{"id":1513,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1513"},"modified":"2015-11-12T18:35:28","modified_gmt":"2015-11-12T18:35:28","slug":"use-compound-interest-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/use-compound-interest-formulas\/","title":{"raw":"Use compound interest formulas","rendered":"Use compound interest formulas"},"content":{"raw":"<p id=\"fs-id1165137447026\">Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use <strong>compound interest<\/strong>. The term <em data-effect=\"italics\">compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.<\/p>\n<p id=\"fs-id1165137447037\">The <strong>annual percentage rate (APR)<\/strong> of an account, also called the <strong>nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0<em data-effect=\"italics\">nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being <em data-effect=\"italics\">greater<\/em> than the nominal rate! This is a powerful tool for investing.<\/p>\n<p id=\"fs-id1165135160118\">We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time <em>t<\/em>, principal <em>P<\/em>, APR <em>r<\/em>, and number of compounding periods in a year\u00a0<em>n<\/em>:<\/p>\n\n<div id=\"eip-986\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137935717\">For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.<\/p>\n\n<table id=\"Table_04_01_03\" summary=\"Six rows and two columns. The first column is labeled,\"><thead><tr><th data-align=\"center\">Frequency<\/th>\n<th data-align=\"center\">Value after 1 year<\/th>\n<\/tr><\/thead><tbody><tr><td>Annually<\/td>\n<td>$1100<\/td>\n<\/tr><tr><td>Semiannually<\/td>\n<td>$1102.50<\/td>\n<\/tr><tr><td>Quarterly<\/td>\n<td>$1103.81<\/td>\n<\/tr><tr><td>Monthly<\/td>\n<td>$1104.71<\/td>\n<\/tr><tr><td>Daily<\/td>\n<td>$1105.16<\/td>\n<\/tr><\/tbody><\/table><div id=\"fs-id1165137793679\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: The Compound Interest Formula<\/h3>\n<p id=\"fs-id1165135184167\"><strong>Compound interest<\/strong> can be calculated using the formula<\/p>\n\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\\\[\/latex]<\/div>\n<p id=\"eip-237\">where<\/p>\n\n<ul id=\"fs-id1165137448453\"><li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\n\t<li><i>t<\/i> is measured in years,<\/li>\n\t<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\n\t<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal, and<\/li>\n\t<li><em>n<\/em>\u00a0is the number of compounding periods in one year.<\/li>\n<\/ul><\/div>\n<div id=\"Example_04_01_08\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 7: Calculating Compound Interest<\/h3>\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137812830\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137812832\">Because we are starting with $3,000, <em>P\u00a0<\/em>= 3000. Our interest rate is 3%, so <em>r<\/em>\u00a0=\u00a00.03. Because we are compounding quarterly, we are compounding 4 times per year, so <em>n\u00a0<\/em>= 4. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\n\n<div id=\"eip-id1402796\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}A\\left(t\\right)\\hfill &amp; =P\\left(1+\\frac{r}{n}\\right)^{nt}\\hfill &amp; \\text{Use the compound interest formula}. \\\\ A\\left(10\\right)\\hfill &amp; =3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}\\hfill &amp; \\text{Substitute using given values}. \\\\ \\text{ }\\hfill &amp; \\approx 4045.05\\hfill &amp; \\text{Round to two decimal places}.\\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137694040\">The account will be worth about $4,045.05 in 10 years.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 7<\/h3>\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<div id=\"Example_04_01_09\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135180446\" class=\"exercise\" data-type=\"exercise\">\n<div class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 8: Using the Compound Interest Formula to Solve for the Principal<\/h3>\n<p id=\"fs-id1165135175327\">A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child\u2019s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135175338\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137664627\">The nominal interest rate is 6%, so <em>r\u00a0<\/em>= 0.06. Interest is compounded twice a year, so <em>k\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165135209414\">We want to find the initial investment, <em>P<\/em>, needed so that the value of the account will be worth $40,000 in 18 years. Substitute the given values into the compound interest formula, and solve for <em>P<\/em>.<\/p>\n\n<div id=\"eip-id1165131884554\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}A\\left(t\\right)\\hfill &amp; =P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\hfill &amp; \\text{Use the compound interest formula}.\\hfill \\\\ 40,000\\hfill &amp; =P{\\left(1+\\frac{0.06}{2}\\right)}^{2\\left(18\\right)}\\hfill &amp; \\text{Substitute using given values }A\\text{, }r, n\\text{, and }t.\\hfill \\\\ 40,000\\hfill &amp; =P{\\left(1.03\\right)}^{36}\\hfill &amp; \\text{Simplify}.\\hfill \\\\ \\frac{40,000}{{\\left(1.03\\right)}^{36}}\\hfill &amp; =P\\hfill &amp; \\text{Isolate }P.\\hfill \\\\ P\\hfill &amp; \\approx 13,801\\hfill &amp; \\text{Divide and round to the nearest dollar}.\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137937589\">Lily will need to invest $13,801 to have $40,000 in 18 years.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 8<\/h3>\n<p id=\"fs-id1165135176729\">Refer to Example 8. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>","rendered":"<p id=\"fs-id1165137447026\">Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use <strong>compound interest<\/strong>. The term <em data-effect=\"italics\">compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.<\/p>\n<p id=\"fs-id1165137447037\">The <strong>annual percentage rate (APR)<\/strong> of an account, also called the <strong>nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0<em data-effect=\"italics\">nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being <em data-effect=\"italics\">greater<\/em> than the nominal rate! This is a powerful tool for investing.<\/p>\n<p id=\"fs-id1165135160118\">We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time <em>t<\/em>, principal <em>P<\/em>, APR <em>r<\/em>, and number of compounding periods in a year\u00a0<em>n<\/em>:<\/p>\n<div id=\"eip-986\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137935717\">For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.<\/p>\n<table id=\"Table_04_01_03\" summary=\"Six rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th data-align=\"center\">Frequency<\/th>\n<th data-align=\"center\">Value after 1 year<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Annually<\/td>\n<td>$1100<\/td>\n<\/tr>\n<tr>\n<td>Semiannually<\/td>\n<td>$1102.50<\/td>\n<\/tr>\n<tr>\n<td>Quarterly<\/td>\n<td>$1103.81<\/td>\n<\/tr>\n<tr>\n<td>Monthly<\/td>\n<td>$1104.71<\/td>\n<\/tr>\n<tr>\n<td>Daily<\/td>\n<td>$1105.16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137793679\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: The Compound Interest Formula<\/h3>\n<p id=\"fs-id1165135184167\"><strong>Compound interest<\/strong> can be calculated using the formula<\/p>\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\\\[\/latex]<\/div>\n<p id=\"eip-237\">where<\/p>\n<ul id=\"fs-id1165137448453\">\n<li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\n<li><i>t<\/i> is measured in years,<\/li>\n<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\n<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal, and<\/li>\n<li><em>n<\/em>\u00a0is the number of compounding periods in one year.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_01_08\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 7: Calculating Compound Interest<\/h3>\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?<\/p>\n<\/div>\n<div id=\"fs-id1165137812830\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137812832\">Because we are starting with $3,000, <em>P\u00a0<\/em>= 3000. Our interest rate is 3%, so <em>r<\/em>\u00a0=\u00a00.03. Because we are compounding quarterly, we are compounding 4 times per year, so <em>n\u00a0<\/em>= 4. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\n<div id=\"eip-id1402796\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}A\\left(t\\right)\\hfill & =P\\left(1+\\frac{r}{n}\\right)^{nt}\\hfill & \\text{Use the compound interest formula}. \\\\ A\\left(10\\right)\\hfill & =3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}\\hfill & \\text{Substitute using given values}. \\\\ \\text{ }\\hfill & \\approx 4045.05\\hfill & \\text{Round to two decimal places}.\\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137694040\">The account will be worth about $4,045.05 in 10 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 7<\/h3>\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"Example_04_01_09\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135180446\" class=\"exercise\" data-type=\"exercise\">\n<div class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 8: Using the Compound Interest Formula to Solve for the Principal<\/h3>\n<p id=\"fs-id1165135175327\">A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child\u2019s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?<\/p>\n<\/div>\n<div id=\"fs-id1165135175338\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137664627\">The nominal interest rate is 6%, so <em>r\u00a0<\/em>= 0.06. Interest is compounded twice a year, so <em>k\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165135209414\">We want to find the initial investment, <em>P<\/em>, needed so that the value of the account will be worth $40,000 in 18 years. Substitute the given values into the compound interest formula, and solve for <em>P<\/em>.<\/p>\n<div id=\"eip-id1165131884554\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}A\\left(t\\right)\\hfill & =P{\\left(1+\\frac{r}{n}\\right)}^{nt}\\hfill & \\text{Use the compound interest formula}.\\hfill \\\\ 40,000\\hfill & =P{\\left(1+\\frac{0.06}{2}\\right)}^{2\\left(18\\right)}\\hfill & \\text{Substitute using given values }A\\text{, }r, n\\text{, and }t.\\hfill \\\\ 40,000\\hfill & =P{\\left(1.03\\right)}^{36}\\hfill & \\text{Simplify}.\\hfill \\\\ \\frac{40,000}{{\\left(1.03\\right)}^{36}}\\hfill & =P\\hfill & \\text{Isolate }P.\\hfill \\\\ P\\hfill & \\approx 13,801\\hfill & \\text{Divide and round to the nearest dollar}.\\hfill \\end{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137937589\">Lily will need to invest $13,801 to have $40,000 in 18 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 8<\/h3>\n<p id=\"fs-id1165135176729\">Refer to Example 8. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a><\/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-1513\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <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>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1513","chapter","type-chapter","status-publish","hentry"],"part":1500,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1513\/revisions"}],"predecessor-version":[{"id":2345,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1513\/revisions\/2345"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1500"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1513\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=1513"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1513"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1513"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=1513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}