu.be\/5NiNcdYytvY[\/embed]<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Car+loan.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCar loan\u201d here (opens in new window).<\/a><\/p>\r\n<p><em>Note: This video uses [latex]k[\/latex] for [latex]n[\/latex] and [latex]N[\/latex] for [latex]t[\/latex].<\/em><br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>In the example above, you computed [latex]P_{0}[\/latex], the initial loan amount.<\/p>\r\n<p>In the example below, you are given the loan amount and must solve for the amount of the monthly payment, [latex]d[\/latex]. Use the same technique that you used in the previous sections.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6942[\/ohm2_question]<\/section>\r\n<h3>Calculating the Balance<\/h3>\r\n<p>With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.<\/p>\r\n<p>To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don\u2019t already know them.\u00a0<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were [latex]$1,000[\/latex] a month, after a year you will <em>not<\/em> have paid off [latex]$12,000[\/latex] of the loan balance.<\/p>\r\n<\/section>\r\n<p>To determine the remaining loan balance, we can think \u201chow much loan will these loan payments be able to pay off in the remaining time on the loan?\u201d<\/p>\r\n<section class=\"textbox example\">If a mortgage at a [latex]6\\%[\/latex] interest rate has payments of [latex]$1,000[\/latex] a month, how much will the loan balance be [latex]10[\/latex] years from the end the loan?<br \/>\r\n[reveal-answer q=\"146377\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"146377\"]To determine this, we are looking for the amount of the loan that can be paid off by [latex]$1,000[\/latex] a month payments in [latex]10[\/latex] years. In other words, we\u2019re looking for [latex]P_0[\/latex] when\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]d = $1,000[\/latex]<\/td>\r\n<td>the monthly loan payment<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]r= 0.06[\/latex]<\/td>\r\n<td>[latex]6\\%[\/latex] annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]n = 12[\/latex]<\/td>\r\n<td>since we\u2019re doing monthly payments, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t = 10[\/latex]<\/td>\r\n<td>\u00a0since we\u2019re making monthly payments for [latex]10[\/latex] more years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{{P}_{0}}=\\frac{1000\\left(1-{{\\left(1+\\frac{0.06}{12}\\right)}^{-10(12)}}\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&amp;{{P}_{0}}=\\frac{1000\\left(1-{{\\left(1.005\\right)}^{-120}}\\right)}{\\left(0.005\\right)}\\\\&amp;{{P}_{0}}=\\frac{1000\\left(1-0.5496\\right)}{\\left(0.005\\right)}=\\$90,073.45 \\\\\\end{align}[\/latex]<\/p>\r\n<p>The loan balance with 10 years remaining on the loan will be [latex]$90,073.45[\/latex].<\/p>\r\n<p>This example is explained in the following video:<\/p>\r\n<p>[embed]https:\/\/youtu.be\/fXLzeyCfAwE[\/embed]<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Calculate+remaining+balance+on+loan+from+payment.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCalculate remaining balance on loan from payment\u201d here (opens in new window).<\/a><\/p>\r\n<p><em>Note: This video uses [latex]k[\/latex] for [latex]n[\/latex] and [latex]N[\/latex] for [latex]t[\/latex].<\/em><br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Oftentimes answering remaining balance questions requires two steps:<\/p>\r\n<ol>\r\n\t<li>Calculating the monthly payments on the loan<\/li>\r\n\t<li>Calculating the remaining loan balance based on the <em>remaining time<\/em> on the loan<\/li>\r\n<\/ol>\r\n<section class=\"textbox example\">A couple purchases a home with a [latex]$180,000[\/latex] mortgage at [latex]4\\%[\/latex] for [latex]30[\/latex] years with monthly payments. What will the remaining balance on their mortgage be after [latex]5[\/latex] years?<br \/>\r\n[reveal-answer q=\"423248\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"423248\"]First we will calculate their monthly payments. We\u2019re looking for [latex]d[\/latex].\r\n\r\n<table style=\"width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]r = 0.04[\/latex]<\/td>\r\n<td style=\"width: 70%;\">[latex]4\\%[\/latex] annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]n = 12[\/latex]<\/td>\r\n<td>since they\u2019re paying monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t = 30[\/latex]<\/td>\r\n<td>[latex]30[\/latex] years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]P_0 = $180,000[\/latex]<\/td>\r\n<td>the starting loan amount<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>We set up the equation and solve for [latex]d[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;180,000=\\frac{d\\left(1-{{\\left(1+\\frac{0.04}{12}\\right)}^{-30(12)}}\\right)}{\\left(\\frac{0.04}{12}\\right)}\\\\&amp;180,000=\\frac{d\\left(1-{{\\left(1.00333\\right)}^{-360}}\\right)}{\\left(0.00333\\right)}\\\\&amp;180,000=d(209.562)\\\\&amp;d=\\frac{180,000}{209.562}=\\$858.93 \\\\\\end{align}[\/latex]<\/p>\r\n<p>Now that we know the monthly payments, we can determine the remaining balance. We want the remaining balance after [latex]5[\/latex] years, when [latex]25[\/latex] years will be remaining on the loan, so we calculate the loan balance that will be paid off with the monthly payments over those [latex]25[\/latex] years.<\/p>\r\n<table style=\"width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]d = $858.93[\/latex]<\/td>\r\n<td style=\"width: 70%;\">the monthly loan payment we calculated above<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]r= 0.04[\/latex]<\/td>\r\n<td>[latex]4\\%[\/latex] annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]n = 12[\/latex]<\/td>\r\n<td>since they\u2019re doing monthly payments<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t = 25[\/latex]<\/td>\r\n<td>since they\u2019d be making monthly payments for [latex]25[\/latex] more years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{{P}_{0}}=\\frac{858.93\\left(1-{{\\left(1+\\frac{0.04}{12}\\right)}^{-25(12)}}\\right)}{\\left(\\frac{0.04}{12}\\right)}\\\\&amp;{{P}_{0}}=\\frac{858.93\\left(1-{{\\left(1.00333\\right)}^{-300}}\\right)}{\\left(0.00333\\right)}\\\\&amp;{{P}_{0}}=\\frac{858.93\\left(1-0.369\\right)}{\\left(0.00333\\right)}=\\$162,758.21 \\\\\\end{align}[\/latex]<\/p>\r\n<p>The loan balance after [latex]5[\/latex] years, with [latex]25[\/latex] years remaining on the loan, will be [latex]$162,758.21[\/latex].<\/p>\r\n<p>Over that [latex]5[\/latex] years, the couple has paid off [latex]$180,000 - $162,758.21 = $17,241.79[\/latex] of the loan balance. They have paid a total of [latex]$858.93[\/latex] a month for [latex]5[\/latex] years ([latex]60[\/latex] months), for a total of [latex]$51,535.80[\/latex], so [latex]$51,535.80 - $17,241.79 = $34,294.01[\/latex] of what they have paid so far has been interest.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Solving for Time<\/h3>\r\n<p>Recall that we have used logarithms to solve for time other exponent interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.<\/p>\r\n<section class=\"textbox example\">Joel is considering putting a [latex]$1,000[\/latex] laptop purchase on his credit card, which has an interest rate of [latex]12\\%[\/latex] compounded monthly. How long will it take him to pay off the purchase if he makes payments of [latex]$30[\/latex] a month?<br \/>\r\n[reveal-answer q=\"427851\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"427851\"]<br \/>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r@{\\hfill}l}<br \/>\r\nd=\\$30 &amp;&amp; \\text{The monthly payments} \\\\<br \/>\r\nr=0.12 &amp;&amp; \\text{12% annual rate} \\\\<br \/>\r\nn=12 &amp;&amp; \\text{since we're making monthly payments} \\\\<br \/>\r\nP_0=\\$1,000 &amp;&amp; \\text{we're starting with a } \\$1,000 \\text{ loan} \\\\<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p>We are solving for [latex]t[\/latex], the time to pay off the loan.<\/p>\r\n<p style=\"text-align: center;\">[latex]1000=\\frac{30\\left(1-\\left(1+\\frac{0.12}{12}\\right)^{-N*12}\\right)}{\\frac{0.12}{12}}[\/latex]<\/p>\r\n<p>Solving for [latex]t[\/latex] gives [latex]3.396[\/latex]. It will take about [latex]3.4[\/latex] years to pay off the purchase.<\/p>\r\n\r\n[\/hidden-answer]<\/section>","rendered":"<h2>Conventional Loans<\/h2>\n<p>You just learned about payout annuities, now, you will learn about <strong>conventional loans<\/strong> (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.<\/p>\n<section class=\"textbox proTip\">\n<p><strong>New topic, same formula!<\/strong><\/p>\n<p>Mathematical formulas sometimes overlap, applying to more than one application. All the exercises and examples in this section use the same formula and techniques that you&#8217;ve already seen.<\/p>\n<\/section>\n<p>One great thing about loans is that they use exactly the same formula as a payout annuity.<\/p>\n<p>To see why, imagine that you had [latex]$10,000[\/latex] invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after [latex]5[\/latex] years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests [latex]$10,000[\/latex] in you. Since you\u2019re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Loans Formula<\/h3>\n<div style=\"text-align: center;\">[latex]P_{0}=\\frac{d\\left(1-\\left(1+\\frac{r}{n}\\right)^{-nt}\\right)}{\\left(\\frac{r}{n}\\right)}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<ul>\n<li>[latex]P_0[\/latex] is the balance in the account at the beginning (the principal, or amount of the loan).<\/li>\n<li>[latex]d[\/latex] is your loan payment (your monthly payment, annual payment, etc)<\/li>\n<li>[latex]r[\/latex] is the annual interest rate in decimal form.<\/li>\n<li>[latex]n[\/latex] is the number of compounding periods in one year.<\/li>\n<li>[latex]t[\/latex] is the length of the loan, in years.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>When do you use this?<\/strong><\/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<ul>\n<li>Compound interest: One deposit<\/li>\n<li>Annuity: Many deposits<\/li>\n<li>Payout Annuity: Many withdrawals<\/li>\n<li>Loans: Many payments<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">You can afford [latex]$200[\/latex] per month as a car payment. If you can get an auto loan at [latex]3\\%[\/latex] interest for [latex]60[\/latex] months ([latex]5[\/latex] years), how expensive a car can you afford? In other words, what loan amount can you pay off with [latex]$200[\/latex] per month?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q129373\">Show Solution<\/button><\/p>\n<div id=\"q129373\" class=\"hidden-answer\" style=\"display: none\">In this example,<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]d = $200[\/latex]<\/td>\n<td>the monthly loan payment<\/td>\n<\/tr>\n<tr>\n<td>[latex]r= 0.03[\/latex]<\/td>\n<td>[latex]3\\%[\/latex] annual rate<\/td>\n<\/tr>\n<tr>\n<td>[latex]n = 12[\/latex]<\/td>\n<td>since we\u2019re doing monthly payments, we\u2019ll compound monthly<\/td>\n<\/tr>\n<tr>\n<td>[latex]t = 5[\/latex]<\/td>\n<td>since we\u2019re making monthly payments for [latex]5[\/latex] years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>We\u2019re looking for [latex]P_0[\/latex], the starting amount of the loan.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{{P}_{0}}=\\frac{200\\left(1-{{\\left(1+\\frac{0.03}{12}\\right)}^{-5(12)}}\\right)}{\\left(\\frac{0.03}{12}\\right)}\\\\&{{P}_{0}}=\\frac{200\\left(1-{{\\left(1.0025\\right)}^{-60}}\\right)}{\\left(0.0025\\right)}\\\\&{{P}_{0}}=\\frac{200\\left(1-0.861\\right)}{\\left(0.0025\\right)}=\\$11,120 \\\\\\end{align}[\/latex]<\/p>\n<p>You can afford a [latex]$11,120[\/latex] loan.<\/p>\n<p>You will pay a total of [latex]$12,000[\/latex] ([latex]$200[\/latex] per month for [latex]60[\/latex] months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, you\u2019re paying [latex]$12,000-$11,120 = $880[\/latex] interest total.<\/p>\n<p>Details of this example are examined in this video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Car loan\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5NiNcdYytvY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Car+loan.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCar loan\u201d here (opens in new window).<\/a><\/p>\n<p><em>Note: This video uses [latex]k[\/latex] for [latex]n[\/latex] and [latex]N[\/latex] for [latex]t[\/latex].<\/em>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>In the example above, you computed [latex]P_{0}[\/latex], the initial loan amount.<\/p>\n<p>In the example below, you are given the loan amount and must solve for the amount of the monthly payment, [latex]d[\/latex]. Use the same technique that you used in the previous sections.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6942\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6942&theme=lumen&iframe_resize_id=ohm6942&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Calculating the Balance<\/h3>\n<p>With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.<\/p>\n<p>To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don\u2019t already know them.\u00a0<\/p>\n<section class=\"textbox proTip\">\n<p>Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were [latex]$1,000[\/latex] a month, after a year you will <em>not<\/em> have paid off [latex]$12,000[\/latex] of the loan balance.<\/p>\n<\/section>\n<p>To determine the remaining loan balance, we can think \u201chow much loan will these loan payments be able to pay off in the remaining time on the loan?\u201d<\/p>\n<section class=\"textbox example\">If a mortgage at a [latex]6\\%[\/latex] interest rate has payments of [latex]$1,000[\/latex] a month, how much will the loan balance be [latex]10[\/latex] years from the end the loan?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q146377\">Show Solution<\/button><\/p>\n<div id=\"q146377\" class=\"hidden-answer\" style=\"display: none\">To determine this, we are looking for the amount of the loan that can be paid off by [latex]$1,000[\/latex] a month payments in [latex]10[\/latex] years. In other words, we\u2019re looking for [latex]P_0[\/latex] when<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]d = $1,000[\/latex]<\/td>\n<td>the monthly loan payment<\/td>\n<\/tr>\n<tr>\n<td>[latex]r= 0.06[\/latex]<\/td>\n<td>[latex]6\\%[\/latex] annual rate<\/td>\n<\/tr>\n<tr>\n<td>[latex]n = 12[\/latex]<\/td>\n<td>since we\u2019re doing monthly payments, we\u2019ll compound monthly<\/td>\n<\/tr>\n<tr>\n<td>[latex]t = 10[\/latex]<\/td>\n<td>\u00a0since we\u2019re making monthly payments for [latex]10[\/latex] more years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{{P}_{0}}=\\frac{1000\\left(1-{{\\left(1+\\frac{0.06}{12}\\right)}^{-10(12)}}\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&{{P}_{0}}=\\frac{1000\\left(1-{{\\left(1.005\\right)}^{-120}}\\right)}{\\left(0.005\\right)}\\\\&{{P}_{0}}=\\frac{1000\\left(1-0.5496\\right)}{\\left(0.005\\right)}=\\$90,073.45 \\\\\\end{align}[\/latex]<\/p>\n<p>The loan balance with 10 years remaining on the loan will be [latex]$90,073.45[\/latex].<\/p>\n<p>This example is explained in the following video:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Calculate remaining balance on loan from payment\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fXLzeyCfAwE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Calculate+remaining+balance+on+loan+from+payment.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCalculate remaining balance on loan from payment\u201d here (opens in new window).<\/a><\/p>\n<p><em>Note: This video uses [latex]k[\/latex] for [latex]n[\/latex] and [latex]N[\/latex] for [latex]t[\/latex].<\/em>\n<\/div>\n<\/div>\n<\/section>\n<p>Oftentimes answering remaining balance questions requires two steps:<\/p>\n<ol>\n<li>Calculating the monthly payments on the loan<\/li>\n<li>Calculating the remaining loan balance based on the <em>remaining time<\/em> on the loan<\/li>\n<\/ol>\n<section class=\"textbox example\">A couple purchases a home with a [latex]$180,000[\/latex] mortgage at [latex]4\\%[\/latex] for [latex]30[\/latex] years with monthly payments. What will the remaining balance on their mortgage be after [latex]5[\/latex] years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q423248\">Show Solution<\/button><\/p>\n<div id=\"q423248\" class=\"hidden-answer\" style=\"display: none\">First we will calculate their monthly payments. We\u2019re looking for [latex]d[\/latex].<\/p>\n<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td>[latex]r = 0.04[\/latex]<\/td>\n<td style=\"width: 70%;\">[latex]4\\%[\/latex] annual rate<\/td>\n<\/tr>\n<tr>\n<td>[latex]n = 12[\/latex]<\/td>\n<td>since they\u2019re paying monthly<\/td>\n<\/tr>\n<tr>\n<td>[latex]t = 30[\/latex]<\/td>\n<td>[latex]30[\/latex] years<\/td>\n<\/tr>\n<tr>\n<td>[latex]P_0 = $180,000[\/latex]<\/td>\n<td>the starting loan amount<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>We set up the equation and solve for [latex]d[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&180,000=\\frac{d\\left(1-{{\\left(1+\\frac{0.04}{12}\\right)}^{-30(12)}}\\right)}{\\left(\\frac{0.04}{12}\\right)}\\\\&180,000=\\frac{d\\left(1-{{\\left(1.00333\\right)}^{-360}}\\right)}{\\left(0.00333\\right)}\\\\&180,000=d(209.562)\\\\&d=\\frac{180,000}{209.562}=\\$858.93 \\\\\\end{align}[\/latex]<\/p>\n<p>Now that we know the monthly payments, we can determine the remaining balance. We want the remaining balance after [latex]5[\/latex] years, when [latex]25[\/latex] years will be remaining on the loan, so we calculate the loan balance that will be paid off with the monthly payments over those [latex]25[\/latex] years.<\/p>\n<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td>[latex]d = $858.93[\/latex]<\/td>\n<td style=\"width: 70%;\">the monthly loan payment we calculated above<\/td>\n<\/tr>\n<tr>\n<td>[latex]r= 0.04[\/latex]<\/td>\n<td>[latex]4\\%[\/latex] annual rate<\/td>\n<\/tr>\n<tr>\n<td>[latex]n = 12[\/latex]<\/td>\n<td>since they\u2019re doing monthly payments<\/td>\n<\/tr>\n<tr>\n<td>[latex]t = 25[\/latex]<\/td>\n<td>since they\u2019d be making monthly payments for [latex]25[\/latex] more years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{{P}_{0}}=\\frac{858.93\\left(1-{{\\left(1+\\frac{0.04}{12}\\right)}^{-25(12)}}\\right)}{\\left(\\frac{0.04}{12}\\right)}\\\\&{{P}_{0}}=\\frac{858.93\\left(1-{{\\left(1.00333\\right)}^{-300}}\\right)}{\\left(0.00333\\right)}\\\\&{{P}_{0}}=\\frac{858.93\\left(1-0.369\\right)}{\\left(0.00333\\right)}=\\$162,758.21 \\\\\\end{align}[\/latex]<\/p>\n<p>The loan balance after [latex]5[\/latex] years, with [latex]25[\/latex] years remaining on the loan, will be [latex]$162,758.21[\/latex].<\/p>\n<p>Over that [latex]5[\/latex] years, the couple has paid off [latex]$180,000 - $162,758.21 = $17,241.79[\/latex] of the loan balance. They have paid a total of [latex]$858.93[\/latex] a month for [latex]5[\/latex] years ([latex]60[\/latex] months), for a total of [latex]$51,535.80[\/latex], so [latex]$51,535.80 - $17,241.79 = $34,294.01[\/latex] of what they have paid so far has been interest.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Solving for Time<\/h3>\n<p>Recall that we have used logarithms to solve for time other exponent interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.<\/p>\n<section class=\"textbox example\">Joel is considering putting a [latex]$1,000[\/latex] laptop purchase on his credit card, which has an interest rate of [latex]12\\%[\/latex] compounded monthly. How long will it take him to pay off the purchase if he makes payments of [latex]$30[\/latex] a month?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427851\">Show Solution<\/button><\/p>\n<div id=\"q427851\" class=\"hidden-answer\" style=\"display: none\"><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r@{\\hfill}l}<br \/>  d=\\$30 && \\text{The monthly payments} \\\\<br \/>  r=0.12 && \\text{12% annual rate} \\\\<br \/>  n=12 && \\text{since we're making monthly payments} \\\\<br \/>  P_0=\\$1,000 && \\text{we're starting with a } \\$1,000 \\text{ loan} \\\\<br \/>  \\end{array}[\/latex]<\/p>\n<p>We are solving for [latex]t[\/latex], the time to pay off the loan.<\/p>\n<p style=\"text-align: center;\">[latex]1000=\\frac{30\\left(1-\\left(1+\\frac{0.12}{12}\\right)^{-N*12}\\right)}{\\frac{0.12}{12}}[\/latex]<\/p>\n<p>Solving for [latex]t[\/latex] gives [latex]3.396[\/latex]. It will take about [latex]3.4[\/latex] years to pay off the purchase.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":28,"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":"learn_it","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\/2547"}],"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":37,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2547\/revisions"}],"predecessor-version":[{"id":14632,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2547\/revisions\/14632"}],"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\/2547\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2547"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2547"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2547"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2547"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}