{"id":23243,"date":"2025-06-17T11:22:41","date_gmt":"2025-06-17T11:22:41","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=23243"},"modified":"2025-06-17T11:22:44","modified_gmt":"2025-06-17T11:22:44","slug":"alan-deposited-5000-into-an-account-with-a-11-annual-interest-rate-compounded-semiannually","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/alan-deposited-5000-into-an-account-with-a-11-annual-interest-rate-compounded-semiannually\/","title":{"rendered":"Alan deposited 5,000 into an account with a 11% annual interest rate, compounded semiannually"},"content":{"rendered":"\n<p>Alan deposited 5,000 into an account with a 11% annual interest rate, compounded semiannually. Assuming that no withdrawals are made, how long will it take for the investment to grow to 29,260? Do not round any intermediate computations, and round your answer to the nearest hundredth.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>To determine how long it will take Alan\u2019s investment to grow from <strong>$5,000 to $29,260<\/strong> at an <strong>11% annual interest rate compounded semiannually<\/strong>, we use the <strong>compound interest formula<\/strong>: A=P(1+rn)ntA = P\\left(1 + \\frac{r}{n}\\right)^{nt}<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A=29,260A = 29,260 (the future amount)<\/li>\n\n\n\n<li>P=5,000P = 5,000 (the principal)<\/li>\n\n\n\n<li>r=0.11r = 0.11 (annual interest rate)<\/li>\n\n\n\n<li>n=2n = 2 (number of compounding periods per year, since it\u2019s semiannually)<\/li>\n\n\n\n<li>tt = number of years (this is what we are solving for)<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Plug in the values into the formula<\/h3>\n\n\n\n<p>29,260=5,000(1+0.112)2t29,260 = 5,000\\left(1 + \\frac{0.11}{2}\\right)^{2t} 29,260=5,000(1.055)2t29,260 = 5,000\\left(1.055\\right)^{2t}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Divide both sides by 5,000<\/h3>\n\n\n\n<p>29,2605,000=(1.055)2t\\frac{29,260}{5,000} = (1.055)^{2t} 5.852=(1.055)2t5.852 = (1.055)^{2t}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Take the natural logarithm (ln) of both sides<\/h3>\n\n\n\n<p>ln\u2061(5.852)=ln\u2061((1.055)2t)\\ln(5.852) = \\ln((1.055)^{2t}) ln\u2061(5.852)=2t\u22c5ln\u2061(1.055)\\ln(5.852) = 2t \\cdot \\ln(1.055) 1.766=2t\u22c50.053551.766 = 2t \\cdot 0.05355<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Solve for tt<\/h3>\n\n\n\n<p>2t=1.7660.05355\u224832.982t = \\frac{1.766}{0.05355} \\approx 32.98 t=32.982=16.49t = \\frac{32.98}{2} = 16.49<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer:<\/h3>\n\n\n\n<p><strong>It will take approximately 16.49 years<\/strong> for Alan\u2019s investment to grow to <strong>$29,260<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (in 300 words):<\/h3>\n\n\n\n<p>Compound interest is a powerful method by which money grows because it earns interest not only on the principal amount but also on the accumulated interest over time. In this problem, the interest is compounded semiannually, meaning interest is added twice per year. The compound interest formula helps us determine either the future value of an investment or the time needed to reach a certain value.<\/p>\n\n\n\n<p>Alan\u2019s investment grows from $5,000 to $29,260 under an 11% annual interest rate, compounded semiannually. We use the formula: A=P(1+rn)ntA = P\\left(1 + \\frac{r}{n}\\right)^{nt}<\/p>\n\n\n\n<p>We plug in the known values and isolate the variable tt, which represents time in years. To solve for tt, we use logarithms, which are necessary when solving equations involving exponents. By taking the natural logarithm of both sides, we linearize the exponential expression, making it solvable through basic algebra.<\/p>\n\n\n\n<p>After carefully computing, we find that the investment will reach the desired amount in approximately <strong>16.49 years<\/strong>. This demonstrates the exponential growth effect of compounding: even a relatively modest principal amount can grow significantly over time with a high interest rate and no withdrawals.<\/p>\n\n\n\n<p>This problem highlights the benefit of early and long-term investing, especially when interest is compounded regularly.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Alan deposited 5,000 into an account with a 11% annual interest rate, compounded semiannually. Assuming that no withdrawals are made, how long will it take for the investment to grow to 29,260? Do not round any intermediate computations, and round your answer to the nearest hundredth. The correct answer and explanation is: To determine how [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-23243","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23243","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=23243"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23243\/revisions"}],"predecessor-version":[{"id":23245,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23243\/revisions\/23245"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=23243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=23243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=23243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}