{"id":1086,"date":"2017-01-11T01:00:24","date_gmt":"2017-01-11T01:00:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1086"},"modified":"2021-05-19T19:08:43","modified_gmt":"2021-05-19T19:08:43","slug":"introduction-how-interest-is-calculated","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cccs-coreq-mathforliberalarts\/chapter\/introduction-how-interest-is-calculated\/","title":{"raw":"3.2: Simple and Compound Interest","rendered":"3.2: Simple and Compound Interest"},"content":{"raw":"<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02232307\/interest.jpg\"><img class=\"size-medium wp-image-1313 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02232307\/interest-300x172.jpg\" alt=\"\" width=\"300\" height=\"172\" \/><\/a>\r\n\r\nWe have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. This section will show you how to solve these types of problems by hand. At the end of the section, there is a video and a calculator you can use to solve these problems as well.\r\n\r\n&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nThe leaning objectives for this section include:\r\n<ol>\r\n \t<li>Calculate one-time simple interest, and simple interest over time<\/li>\r\n \t<li>Calculate compound interest<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h1 style=\"text-align: center\">1. Simple\u00a0Interest<\/h1>\r\nDiscussing interest starts with the <strong>principal<\/strong>, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest.\r\n\r\n[embed]https:\/\/youtu.be\/7gRgZW3yxjY[\/embed]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200620\/money-1604921_1280.jpg\"><img class=\"aligncenter wp-image-553 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200620\/money-1604921_1280-1024x682.jpg\" alt=\"four rolled-up dollar bills seeming to grow out of dirt, with a miniature rake lying in between them\" width=\"1024\" height=\"682\" \/><\/a>\r\n<div class=\"textbox\">\r\n<h3>Simple One-time Interest<\/h3>\r\n[latex]\\begin{align}&amp;I={{P}_{0}}r\\\\&amp; \\\\&amp;A={{P}_{0}}+I\\\\&amp;A={{P}_{0}}+{{P}_{0}}r\\\\&amp;A={{P}_{0}}(1+r)\\\\\\end{align}[\/latex]\r\n\r\n&nbsp;\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. Example: 5% = 0.05)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nA friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest (simple) will you earn?\r\n[reveal-answer q=\"227650\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"227650\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] = $300<\/td>\r\n<td>the principal<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.03<\/td>\r\n<td>3% rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>I<\/em> = $300(0.03) = $9.<\/td>\r\n<td>You will earn $9 interest.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nThe following video works through this example in detail.\r\n\r\nhttps:\/\/youtu.be\/TJYq7XGB8EY\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nOne-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly.\r\n\r\nFor example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nSuppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?\r\n[reveal-answer q=\"14596\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"14596\"]Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.[\/hidden-answer]\r\n\r\nFurther explanation about solving this example can be seen here.\r\n\r\nhttps:\/\/youtu.be\/rNOEYPCnGwg\r\n\r\n<\/div>\r\nWe can generalize this idea of simple interest over time.\r\n<div class=\"textbox\">\r\n<h3>Simple Interest over Time<\/h3>\r\n[latex]\\begin{align}&amp;I={{P}_{0}}rt\\\\&amp; \\\\&amp;A={{P}_{0}}+I\\\\&amp;A={{P}_{0}}+{{P}_{0}}rt\\\\&amp;={{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<div class=\"textbox\">\r\n<h3>APR \u2013 Annual Percentage Rate<\/h3>\r\nInterest rates are usually given as an <strong>annual percentage rate (APR)<\/strong> \u2013 the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.\r\n\r\nFor example, a 6% APR paid monthly would be divided into twelve 0.5% payments.\r\n[latex]6\\div{12}=0.5[\/latex]\r\n\r\nA 4% annual rate paid quarterly would be divided into four 1% payments.\r\n[latex]4\\div{4}=1[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTreasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?\r\n[reveal-answer q=\"529216\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"529216\"]\r\n\r\nSince interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] = $1000<\/td>\r\n<td>the principal<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.02<\/td>\r\n<td>2% rate per half-year<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>t<\/em> = 8<\/td>\r\n<td>4 years = 8 half-years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>I<\/em> = $1000(0.02)(8) = $160.<\/td>\r\n<td>\u00a0You will earn $160 interest total over the four years.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nThis video explains the solution.\r\n\r\nhttps:\/\/youtu.be\/IfVn20go7-Y\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\n[embed]https:\/\/www.myopenmath.com\/multiembedq.php?id=72467&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\nA loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.\r\n[reveal-answer q=\"288479\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"288479\"]\r\n\r\nI = $30 of interest\r\n[latex]P_0[\/latex] = $500 principal\r\nr = unknown\r\nt = 1 month\r\n\r\nUsing [latex]I = P_0rt[\/latex], we get [latex]30 = 500\u00b7r\u00b71[\/latex]. Solving, we get r = 0.06, or 6%. Since the time was monthly, this is the monthly interest. The annual rate would be 12 times this: 72% interest.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\n[embed]https:\/\/www.myopenmath.com\/multiembedq.php?id=929&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\n<h1 style=\"text-align: center\">2. Compound Interest<\/h1>\r\nWith simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called <strong>compounding<\/strong>.\r\n\r\n[embed]https:\/\/youtu.be\/DgLMeA8Tr6A[\/embed]\r\n<h2>TIME VALUE OF MONEY CALCULATOR<\/h2>\r\nThe following <a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a> application\/calculator will be a helpful tool in checking your answers:\r\n\r\nThe Time Value of Money Solver asks you to provide N, I%, PV, PMT, FV, P\/Y, and C\/Y.\r\n\r\nN is the number of years, i% is the interest rate as a number (not a decimal), PV is the present value or\u00a0<em>P<sub>0<\/sub><\/em>, PMT is the payment each period (if there is no payment, then put in zero), FV is the future value or A, P\/Y is periods per year (so if there is monthly compounding, this would be 12), and C\/Y is how often compounding occurs (so if compounded quarterly, then put in 4). Note, that P\/Y and C\/Y will be the same for any problem you will see in this course. Lastly, PV and FV always have opposite signs (if PV is positive, FV will be negative)\r\n\r\nHere is a video walking you through one of these types of problems:\r\n\r\nhttps:\/\/youtu.be\/ohyUCZL97-8\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200323\/achievement-18134_1280.jpg\"><img class=\"aligncenter wp-image-552 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200323\/achievement-18134_1280-1024x759.jpg\" alt=\"a row of gold coin stacks. From left to right, they grown from one coin, to two, to four, ending with a stack of 32 coins\" width=\"1024\" height=\"759\" \/><\/a>\r\n\r\nSuppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?\r\n\r\nThe 3% interest is an annual percentage rate (APR) \u2013 the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\\frac{3%}{12}[\/latex]= 0.25% per month.\r\n\r\nIn the first month,\r\n<ul>\r\n \t<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\r\n \t<li><em>r<\/em> = 0.0025 (0.25%)<\/li>\r\n \t<li><em>I <\/em>= $1000 (0.0025) = $2.50<\/li>\r\n \t<li><em>A<\/em> = $1000 + $2.50 = $1002.50<\/li>\r\n<\/ul>\r\nIn the first month, we will earn $2.50 in interest, raising our account balance to $1002.50.\r\n\r\n&nbsp;\r\n\r\nIn the second month,\r\n<ul>\r\n \t<li><em>P<sub>0<\/sub><\/em> = $1002.50<\/li>\r\n \t<li><em>I <\/em>= $1002.50 (0.0025) = $2.51 (rounded)<\/li>\r\n \t<li><em>A<\/em> = $1002.50 + $2.51 = $1005.01<\/li>\r\n<\/ul>\r\nNotice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original $1000 we deposited, but we also earned interest on the $2.50 of interest we earned the first month. This is the key advantage that <strong>compounding<\/strong>\u00a0interest gives us.\r\n\r\nCalculating out a few more months gives the following:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Month<\/strong><\/td>\r\n<td><strong>Starting balance<\/strong><\/td>\r\n<td><strong>Interest earned<\/strong><\/td>\r\n<td><strong>Ending Balance<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>1000.00<\/td>\r\n<td>2.50<\/td>\r\n<td>1002.50<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>1002.50<\/td>\r\n<td>2.51<\/td>\r\n<td>1005.01<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>1005.01<\/td>\r\n<td>2.51<\/td>\r\n<td>1007.52<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>1007.52<\/td>\r\n<td>2.52<\/td>\r\n<td>1010.04<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>1010.04<\/td>\r\n<td>2.53<\/td>\r\n<td>1012.57<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>1012.57<\/td>\r\n<td>2.53<\/td>\r\n<td>1015.10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7<\/td>\r\n<td>1015.10<\/td>\r\n<td>2.54<\/td>\r\n<td>1017.64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>1017.64<\/td>\r\n<td>2.54<\/td>\r\n<td>1020.18<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9<\/td>\r\n<td>1020.18<\/td>\r\n<td>2.55<\/td>\r\n<td>1022.73<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>1022.73<\/td>\r\n<td>2.56<\/td>\r\n<td>1025.29<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>11<\/td>\r\n<td>1025.29<\/td>\r\n<td>2.56<\/td>\r\n<td>1027.85<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12<\/td>\r\n<td>1027.85<\/td>\r\n<td>2.57<\/td>\r\n<td>1030.42<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nCALCULATOR: See if you can recreate the Ending Balance column using the calculator (<a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a>)\r\n\r\nFor example, Month 1, starting balance of 1000 means you would put the following into the calculator:\r\n\r\nN=1\/12 (because we are only looking at 1 month and this box is the number of years)\r\n\r\nI=3 (interest rate of 3%)\r\n\r\nPV=1000 (the money we started with)\r\n\r\nPMT=0 (there were no regular, recurring payments)\r\n\r\nFV = (what we are solving for) - push this button and make sure it matches the above table (note that it will be negative because PV and FV are always opposite signs)\r\n\r\nP\/Y=12\r\n\r\nC\/Y=12\r\n\r\nFORMULA: We want to simplify the process for calculating compounding, because creating a table like the one above is time consuming. Luckily, math is good at giving you ways to take shortcuts. To find an equation to represent this, if <em>P<sub>m <\/sub><\/em>represents the amount of money after <em>m<\/em> months, then we could write the recursive equation:\r\n\r\n<em>P<sub>0<\/sub><\/em> = $1000\r\n\r\n<em>P<sub>m<\/sub><\/em> = (1+0.0025)<em>P<sub>m-1<\/sub><\/em>\r\n\r\nYou probably recognize this as the recursive form of exponential growth. If not, we go through the steps to build an explicit equation for the growth in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nBuild an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly.\r\n[reveal-answer q=\"530288\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"530288\"]\r\n<ul>\r\n \t<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\r\n \t<li><em>P\u00ad<sub>1<\/sub><\/em> = 1.0025<em>P\u00ad<sub>0<\/sub><\/em> = 1.0025 (1000)<\/li>\r\n \t<li><em>P\u00ad<sub>2<\/sub><\/em> = 1.0025<em>P\u00ad<sub>1<\/sub><\/em> = 1.0025 (1.0025 (1000)) = 1.0025 2(1000)<\/li>\r\n \t<li><em>P\u00ad<sub>3<\/sub><\/em> = 1.0025<em>P\u00ad<sub>2<\/sub><\/em> = 1.0025 (1.00252(1000)) = 1.00253(1000)<\/li>\r\n \t<li><em>P\u00ad<sub>4<\/sub><\/em> = 1.0025<em>P\u00ad<sub>3<\/sub><\/em> = 1.0025 (1.00253(1000)) = 1.00254(1000)<\/li>\r\n<\/ul>\r\nObserving a pattern, we could conclude\r\n<ul>\r\n \t<li><em>P<sub>m<\/sub><\/em> = (1.0025)<sup><em>m<\/em><\/sup>($1000)<\/li>\r\n<\/ul>\r\nNotice that the $1000 in the equation was <em>P<sub>0<\/sub><\/em>, the starting amount. We found 1.0025 by adding one to the growth rate divided by 12, since we were compounding 12 times per year.\r\n\r\n&nbsp;\r\n\r\nGeneralizing our result, we could write\r\n\r\n[latex]{{P}_{m}}={{P}_{0}}{{\\left(1+\\frac{r}{k}\\right)}^{m}}[\/latex]\r\n\r\n&nbsp;\r\n\r\nIn this formula:\r\n<ul>\r\n \t<li><em>m<\/em> is the number of compounding periods (months in our example)<\/li>\r\n \t<li><em>r<\/em> is the annual interest rate<\/li>\r\n \t<li><em>k<\/em> is the number of compounds per year.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\nView this video for a walkthrough of the concept of compound interest.\r\n\r\nhttps:\/\/youtu.be\/xuQTFmP9nNg\r\n\r\n<\/div>\r\nWhile this formula works fine, it is more common to use a formula that involves the number of years, rather than the number of compounding periods. If <em>N<\/em> is the number of years, then <em>m = N k<\/em>. Making this change gives us the standard formula for compound interest.\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<div class=\"textbox shaded\"><strong><em>The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest.\u00a0<\/em><\/strong><\/div>\r\nIn the next example, we show how to use the compound interest formula to find the balance on a certificate of deposit after 20 years.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years?\r\n[reveal-answer q=\"788137\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"788137\"]\r\n\r\nIn this example,\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>P<sub>0<\/sub><\/em> = $3000<\/td>\r\n<td>the initial deposit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.06<\/td>\r\n<td>6% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>12 months in 1 year<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>N<\/em> = 20<\/td>\r\n<td>\u00a0since we\u2019re looking for how much we\u2019ll have after 20 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]{{P}_{20}}=3000{{\\left(1+\\frac{0.06}{12}\\right)}^{20\\times12}}=\\$9930.61[\/latex] (round your answer to the nearest penny)\r\n\r\n[\/hidden-answer]\r\n\r\nA video walkthrough of this example problem\u00a0is available below.\r\n\r\nHere is how you would enter it into the calculator (https:\/\/www.geogebra.org\/m\/XKxv7Xc2)\r\n\r\nN = 20\r\n\r\nI = 6\r\n\r\nPV = 3000\r\n\r\nFV = what you want to find (note that the answer is negative - PV and FV will always have opposite signs)\r\n\r\nP\/Y = 12\r\n\r\nC\/Y = 12\r\n\r\nhttps:\/\/youtu.be\/8NazxAjhpJw\r\n\r\n<\/div>\r\nLet us compare the amount of money earned from compounding against the amount you would earn from simple interest\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Years<\/td>\r\n<td>Simple Interest ($15 per month)<\/td>\r\n<td>6% compounded monthly = 0.5% each month.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>$3900<\/td>\r\n<td>$4046.55<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>$4800<\/td>\r\n<td>$5458.19<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>15<\/td>\r\n<td>$5700<\/td>\r\n<td>$7362.28<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<\/td>\r\n<td>$6600<\/td>\r\n<td>$9930.61<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>25<\/td>\r\n<td>$7500<\/td>\r\n<td>$13394.91<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>30<\/td>\r\n<td>$8400<\/td>\r\n<td>$18067.73<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>35<\/td>\r\n<td>$9300<\/td>\r\n<td>$24370.65<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter wp-image-381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/11224717\/accountbalanceyears.png\" alt=\"Line graph. Vertical axis: Account Balance ($), in increments of 5000 from 5000 to 25000. Horizontal axis: years, in increments of five, from 0 to 25. A blue dotted line shows a gradual increase over time, from roughly $2500 at year 0 to roughly $10000 at year 35. A pink dotted line shows a more dramatic increase, from roughly $2500 at year 0 to $25000 at year 35.\" width=\"427\" height=\"345\" \/>\r\n\r\nAs you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\n[embed]https:\/\/www.myopenmath.com\/multiembedq.php?id=6693&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Evaluating exponents on the calculator<\/h3>\r\nHere are some examples of how to solve compound interest problems using online time value of money calculators such as: <a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<strong>You know that you will need $40,000 for your child\u2019s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal? Using the time value of money online calculator:\u00a0https:\/\/www.geogebra.org\/m\/XKxv7Xc2<\/strong>\r\n[reveal-answer q=\"842460\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"842460\"]\r\n\r\nIn this example, we\u2019re looking for <i>the present value (PV)<\/i>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>N<\/em> = 18<\/td>\r\n<td>18 years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>I% = 4<\/em><\/td>\r\n<td>4% interest rage<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>FV = 40000<\/i><\/td>\r\n<td>$40,000 is the amount we need in the future<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>P\/Y and C\/Y = 4<\/i><\/td>\r\n<td>Because we are compounding quarterly, this means there are 4 periods per year<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nClick the \"Solve for FV\r\n\r\nSo you would need to deposit $19,539.84 now to have $40,000 in 18 years.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It Now<\/h3>\r\n[embed]https:\/\/www.myopenmath.com\/multiembedq.php?id=6692&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Rounding<\/h3>\r\nIt is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to <strong>keep at least 3 significant digits<\/strong> (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a \u201cclose enough\u201d answer, but keeping more digits is always better.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<strong>To see why not over-rounding is so important, suppose you were investing $1000 at 5\/12% interest compounded monthly for 30 years. Using an online time value of money calculator:\u00a0https:\/\/www.geogebra.org\/m\/XKxv7Xc2<\/strong>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><i>N=30<\/i><\/td>\r\n<td>30 years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>I% = 5\/12<\/td>\r\n<td>For the 5\/12% interest rage<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>PV=1000<\/td>\r\n<td>The $1,000 you are investing<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>P\/Y and C\/Y = 12<\/td>\r\n<td>since we\u2019re compounding monthly<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen we compute the future value (the amount the $1,000 grows to, leaving PMT at zero because no payments are mentioned), we get $1,133.12\r\n\r\nHere is the effect of rounding this to different values:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0<i>5\/12%<\/i><strong>\u00a0rounded to:<\/strong><\/td>\r\n<td><strong>Gives <i>FV\u00a0<\/i>to be:<\/strong><\/td>\r\n<td><strong>Error<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.4<\/td>\r\n<td>$1,127.47<\/td>\r\n<td>$259.15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.42<\/td>\r\n<td>$1,134.26<\/td>\r\n<td>$53.71<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.417<\/td>\r\n<td>$1,133.24<\/td>\r\n<td>$5.35<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.4167<\/td>\r\n<td>$1,133.14<\/td>\r\n<td>$0.54<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.41667<\/td>\r\n<td>$1,133.13<\/td>\r\n<td>$0.06<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>no rounding<\/td>\r\n<td>$1,133.13<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf you\u2019re working in a bank, of course you wouldn\u2019t round at all. For our purposes, the answer we got by rounding to five decimals, is close enough. Certainly keeping that fourth decimal place wouldn\u2019t have hurt.\r\n\r\nView the following for a demonstration of this example.\r\n\r\nhttps:\/\/youtu.be\/VhhYtaMN6mo\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Using your calculator<\/h3>\r\nIn many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{12\\times30}}[\/latex]\r\n\r\nWe can quickly calculate 12\u00d730 = 360, giving [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{360}}[\/latex].\r\n\r\nNow we can use the calculator.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Type this<\/strong><\/td>\r\n<td><strong>Calculator shows<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.05 \u00f7 12 = .<\/td>\r\n<td>0.00416666666667<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+ 1 = .<\/td>\r\n<td>1.00416666666667<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>yx 360 = .<\/td>\r\n<td>4.46774431400613<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00d7 1000 = .<\/td>\r\n<td>4467.74431400613<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Using your calculator continued<\/h3>\r\nThe previous steps were assuming you have a \u201cone operation at a time\u201d calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:\r\n\r\n1000 \u00d7 \u00a0( 1 + 0.05 \u00f7 12 ) y<sup>x<\/sup> 360 =\r\n\r\n<\/div>\r\n<h2>Solving For Time<\/h2>\r\n<div class=\"textbox shaded\">Note: This section assumes you\u2019ve covered solving exponential equations using logarithms, either in prior classes or in the growth models chapter.<\/div>\r\nOften we are interested in how long it will take to accumulate money or how long we\u2019d need to extend a loan to bring payments down to a reasonable level.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nIf you invest $2000 at 6% compounded monthly, how long will it take the account to double in value?\r\n[reveal-answer q=\"610603\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"610603\"]\r\n\r\nThis is a compound interest problem, since we are depositing money once and allowing it to grow. In this problem,\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>P<sub>0<\/sub><\/em> = $2000<\/td>\r\n<td>the initial deposit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.06<\/td>\r\n<td>6% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>12 months in 1 year<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo our general equation is [latex]{{P}_{N}}=2000{{\\left(1+\\frac{0.06}{12}\\right)}^{N\\times12}}[\/latex]. We also know that we want our ending amount to be double of $2000, which is $4000, so we\u2019re looking for <em>N<\/em> so that <em>P<sub>N<\/sub><\/em> = 4000. To solve this, we set our equation for <em>P<sub>N<\/sub><\/em> equal to 4000.\r\n<p style=\"text-align: center\">[latex]4000=2000{{\\left(1+\\frac{0.06}{12}\\right)}^{N\\times12}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide both sides by 2000<\/p>\r\n<p style=\"text-align: center\">[latex]2={{\\left(1.005\\right)}^{12N}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 To solve for the exponent, take the log of both sides<\/p>\r\n<p style=\"text-align: center\">[latex]\\log\\left(2\\right)=\\log\\left({{\\left(1.005\\right)}^{12N}}\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Use the exponent property of logs on the right side<\/p>\r\n<p style=\"text-align: center\">[latex]\\log\\left(2\\right)=12N\\log\\left(1.005\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Now we can divide both sides by 12log(1.005)<\/p>\r\n<p style=\"text-align: center\">[latex]\\frac{\\log\\left(2\\right)}{12\\log\\left(1.005\\right)}=N[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Approximating this to a decimal<\/p>\r\n<em>N<\/em> = 11.581\r\n\r\nIt will take about 11.581 years for the account to double in value. Note that your answer may come out slightly differently if you had evaluated the logs to decimals and rounded during your calculations, but your answer should be close. For example if you rounded log(2) to 0.301 and log(1.005) to 0.00217, then your final answer would have been about 11.577 years.\r\n\r\n[\/hidden-answer]\r\n\r\nGet additional guidance for this example in the following:\r\n\r\nhttps:\/\/youtu.be\/zHRTxtFiyxc\r\n\r\n<\/div>","rendered":"<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02232307\/interest.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1313 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02232307\/interest-300x172.jpg\" alt=\"\" width=\"300\" height=\"172\" \/><\/a><\/p>\n<p>We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. This section will show you how to solve these types of problems by hand. At the end of the section, there is a video and a calculator you can use to solve these problems as well.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>The leaning objectives for this section include:<\/p>\n<ol>\n<li>Calculate one-time simple interest, and simple interest over time<\/li>\n<li>Calculate compound interest<\/li>\n<\/ol>\n<\/div>\n<h1 style=\"text-align: center\">1. Simple\u00a0Interest<\/h1>\n<p>Discussing interest starts with the <strong>principal<\/strong>, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simple Interest\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7gRgZW3yxjY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200620\/money-1604921_1280.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-553 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200620\/money-1604921_1280-1024x682.jpg\" alt=\"four rolled-up dollar bills seeming to grow out of dirt, with a miniature rake lying in between them\" width=\"1024\" height=\"682\" \/><\/a><\/p>\n<div class=\"textbox\">\n<h3>Simple One-time Interest<\/h3>\n<p>[latex]\\begin{align}&I={{P}_{0}}r\\\\& \\\\&A={{P}_{0}}+I\\\\&A={{P}_{0}}+{{P}_{0}}r\\\\&A={{P}_{0}}(1+r)\\\\\\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/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. Example: 5% = 0.05)<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest (simple) will you earn?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q227650\">Show Answer<\/span><\/p>\n<div id=\"q227650\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] = $300<\/td>\n<td>the principal<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.03<\/td>\n<td>3% rate<\/td>\n<\/tr>\n<tr>\n<td><em>I<\/em> = $300(0.03) = $9.<\/td>\n<td>You will earn $9 interest.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>The following video works through this example in detail.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"One time simple interest\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TJYq7XGB8EY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly.<\/p>\n<p>For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q14596\">Show Answer<\/span><\/p>\n<div id=\"q14596\" class=\"hidden-answer\" style=\"display: none\">Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.<\/div>\n<\/div>\n<p>Further explanation about solving this example can be seen here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Simple interest over time\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rNOEYPCnGwg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>We can generalize this idea of simple interest over time.<\/p>\n<div class=\"textbox\">\n<h3>Simple Interest over Time<\/h3>\n<p>[latex]\\begin{align}&I={{P}_{0}}rt\\\\& \\\\&A={{P}_{0}}+I\\\\&A={{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<div class=\"textbox\">\n<h3>APR \u2013 Annual Percentage Rate<\/h3>\n<p>Interest rates are usually given as an <strong>annual percentage rate (APR)<\/strong> \u2013 the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.<\/p>\n<p>For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.<br \/>\n[latex]6\\div{12}=0.5[\/latex]<\/p>\n<p>A 4% annual rate paid quarterly would be divided into four 1% payments.<br \/>\n[latex]4\\div{4}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q529216\">Show Answer<\/span><\/p>\n<div id=\"q529216\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] = $1000<\/td>\n<td>the principal<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.02<\/td>\n<td>2% rate per half-year<\/td>\n<\/tr>\n<tr>\n<td><em>t<\/em> = 8<\/td>\n<td>4 years = 8 half-years<\/td>\n<\/tr>\n<tr>\n<td><em>I<\/em> = $1000(0.02)(8) = $160.<\/td>\n<td>\u00a0You will earn $160 interest total over the four years.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>This video explains the solution.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Simple interest T-note example\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IfVn20go7-Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=72467&#38;theme=oea&#38;iframe_resize_id=mom2\">https:\/\/www.myopenmath.com\/multiembedq.php?id=72467&amp;theme=oea&amp;iframe_resize_id=mom2<\/a><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p>A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288479\">Show Answer<\/span><\/p>\n<div id=\"q288479\" class=\"hidden-answer\" style=\"display: none\">\n<p>I = $30 of interest<br \/>\n[latex]P_0[\/latex] = $500 principal<br \/>\nr = unknown<br \/>\nt = 1 month<\/p>\n<p>Using [latex]I = P_0rt[\/latex], we get [latex]30 = 500\u00b7r\u00b71[\/latex]. Solving, we get r = 0.06, or 6%. Since the time was monthly, this is the monthly interest. The annual rate would be 12 times this: 72% interest.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=929&#38;theme=oea&#38;iframe_resize_id=mom5\">https:\/\/www.myopenmath.com\/multiembedq.php?id=929&amp;theme=oea&amp;iframe_resize_id=mom5<\/a><\/p>\n<\/div>\n<h1 style=\"text-align: center\">2. Compound Interest<\/h1>\n<p>With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called <strong>compounding<\/strong>.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"How to use the Free Geogebra TVM (Time Value of Money) Solver for compound interest problems\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/DgLMeA8Tr6A?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>TIME VALUE OF MONEY CALCULATOR<\/h2>\n<p>The following <a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a> application\/calculator will be a helpful tool in checking your answers:<\/p>\n<p>The Time Value of Money Solver asks you to provide N, I%, PV, PMT, FV, P\/Y, and C\/Y.<\/p>\n<p>N is the number of years, i% is the interest rate as a number (not a decimal), PV is the present value or\u00a0<em>P<sub>0<\/sub><\/em>, PMT is the payment each period (if there is no payment, then put in zero), FV is the future value or A, P\/Y is periods per year (so if there is monthly compounding, this would be 12), and C\/Y is how often compounding occurs (so if compounded quarterly, then put in 4). Note, that P\/Y and C\/Y will be the same for any problem you will see in this course. Lastly, PV and FV always have opposite signs (if PV is positive, FV will be negative)<\/p>\n<p>Here is a video walking you through one of these types of problems:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"MontlyMortgagePayment\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ohyUCZL97-8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200323\/achievement-18134_1280.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-552 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/23200323\/achievement-18134_1280-1024x759.jpg\" alt=\"a row of gold coin stacks. From left to right, they grown from one coin, to two, to four, ending with a stack of 32 coins\" width=\"1024\" height=\"759\" \/><\/a><\/p>\n<p>Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?<\/p>\n<p>The 3% interest is an annual percentage rate (APR) \u2013 the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\\frac{3%}{12}[\/latex]= 0.25% per month.<\/p>\n<p>In the first month,<\/p>\n<ul>\n<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\n<li><em>r<\/em> = 0.0025 (0.25%)<\/li>\n<li><em>I <\/em>= $1000 (0.0025) = $2.50<\/li>\n<li><em>A<\/em> = $1000 + $2.50 = $1002.50<\/li>\n<\/ul>\n<p>In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50.<\/p>\n<p>&nbsp;<\/p>\n<p>In the second month,<\/p>\n<ul>\n<li><em>P<sub>0<\/sub><\/em> = $1002.50<\/li>\n<li><em>I <\/em>= $1002.50 (0.0025) = $2.51 (rounded)<\/li>\n<li><em>A<\/em> = $1002.50 + $2.51 = $1005.01<\/li>\n<\/ul>\n<p>Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original $1000 we deposited, but we also earned interest on the $2.50 of interest we earned the first month. This is the key advantage that <strong>compounding<\/strong>\u00a0interest gives us.<\/p>\n<p>Calculating out a few more months gives the following:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Month<\/strong><\/td>\n<td><strong>Starting balance<\/strong><\/td>\n<td><strong>Interest earned<\/strong><\/td>\n<td><strong>Ending Balance<\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1000.00<\/td>\n<td>2.50<\/td>\n<td>1002.50<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>1002.50<\/td>\n<td>2.51<\/td>\n<td>1005.01<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>1005.01<\/td>\n<td>2.51<\/td>\n<td>1007.52<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>1007.52<\/td>\n<td>2.52<\/td>\n<td>1010.04<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>1010.04<\/td>\n<td>2.53<\/td>\n<td>1012.57<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>1012.57<\/td>\n<td>2.53<\/td>\n<td>1015.10<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>1015.10<\/td>\n<td>2.54<\/td>\n<td>1017.64<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>1017.64<\/td>\n<td>2.54<\/td>\n<td>1020.18<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>1020.18<\/td>\n<td>2.55<\/td>\n<td>1022.73<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>1022.73<\/td>\n<td>2.56<\/td>\n<td>1025.29<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>1025.29<\/td>\n<td>2.56<\/td>\n<td>1027.85<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>1027.85<\/td>\n<td>2.57<\/td>\n<td>1030.42<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>CALCULATOR: See if you can recreate the Ending Balance column using the calculator (<a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a>)<\/p>\n<p>For example, Month 1, starting balance of 1000 means you would put the following into the calculator:<\/p>\n<p>N=1\/12 (because we are only looking at 1 month and this box is the number of years)<\/p>\n<p>I=3 (interest rate of 3%)<\/p>\n<p>PV=1000 (the money we started with)<\/p>\n<p>PMT=0 (there were no regular, recurring payments)<\/p>\n<p>FV = (what we are solving for) &#8211; push this button and make sure it matches the above table (note that it will be negative because PV and FV are always opposite signs)<\/p>\n<p>P\/Y=12<\/p>\n<p>C\/Y=12<\/p>\n<p>FORMULA: We want to simplify the process for calculating compounding, because creating a table like the one above is time consuming. Luckily, math is good at giving you ways to take shortcuts. To find an equation to represent this, if <em>P<sub>m <\/sub><\/em>represents the amount of money after <em>m<\/em> months, then we could write the recursive equation:<\/p>\n<p><em>P<sub>0<\/sub><\/em> = $1000<\/p>\n<p><em>P<sub>m<\/sub><\/em> = (1+0.0025)<em>P<sub>m-1<\/sub><\/em><\/p>\n<p>You probably recognize this as the recursive form of exponential growth. If not, we go through the steps to build an explicit equation for the growth in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Build an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q530288\">Show Answer<\/span><\/p>\n<div id=\"q530288\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\n<li><em>P\u00ad<sub>1<\/sub><\/em> = 1.0025<em>P\u00ad<sub>0<\/sub><\/em> = 1.0025 (1000)<\/li>\n<li><em>P\u00ad<sub>2<\/sub><\/em> = 1.0025<em>P\u00ad<sub>1<\/sub><\/em> = 1.0025 (1.0025 (1000)) = 1.0025 2(1000)<\/li>\n<li><em>P\u00ad<sub>3<\/sub><\/em> = 1.0025<em>P\u00ad<sub>2<\/sub><\/em> = 1.0025 (1.00252(1000)) = 1.00253(1000)<\/li>\n<li><em>P\u00ad<sub>4<\/sub><\/em> = 1.0025<em>P\u00ad<sub>3<\/sub><\/em> = 1.0025 (1.00253(1000)) = 1.00254(1000)<\/li>\n<\/ul>\n<p>Observing a pattern, we could conclude<\/p>\n<ul>\n<li><em>P<sub>m<\/sub><\/em> = (1.0025)<sup><em>m<\/em><\/sup>($1000)<\/li>\n<\/ul>\n<p>Notice that the $1000 in the equation was <em>P<sub>0<\/sub><\/em>, the starting amount. We found 1.0025 by adding one to the growth rate divided by 12, since we were compounding 12 times per year.<\/p>\n<p>&nbsp;<\/p>\n<p>Generalizing our result, we could write<\/p>\n<p>[latex]{{P}_{m}}={{P}_{0}}{{\\left(1+\\frac{r}{k}\\right)}^{m}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>In this formula:<\/p>\n<ul>\n<li><em>m<\/em> is the number of compounding periods (months in our example)<\/li>\n<li><em>r<\/em> is the annual interest rate<\/li>\n<li><em>k<\/em> is the number of compounds per year.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>View this video for a walkthrough of the concept of compound interest.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Compound Interest\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xuQTFmP9nNg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>While this formula works fine, it is more common to use a formula that involves the number of years, rather than the number of compounding periods. If <em>N<\/em> is the number of years, then <em>m = N k<\/em>. Making this change gives us the standard formula for compound interest.<\/p>\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<div class=\"textbox shaded\"><strong><em>The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest.\u00a0<\/em><\/strong><\/div>\n<p>In the next example, we show how to use the compound interest formula to find the balance on a certificate of deposit after 20 years.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788137\">Show Answer<\/span><\/p>\n<div id=\"q788137\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example,<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>P<sub>0<\/sub><\/em> = $3000<\/td>\n<td>the initial deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.06<\/td>\n<td>6% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>12 months in 1 year<\/td>\n<\/tr>\n<tr>\n<td><em>N<\/em> = 20<\/td>\n<td>\u00a0since we\u2019re looking for how much we\u2019ll have after 20 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]{{P}_{20}}=3000{{\\left(1+\\frac{0.06}{12}\\right)}^{20\\times12}}=\\$9930.61[\/latex] (round your answer to the nearest penny)<\/p>\n<\/div>\n<\/div>\n<p>A video walkthrough of this example problem\u00a0is available below.<\/p>\n<p>Here is how you would enter it into the calculator (https:\/\/www.geogebra.org\/m\/XKxv7Xc2)<\/p>\n<p>N = 20<\/p>\n<p>I = 6<\/p>\n<p>PV = 3000<\/p>\n<p>FV = what you want to find (note that the answer is negative &#8211; PV and FV will always have opposite signs)<\/p>\n<p>P\/Y = 12<\/p>\n<p>C\/Y = 12<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Compound interest CD example\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8NazxAjhpJw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>Let us compare the amount of money earned from compounding against the amount you would earn from simple interest<\/p>\n<table>\n<tbody>\n<tr>\n<td>Years<\/td>\n<td>Simple Interest ($15 per month)<\/td>\n<td>6% compounded monthly = 0.5% each month.<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>$3900<\/td>\n<td>$4046.55<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>$4800<\/td>\n<td>$5458.19<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>$5700<\/td>\n<td>$7362.28<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>$6600<\/td>\n<td>$9930.61<\/td>\n<\/tr>\n<tr>\n<td>25<\/td>\n<td>$7500<\/td>\n<td>$13394.91<\/td>\n<\/tr>\n<tr>\n<td>30<\/td>\n<td>$8400<\/td>\n<td>$18067.73<\/td>\n<\/tr>\n<tr>\n<td>35<\/td>\n<td>$9300<\/td>\n<td>$24370.65<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/11224717\/accountbalanceyears.png\" alt=\"Line graph. Vertical axis: Account Balance ($), in increments of 5000 from 5000 to 25000. Horizontal axis: years, in increments of five, from 0 to 25. A blue dotted line shows a gradual increase over time, from roughly $2500 at year 0 to roughly $10000 at year 35. A pink dotted line shows a more dramatic increase, from roughly $2500 at year 0 to $25000 at year 35.\" width=\"427\" height=\"345\" \/><\/p>\n<p>As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6693&#38;theme=oea&#38;iframe_resize_id=mom1\">https:\/\/www.myopenmath.com\/multiembedq.php?id=6693&amp;theme=oea&amp;iframe_resize_id=mom1<\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Evaluating exponents on the calculator<\/h3>\n<p>Here are some examples of how to solve compound interest problems using online time value of money calculators such as: <a href=\"https:\/\/www.geogebra.org\/m\/jhyUqg2A\">Time Value of Money Solver<\/a><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><strong>You know that you will need $40,000 for your child\u2019s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal? Using the time value of money online calculator:\u00a0https:\/\/www.geogebra.org\/m\/XKxv7Xc2<\/strong><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q842460\">Show Answer<\/span><\/p>\n<div id=\"q842460\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example, we\u2019re looking for <i>the present value (PV)<\/i>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>N<\/em> = 18<\/td>\n<td>18 years<\/td>\n<\/tr>\n<tr>\n<td><em>I% = 4<\/em><\/td>\n<td>4% interest rage<\/td>\n<\/tr>\n<tr>\n<td><i>FV = 40000<\/i><\/td>\n<td>$40,000 is the amount we need in the future<\/td>\n<\/tr>\n<tr>\n<td><i>P\/Y and C\/Y = 4<\/i><\/td>\n<td>Because we are compounding quarterly, this means there are 4 periods per year<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Click the &#8220;Solve for FV<\/p>\n<p>So you would need to deposit $19,539.84 now to have $40,000 in 18 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It Now<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6692&#38;theme=oea&#38;iframe_resize_id=mom2\">https:\/\/www.myopenmath.com\/multiembedq.php?id=6692&amp;theme=oea&amp;iframe_resize_id=mom2<\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Rounding<\/h3>\n<p>It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to <strong>keep at least 3 significant digits<\/strong> (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a \u201cclose enough\u201d answer, but keeping more digits is always better.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><strong>To see why not over-rounding is so important, suppose you were investing $1000 at 5\/12% interest compounded monthly for 30 years. Using an online time value of money calculator:\u00a0https:\/\/www.geogebra.org\/m\/XKxv7Xc2<\/strong><\/p>\n<table>\n<tbody>\n<tr>\n<td><i>N=30<\/i><\/td>\n<td>30 years<\/td>\n<\/tr>\n<tr>\n<td>I% = 5\/12<\/td>\n<td>For the 5\/12% interest rage<\/td>\n<\/tr>\n<tr>\n<td>PV=1000<\/td>\n<td>The $1,000 you are investing<\/td>\n<\/tr>\n<tr>\n<td>P\/Y and C\/Y = 12<\/td>\n<td>since we\u2019re compounding monthly<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When we compute the future value (the amount the $1,000 grows to, leaving PMT at zero because no payments are mentioned), we get $1,133.12<\/p>\n<p>Here is the effect of rounding this to different values:<\/p>\n<table>\n<tbody>\n<tr>\n<td>\u00a0<i>5\/12%<\/i><strong>\u00a0rounded to:<\/strong><\/td>\n<td><strong>Gives <i>FV\u00a0<\/i>to be:<\/strong><\/td>\n<td><strong>Error<\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.4<\/td>\n<td>$1,127.47<\/td>\n<td>$259.15<\/td>\n<\/tr>\n<tr>\n<td>0.42<\/td>\n<td>$1,134.26<\/td>\n<td>$53.71<\/td>\n<\/tr>\n<tr>\n<td>0.417<\/td>\n<td>$1,133.24<\/td>\n<td>$5.35<\/td>\n<\/tr>\n<tr>\n<td>0.4167<\/td>\n<td>$1,133.14<\/td>\n<td>$0.54<\/td>\n<\/tr>\n<tr>\n<td>0.41667<\/td>\n<td>$1,133.13<\/td>\n<td>$0.06<\/td>\n<\/tr>\n<tr>\n<td>no rounding<\/td>\n<td>$1,133.13<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If you\u2019re working in a bank, of course you wouldn\u2019t round at all. For our purposes, the answer we got by rounding to five decimals, is close enough. Certainly keeping that fourth decimal place wouldn\u2019t have hurt.<\/p>\n<p>View the following for a demonstration of this example.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Compound interest - the importance of rounding\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VhhYtaMN6mo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Using your calculator<\/h3>\n<p>In many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{12\\times30}}[\/latex]<\/p>\n<p>We can quickly calculate 12\u00d730 = 360, giving [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{360}}[\/latex].<\/p>\n<p>Now we can use the calculator.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Type this<\/strong><\/td>\n<td><strong>Calculator shows<\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.05 \u00f7 12 = .<\/td>\n<td>0.00416666666667<\/td>\n<\/tr>\n<tr>\n<td>+ 1 = .<\/td>\n<td>1.00416666666667<\/td>\n<\/tr>\n<tr>\n<td>yx 360 = .<\/td>\n<td>4.46774431400613<\/td>\n<\/tr>\n<tr>\n<td>\u00d7 1000 = .<\/td>\n<td>4467.74431400613<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Using your calculator continued<\/h3>\n<p>The previous steps were assuming you have a \u201cone operation at a time\u201d calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:<\/p>\n<p>1000 \u00d7 \u00a0( 1 + 0.05 \u00f7 12 ) y<sup>x<\/sup> 360 =<\/p>\n<\/div>\n<h2>Solving For Time<\/h2>\n<div class=\"textbox shaded\">Note: This section assumes you\u2019ve covered solving exponential equations using logarithms, either in prior classes or in the growth models chapter.<\/div>\n<p>Often we are interested in how long it will take to accumulate money or how long we\u2019d need to extend a loan to bring payments down to a reasonable level.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>If you invest $2000 at 6% compounded monthly, how long will it take the account to double in value?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q610603\">Show Answer<\/span><\/p>\n<div id=\"q610603\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a compound interest problem, since we are depositing money once and allowing it to grow. In this problem,<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>P<sub>0<\/sub><\/em> = $2000<\/td>\n<td>the initial deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.06<\/td>\n<td>6% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>12 months in 1 year<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So our general equation is [latex]{{P}_{N}}=2000{{\\left(1+\\frac{0.06}{12}\\right)}^{N\\times12}}[\/latex]. We also know that we want our ending amount to be double of $2000, which is $4000, so we\u2019re looking for <em>N<\/em> so that <em>P<sub>N<\/sub><\/em> = 4000. To solve this, we set our equation for <em>P<sub>N<\/sub><\/em> equal to 4000.<\/p>\n<p style=\"text-align: center\">[latex]4000=2000{{\\left(1+\\frac{0.06}{12}\\right)}^{N\\times12}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Divide both sides by 2000<\/p>\n<p style=\"text-align: center\">[latex]2={{\\left(1.005\\right)}^{12N}}[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 To solve for the exponent, take the log of both sides<\/p>\n<p style=\"text-align: center\">[latex]\\log\\left(2\\right)=\\log\\left({{\\left(1.005\\right)}^{12N}}\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Use the exponent property of logs on the right side<\/p>\n<p style=\"text-align: center\">[latex]\\log\\left(2\\right)=12N\\log\\left(1.005\\right)[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Now we can divide both sides by 12log(1.005)<\/p>\n<p style=\"text-align: center\">[latex]\\frac{\\log\\left(2\\right)}{12\\log\\left(1.005\\right)}=N[\/latex]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Approximating this to a decimal<\/p>\n<p><em>N<\/em> = 11.581<\/p>\n<p>It will take about 11.581 years for the account to double in value. Note that your answer may come out slightly differently if you had evaluated the logs to decimals and rounded during your calculations, but your answer should be close. For example if you rounded log(2) to 0.301 and log(1.005) to 0.00217, then your final answer would have been about 11.577 years.<\/p>\n<\/div>\n<\/div>\n<p>Get additional guidance for this example in the following:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Find doubling time for compound interest\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zHRTxtFiyxc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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-1086\">\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>Introduction and Learning Objectives. <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><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><li>Question ID 6692. <strong>Authored by<\/strong>: Lippman, David. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>interest. <strong>Authored by<\/strong>: NY. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.picserver.org\/i\/interest.html\">http:\/\/www.picserver.org\/i\/interest.html<\/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>Math in Society. <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>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>money-grow-interest-save-invest-1604921. <strong>Authored by<\/strong>: TheDigitalWay. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/money-grow-interest-save-invest-1604921\/\">https:\/\/pixabay.com\/en\/money-grow-interest-save-invest-1604921\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>One time simple interest. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/TJYq7XGB8EY\">https:\/\/youtu.be\/TJYq7XGB8EY<\/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>Simple interest over time. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rNOEYPCnGwg\">https:\/\/youtu.be\/rNOEYPCnGwg<\/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>Simple interest T-note example. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IfVn20go7-Y\">https:\/\/youtu.be\/IfVn20go7-Y<\/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 929. <strong>Authored by<\/strong>: Lippman, David. <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><li>Question ID 72476. <strong>Authored by<\/strong>: Day, Alyson. <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><li>achievement-bar-business-chart-18134. <strong>Authored by<\/strong>: PublicDomainPictures. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/achievement-bar-business-chart-18134\/\">https:\/\/pixabay.com\/en\/achievement-bar-business-chart-18134\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Compound Interest. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xuQTFmP9nNg\">https:\/\/youtu.be\/xuQTFmP9nNg<\/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>Compound interest CD example. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8NazxAjhpJw\">https:\/\/youtu.be\/8NazxAjhpJw<\/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>Compound interest - the importance of rounding. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/VhhYtaMN6mo\">https:\/\/youtu.be\/VhhYtaMN6mo<\/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 6693. <strong>Authored by<\/strong>: Lippman, David. <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":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"interest\",\"author\":\"NY\",\"organization\":\"\",\"url\":\"http:\/\/www.picserver.org\/i\/interest.html\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Introduction and Learning Objectives\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David 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