{"id":174,"date":"2017-01-11T01:00:24","date_gmt":"2017-01-11T01:00:24","guid":{"rendered":"https:\/\/pressbooks.usnh.edu\/math502\/chapter\/simple-and-compound-interest\/"},"modified":"2023-03-24T14:26:19","modified_gmt":"2023-03-24T14:26:19","slug":"simple-and-compound-interest","status":"publish","type":"chapter","link":"https:\/\/pressbooks.usnh.edu\/mth402\/chapter\/simple-and-compound-interest\/","title":{"raw":"Simple and Compound Interest","rendered":"Simple and Compound Interest"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Calculate one-time simple interest, and simple interest over time<\/li>\n \t<li>Determine APY given an interest scenario<\/li>\n \t<li>Calculate compound interest<\/li>\n<\/ul>\n<\/div>\n<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>\n\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.\n\n&nbsp;\n<h2>Simple\u00a0Interest<\/h2>\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.\n\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>\n<div class=\"textbox\">\n<h3>Simple One-time Interest<\/h3>\n[latex]\\begin{align}&amp;I={{P}_{0}}r\\\\&amp;A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\\\\end{align}[\/latex]\n<ul>\n \t<li><em>I<\/em> is the interest<\/li>\n \t<li><em>A<\/em> is the end amount: principal plus interest<\/li>\n \t<li>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] is the principal (starting amount)<\/li>\n \t<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>\nA friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?\n\n<em><strong>Solution:<\/strong><\/em>\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&nbsp;\n\nThe following video works through this example in detail.\n\nhttps:\/\/youtu.be\/TJYq7XGB8EY\n\n&nbsp;\n\n<\/div>\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.\n\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.\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\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?\n[reveal-answer q=\"14596\"]Show Solution[\/reveal-answer]\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]\n\nFurther explanation about solving this example can be seen here.\n\nhttps:\/\/youtu.be\/rNOEYPCnGwg\n\n<\/div>\nWe can generalize this idea of simple interest over time.\n<div class=\"textbox\">\n<h3>Simple Interest over Time<\/h3>\n[latex]\\begin{align}&amp;I={{P}_{0}}rt\\\\&amp;A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\\\\end{align}[\/latex]\n<ul>\n \t<li><em>I<\/em> is the interest<\/li>\n \t<li><em>A<\/em> is the end amount: principal plus interest<\/li>\n \t<li>[latex]\\begin{align}{{P}_{0}}\\\\\\end{align}[\/latex] is the principal (starting amount)<\/li>\n \t<li><em>r<\/em> is the interest rate in decimal form<\/li>\n \t<li><em>t<\/em> is time<\/li>\n<\/ul>\nThe units of measurement (years, months, etc.) for the time should match the time period for the interest rate.\n\n<\/div>\n&nbsp;\n<div class=\"textbox\">\n<h3>APR \u2013 Annual Percentage Rate<\/h3>\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.\n\nFor example, a 6% APR paid monthly would be divided into twelve 0.5% payments.\n[latex]6\\div{12}=0.5[\/latex]\n\nA 4% annual rate paid quarterly would be divided into four 1% payments.\n[latex]4\\div{4}=1[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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?\n\n<em><strong>Solution:<\/strong><\/em>\n\nSince interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.\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&nbsp;\n\nThis video explains the solution.\n\nhttps:\/\/youtu.be\/IfVn20go7-Y\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=72467&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom2\">Click here to try this problem.<\/a>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.\n\n<em><strong>Solution:<\/strong><\/em>\n\nI = $30 of interest\n[latex]P_0[\/latex] = $500 principal\nr = unknown\nt = 1 month\n\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.\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=929&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom5\">Click here to try this problem.<\/a>\n\n<\/div>\n<h2>Compound Interest<\/h2>\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>.\n\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>\n\nSuppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?\n\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 3% \u00f7 12 = 0.25% per month.\n\nIn the first month,\n<ul>\n \t<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\n \t<li><em>r<\/em> = 0.0025 (0.25%)<\/li>\n \t<li><em>I <\/em>= $1000 (0.0025) = $2.50<\/li>\n \t<li><em>A<\/em> = $1000 + $2.50 = $1002.50<\/li>\n<\/ul>\nIn the first month, we will earn $2.50 in interest, raising our account balance to $1002.50.\n\n&nbsp;\n\nIn the second month,\n<ul>\n \t<li><em>P<sub>0<\/sub><\/em> = $1002.50<\/li>\n \t<li><em>I <\/em>= $1002.50 (0.0025) = $2.51 (rounded)<\/li>\n \t<li><em>A<\/em> = $1002.50 + $2.51 = $1005.01<\/li>\n<\/ul>\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.\n\nCalculating out a few more months gives the following:\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>\nWe 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:\n\n<em>P<sub>0<\/sub><\/em> = $1000\n\n<em>P<sub>m<\/sub><\/em> = (1+0.0025)<em>P<sub>m-1<\/sub><\/em>\n\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.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nBuild an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly.\n\n<em><strong>Solution:<\/strong><\/em>\n<ul>\n \t<li><em>P<sub>0<\/sub><\/em> = $1000<\/li>\n \t<li><em>P\u00ad<sub>1<\/sub><\/em> = 1.0025<em>P\u00ad<sub>0<\/sub><\/em> = 1.0025 (1000)<\/li>\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>\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>\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>\n<\/ul>\nObserving a pattern, we could conclude\n<ul>\n \t<li><em>P<sub>m<\/sub><\/em> = (1.0025)<sup><em>m<\/em><\/sup>($1000)<\/li>\n<\/ul>\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.\n\n&nbsp;\n\nGeneralizing our result, we could write\n\n[latex]{{P}_{m}}={{P}_{0}}{{\\left(1+\\frac{r}{k}\\right)}^{m}}[\/latex]\n\n&nbsp;\n\nIn this formula:\n<ul>\n \t<li><em>m<\/em> is the number of compounding periods (months in our example)<\/li>\n \t<li><em>r<\/em> is the annual interest rate<\/li>\n \t<li><em>k<\/em> is the number of compounds per year.<\/li>\n<\/ul>\nView this video for a walkthrough of the concept of compound interest.\n\nhttps:\/\/youtu.be\/xuQTFmP9nNg\n\n<\/div>\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.\n<div class=\"textbox\">\n<h3>Compound Interest<\/h3>\n[latex]P_{N}=P_{0}\\left(1+\\frac{r}{k}\\right)^{Nk}[\/latex]\n<ul>\n \t<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\n \t<li><em>P<sub>0 <\/sub><\/em> is the starting balance of the account (also called initial deposit, or principal)<\/li>\n \t<li><em>r<\/em> is the annual interest rate in decimal form<\/li>\n \t<li><em>k<\/em> is the number of compounding periods in one year\n<ul>\n \t<li>If the compounding is done annually (once a year), <em>k<\/em> = 1.<\/li>\n \t<li>If the compounding is done quarterly, <em>k<\/em> = 4.<\/li>\n \t<li>If the compounding is done monthly, <em>k<\/em> = 12.<\/li>\n \t<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\"><span style=\"color: #ff0000\"><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><\/span><\/div>\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.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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?\n\n<em><strong>Solution:<\/strong><\/em>\n\nIn this example,\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>\nSo [latex]{{P}_{20}}=3000{{\\left(1+\\frac{0.06}{12}\\right)}^{20\\times12}}=\\$9930.61[\/latex] (round your answer to the nearest penny)\n\nA video walkthrough of this example problem\u00a0is available below.\n\nhttps:\/\/youtu.be\/8NazxAjhpJw\n\n<\/div>\nLet us compare the amount of money earned from compounding against the amount you would earn from simple interest\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>$24370.65<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<img class=\"aligncenter wp-image-381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/11224717\/accountbalanceyears.png\" alt=\"Line graph. Vertical axis: Account Balance ($), 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\">\n\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.\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6693&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom1\">Click here to try this problem.<\/a>\n\n<\/div>\n&nbsp;\n<div class=\"textbox\">\n<h3>Evaluating exponents on the Desmos calculator<\/h3>\nWhen we need to calculate something like [latex]5^3[\/latex] it is easy enough to just multiply [latex]5\\cdot{5}\\cdot{5}=125[\/latex].\u00a0 But when we need to calculate something like [latex]1.005^{240}[\/latex], it would be very tedious to calculate this by multiplying [latex]1.005[\/latex] by itself [latex]240[\/latex] times!\u00a0 So to make things easier, we can harness the power of our scientific calculators.\u00a0 In this class, we are using the <a href=\"https:\/\/www.desmos.com\/scientific\">Desmos calculator<\/a>.\u00a0 If you just want to square a number, the key is <span style=\"color: #ff0000\"><em><strong>a<sup>2<\/sup><\/strong><\/em><\/span>.\u00a0 If you want to raise a number to another power, you use the key <em><strong><span style=\"color: #ff0000\">a<sup>b<\/sup><\/span><\/strong><\/em> on the main menu.\n\nTo evaluate [latex]1.005^{240}[\/latex] we'd type 1.005 <em><strong><span style=\"color: #ff0000\">a<sup>b\u00a0<\/sup><\/span><\/strong><\/em><span style=\"color: #ff0000\"><span style=\"color: #000000\">240<\/span><\/span> .\u00a0 Try it out - you should get the answer in the figure below:\n\n&nbsp;\n\n<img class=\"aligncenter wp-image-172 size-full\" src=\"http:\/\/pressbooks.usnh.edu\/math502\/wp-content\/blogs.dir\/12\/files\/2016\/12\/Desmos-Exponent-Entry.png\" alt=\"This figure shows how an exponent is evaluated in Desmos.\" width=\"1316\" height=\"1210\">\n\nMost scientific calculators have a button for exponents.\u00a0 If you are not using the Desmos calculator, it is typically either labeled like:\n<p style=\"text-align: center\">^ , \u00a0 [latex]y^x[\/latex] ,\u00a0\u00a0 or [latex]x^y[\/latex] .<\/p>\n&nbsp;\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nYou 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?\n\n<em><strong>Solution:<\/strong><\/em>\n\nIn this example, we\u2019re looking for <em>P<sub>0<\/sub><\/em>.\n<table>\n<tbody>\n<tr>\n<td><em>r<\/em> = 0.04<\/td>\n<td>4%<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 4<\/td>\n<td>4 quarters in 1 year<\/td>\n<\/tr>\n<tr>\n<td><em>N <\/em>= 18<\/td>\n<td>Since we know the balance in 18 years<\/td>\n<\/tr>\n<tr>\n<td><em>P<sub>18<\/sub><\/em> = $40,000<\/td>\n<td>The amount we have in 18 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIn this case, we\u2019re going to have to set up the equation, and solve for <em>P<sub>0<\/sub><\/em>.\n\n[latex]\\begin{align}&amp;40000={{P}_{0}}{{\\left(1+\\frac{0.04}{4}\\right)}^{18\\times4}}\\\\&amp;40000={{P}_{0}}(2.0471)\\\\&amp;{{P}_{0}}=\\frac{40000}{2.0471}=\\$19539.84 \\\\\\end{align}[\/latex]\n\nSo you would need to deposit $19,539.84 now to have $40,000 in 18 years.\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6692&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom2\">Click here to try this problem.<\/a>\n\n<\/div>\n&nbsp;\n<div class=\"textbox\">\n<h3>Rounding<\/h3>\nIf you are not inputting your entire formula into Desmos and choose to do it section by section, 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.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nTo see why not over-rounding is so important, if you choose not to enter your formula all at once into Desmos, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.\n<table>\n<tbody>\n<tr>\n<td><em>P<sub>0<\/sub><\/em> = $1000<\/td>\n<td>the initial deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.05<\/td>\n<td>5%<\/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> = 30<\/td>\n<td>since we\u2019re looking for the amount after 30 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIf we first compute <em>r\/k<\/em>, we find 0.05\/12 = 0.00416666666667\n\nHere is the effect of rounding this to different values:\n<table>\n<tbody>\n<tr>\n<td>&nbsp;\n\n<strong><em>r\/k<\/em> rounded to:<\/strong><\/td>\n<td><strong>Gives <em>P\u00ad30<\/em>\u00ad to be:<\/strong><\/td>\n<td><strong>Error<\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.004<\/td>\n<td>$4208.59<\/td>\n<td>$259.15<\/td>\n<\/tr>\n<tr>\n<td>0.0042<\/td>\n<td>$4521.45<\/td>\n<td>$53.71<\/td>\n<\/tr>\n<tr>\n<td>0.00417<\/td>\n<td>$4473.09<\/td>\n<td>$5.35<\/td>\n<\/tr>\n<tr>\n<td>0.004167<\/td>\n<td>$4468.28<\/td>\n<td>$0.54<\/td>\n<\/tr>\n<tr>\n<td>0.0041667<\/td>\n<td>$4467.80<\/td>\n<td>$0.06<\/td>\n<\/tr>\n<tr>\n<td>no rounding<\/td>\n<td>$4467.74<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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 0.00417, three significant digits, is close enough - $5 off of $4500 isn\u2019t too bad. Certainly keeping that fourth decimal place wouldn\u2019t have hurt.\n\nView the following for a demonstration of this example.\n\nhttps:\/\/youtu.be\/VhhYtaMN6mo\n\n<\/div>\n&nbsp;\n<div class=\"textbox\">\n<h3>Using your\u00a0 Desmos calculator<\/h3>\n&nbsp;\n\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]\n\nWe can quickly calculate this on the Desmos Calculator by putting in the formula all at once:\n\n<img class=\"aligncenter wp-image-173 size-full\" src=\"http:\/\/pressbooks.usnh.edu\/math502\/wp-content\/blogs.dir\/12\/files\/2023\/03\/Compound-Interest.png\" alt=\"\" width=\"1360\" height=\"1198\">\n\nTo enter this into the calculator, type in the following:\n\n1000 * (1 + .05\/12) <span style=\"color: #ff0000\"><strong><em>a<sup>b<\/sup><\/em><\/strong><\/span> (12 * 30)\n\nNote:\u00a0 <span style=\"color: #ff0000\"><em><strong>a<sup>b<\/sup><\/strong><\/em><\/span> is in the first row, second column of the main menu above. Now you can round your final answer to the nearest cent.\n\n<\/div>\n<div class=\"textbox exercises\">\n<p style=\"text-align: center\">Attributions<\/p>\nThis chapter contains material taken from <em>Math in Society<\/em> (on <a href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/2.5\/MathinSociety.pdf\">OpenTextBookStore<\/a>) by David Lippman, and is used under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/us\/\">CC Attribution-Share Alike 3.0 United States<\/a>\u00a0(CC BY-SA 3.0 US) license.\n\nThis chapter contains material taken from of <em>Math for the Liberal Arts<\/em> (on <a href=\"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/\">Lumen Learning<\/a>) by Lumen Learning, and is used under a <em><a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\" target=\"_blank\" rel=\"noopener noreferrer\">CC BY: Attribution<\/a><\/em> license.\n\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate one-time simple interest, and simple interest over time<\/li>\n<li>Determine APY given an interest scenario<\/li>\n<li>Calculate compound interest<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/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.<\/p>\n<p>&nbsp;<\/p>\n<h2>Simple\u00a0Interest<\/h2>\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><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={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\\\\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. 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 will you earn?<\/p>\n<p><em><strong>Solution:<\/strong><\/em><\/p>\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<p>&nbsp;<\/p>\n<p>The following video works through this example in detail.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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?<br \/>\n[reveal-answer q=&#8221;14596&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;14596&#8243;]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]<\/p>\n<p>Further explanation about solving this example can be seen here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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={{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<p><em><strong>Solution:<\/strong><\/em><\/p>\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<p>&nbsp;<\/p>\n<p>This video explains the solution.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=72467&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom2\">Click here to try this problem.<\/a><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/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<p><em><strong>Solution:<\/strong><\/em><\/p>\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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=929&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom5\">Click here to try this problem.<\/a><\/p>\n<\/div>\n<h2>Compound Interest<\/h2>\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><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 3% \u00f7 12 = 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>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<p><em><strong>Solution:<\/strong><\/em><\/p>\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<p>View this video for a walkthrough of the concept of compound interest.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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\"><span style=\"color: #ff0000\"><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><\/span><\/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<p><em><strong>Solution:<\/strong><\/em><\/p>\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<p>A video walkthrough of this example problem\u00a0is available below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6693&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom1\">Click here to try this problem.<\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Evaluating exponents on the Desmos calculator<\/h3>\n<p>When we need to calculate something like [latex]5^3[\/latex] it is easy enough to just multiply [latex]5\\cdot{5}\\cdot{5}=125[\/latex].\u00a0 But when we need to calculate something like [latex]1.005^{240}[\/latex], it would be very tedious to calculate this by multiplying [latex]1.005[\/latex] by itself [latex]240[\/latex] times!\u00a0 So to make things easier, we can harness the power of our scientific calculators.\u00a0 In this class, we are using the <a href=\"https:\/\/www.desmos.com\/scientific\">Desmos calculator<\/a>.\u00a0 If you just want to square a number, the key is <span style=\"color: #ff0000\"><em><strong>a<sup>2<\/sup><\/strong><\/em><\/span>.\u00a0 If you want to raise a number to another power, you use the key <em><strong><span style=\"color: #ff0000\">a<sup>b<\/sup><\/span><\/strong><\/em> on the main menu.<\/p>\n<p>To evaluate [latex]1.005^{240}[\/latex] we&#8217;d type 1.005 <em><strong><span style=\"color: #ff0000\">a<sup>b\u00a0<\/sup><\/span><\/strong><\/em><span style=\"color: #ff0000\"><span style=\"color: #000000\">240<\/span><\/span> .\u00a0 Try it out &#8211; you should get the answer in the figure below:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-172 size-full\" src=\"http:\/\/pressbooks.usnh.edu\/math502\/wp-content\/blogs.dir\/12\/files\/2016\/12\/Desmos-Exponent-Entry.png\" alt=\"This figure shows how an exponent is evaluated in Desmos.\" width=\"1316\" height=\"1210\" srcset=\"https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry.png 1316w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-300x276.png 300w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-1024x942.png 1024w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-768x706.png 768w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-65x60.png 65w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-225x207.png 225w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2016\/12\/Desmos-Exponent-Entry-350x322.png 350w\" sizes=\"auto, (max-width: 1316px) 100vw, 1316px\" \/><\/p>\n<p>Most scientific calculators have a button for exponents.\u00a0 If you are not using the Desmos calculator, it is typically either labeled like:<\/p>\n<p style=\"text-align: center\">^ , \u00a0 [latex]y^x[\/latex] ,\u00a0\u00a0 or [latex]x^y[\/latex] .<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>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?<\/p>\n<p><em><strong>Solution:<\/strong><\/em><\/p>\n<p>In this example, we\u2019re looking for <em>P<sub>0<\/sub><\/em>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>r<\/em> = 0.04<\/td>\n<td>4%<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 4<\/td>\n<td>4 quarters in 1 year<\/td>\n<\/tr>\n<tr>\n<td><em>N <\/em>= 18<\/td>\n<td>Since we know the balance in 18 years<\/td>\n<\/tr>\n<tr>\n<td><em>P<sub>18<\/sub><\/em> = $40,000<\/td>\n<td>The amount we have in 18 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In this case, we\u2019re going to have to set up the equation, and solve for <em>P<sub>0<\/sub><\/em>.<\/p>\n<p>[latex]\\begin{align}&40000={{P}_{0}}{{\\left(1+\\frac{0.04}{4}\\right)}^{18\\times4}}\\\\&40000={{P}_{0}}(2.0471)\\\\&{{P}_{0}}=\\frac{40000}{2.0471}=\\$19539.84 \\\\\\end{align}[\/latex]<\/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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><a href=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6692&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom2\">Click here to try this problem.<\/a><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Rounding<\/h3>\n<p>If you are not inputting your entire formula into Desmos and choose to do it section by section, 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>To see why not over-rounding is so important, if you choose not to enter your formula all at once into Desmos, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>P<sub>0<\/sub><\/em> = $1000<\/td>\n<td>the initial deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.05<\/td>\n<td>5%<\/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> = 30<\/td>\n<td>since we\u2019re looking for the amount after 30 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If we first compute <em>r\/k<\/em>, we find 0.05\/12 = 0.00416666666667<\/p>\n<p>Here is the effect of rounding this to different values:<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/p>\n<p><strong><em>r\/k<\/em> rounded to:<\/strong><\/td>\n<td><strong>Gives <em>P\u00ad30<\/em>\u00ad to be:<\/strong><\/td>\n<td><strong>Error<\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.004<\/td>\n<td>$4208.59<\/td>\n<td>$259.15<\/td>\n<\/tr>\n<tr>\n<td>0.0042<\/td>\n<td>$4521.45<\/td>\n<td>$53.71<\/td>\n<\/tr>\n<tr>\n<td>0.00417<\/td>\n<td>$4473.09<\/td>\n<td>$5.35<\/td>\n<\/tr>\n<tr>\n<td>0.004167<\/td>\n<td>$4468.28<\/td>\n<td>$0.54<\/td>\n<\/tr>\n<tr>\n<td>0.0041667<\/td>\n<td>$4467.80<\/td>\n<td>$0.06<\/td>\n<\/tr>\n<tr>\n<td>no rounding<\/td>\n<td>$4467.74<\/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 0.00417, three significant digits, is close enough &#8211; $5 off of $4500 isn\u2019t too bad. 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-6\" 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\u00a0 Desmos calculator<\/h3>\n<p>&nbsp;<\/p>\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 this on the Desmos Calculator by putting in the formula all at once:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-173 size-full\" src=\"http:\/\/pressbooks.usnh.edu\/math502\/wp-content\/blogs.dir\/12\/files\/2023\/03\/Compound-Interest.png\" alt=\"\" width=\"1360\" height=\"1198\" srcset=\"https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest.png 1360w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-300x264.png 300w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-1024x902.png 1024w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-768x677.png 768w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-65x57.png 65w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-225x198.png 225w, https:\/\/pressbooks.usnh.edu\/mth402\/files\/2023\/03\/Compound-Interest-350x308.png 350w\" sizes=\"auto, (max-width: 1360px) 100vw, 1360px\" \/><\/p>\n<p>To enter this into the calculator, type in the following:<\/p>\n<p>1000 * (1 + .05\/12) <span style=\"color: #ff0000\"><strong><em>a<sup>b<\/sup><\/em><\/strong><\/span> (12 * 30)<\/p>\n<p>Note:\u00a0 <span style=\"color: #ff0000\"><em><strong>a<sup>b<\/sup><\/strong><\/em><\/span> is in the first row, second column of the main menu above. Now you can round your final answer to the nearest cent.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<p style=\"text-align: center\">Attributions<\/p>\n<p>This chapter contains material taken from <em>Math in Society<\/em> (on <a href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/2.5\/MathinSociety.pdf\">OpenTextBookStore<\/a>) by David Lippman, and is used under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/us\/\">CC Attribution-Share Alike 3.0 United States<\/a>\u00a0(CC BY-SA 3.0 US) license.<\/p>\n<p>This chapter contains material taken from of <em>Math for the Liberal Arts<\/em> (on <a href=\"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/\">Lumen Learning<\/a>) by Lumen Learning, and is used under a <em><a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\" target=\"_blank\" rel=\"noopener noreferrer\">CC BY: Attribution<\/a><\/em> license.<\/p>\n<\/div>\n","protected":false},"author":11,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by"},"chapter-type":[49],"contributor":[],"license":[53],"class_list":["post-174","chapter","type-chapter","status-publish","hentry","chapter-type-numberless","license-cc-by"],"part":169,"_links":{"self":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapters\/174","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/wp\/v2\/users\/11"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapters\/174\/revisions"}],"predecessor-version":[{"id":175,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapters\/174\/revisions\/175"}],"part":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/parts\/169"}],"metadata":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapters\/174\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/wp\/v2\/media?parent=174"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/pressbooks\/v2\/chapter-type?post=174"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/wp\/v2\/contributor?post=174"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.usnh.edu\/mth402\/wp-json\/wp\/v2\/license?post=174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}