{"id":236084,"date":"2025-06-16T04:22:03","date_gmt":"2025-06-16T04:22:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236084"},"modified":"2025-06-16T04:22:05","modified_gmt":"2025-06-16T04:22:05","slug":"a-property-worth-50000-can-be-purchased-for-20-down-and-quarterly-mortgage-payments-of-1000-for-25-years","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/a-property-worth-50000-can-be-purchased-for-20-down-and-quarterly-mortgage-payments-of-1000-for-25-years\/","title":{"rendered":"A property worth 50,000 can be purchased for 20% down and quarterly mortgage payments of $ 1000 for 25 years"},"content":{"rendered":"\n<p>A property worth 50,000 can be purchased for 20% down and quarterly mortgage payments of $ 1000 for 25 years. What nominal rate of interest compounded monthly is charged?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>nominal rate of interest compounded monthly<\/strong>, first identify key values from the problem:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Property Value<\/strong>: $50,000<\/li>\n\n\n\n<li><strong>Down Payment<\/strong>: 20% of $50,000 = $10,000<\/li>\n\n\n\n<li><strong>Loan Amount (Principal)<\/strong>: $50,000 \u2212 $10,000 = $40,000<\/li>\n\n\n\n<li><strong>Quarterly Payment<\/strong>: $1,000<\/li>\n\n\n\n<li><strong>Loan Term<\/strong>: 25 years = 100 quarters (since 4 quarters per year)<\/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: Set Up the Present Value of an Annuity Formula<\/h3>\n\n\n\n<p>The present value of an ordinary annuity is:P=R\u22c51\u2212(1+i)\u2212niP = R \\cdot \\frac{1 &#8211; (1 + i)^{-n}}{i}P=R\u22c5i1\u2212(1+i)\u2212n\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P=40,000P = 40,000P=40,000 (present value of the loan)<\/li>\n\n\n\n<li>R=1,000R = 1,000R=1,000 (quarterly payment)<\/li>\n\n\n\n<li>n=100n = 100n=100 (total number of payments)<\/li>\n\n\n\n<li>iii = unknown <strong>quarterly interest rate<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Solve for iii numerically.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use a Financial Calculator or Iterative Method<\/h3>\n\n\n\n<p>Using a financial calculator or numerical solver:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>PV = 40,000<\/li>\n\n\n\n<li>PMT = \u22121,000<\/li>\n\n\n\n<li>N = 100<\/li>\n\n\n\n<li>FV = 0<\/li>\n<\/ul>\n\n\n\n<p>Solving gives i\u22480.006126i \\approx 0.006126i\u22480.006126 (quarterly interest rate \u2248 0.6126%)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Convert to Nominal Annual Rate Compounded Monthly<\/h3>\n\n\n\n<p>Convert the quarterly rate to the nominal annual rate <strong>compounded monthly<\/strong>. This requires first converting the effective quarterly rate to a monthly rate:(1+iq)=(1+im)3\u21d2(1+0.006126)=(1+im)3(1 + i_q) = (1 + i_m)^{3} \\Rightarrow (1 + 0.006126) = (1 + i_m)^3(1+iq\u200b)=(1+im\u200b)3\u21d2(1+0.006126)=(1+im\u200b)3<\/p>\n\n\n\n<p>Solving:1+im=(1.006126)1\/3\u22481.002037\u21d2im\u22480.002037&nbsp;(monthly&nbsp;rate)1 + i_m = (1.006126)^{1\/3} \\approx 1.002037 \\Rightarrow i_m \\approx 0.002037 \\text{ (monthly rate)}1+im\u200b=(1.006126)1\/3\u22481.002037\u21d2im\u200b\u22480.002037&nbsp;(monthly&nbsp;rate)<\/p>\n\n\n\n<p>Nominal annual rate compounded monthly:r=im\u00d712=0.002037\u00d712=0.024444\u22482.44%r = i_m \\times 12 = 0.002037 \\times 12 = 0.024444 \\approx 2.44\\%r=im\u200b\u00d712=0.002037\u00d712=0.024444\u22482.44%<\/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>Nominal interest rate compounded monthly \u2248 2.44%<\/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 (300 words):<\/h3>\n\n\n\n<p>A real estate purchase includes a property valued at $50,000, with a 20% down payment reducing the actual borrowed amount to $40,000. The buyer agrees to repay the loan via quarterly installments of $1,000 over a period of 25 years, totaling 100 payments.<\/p>\n\n\n\n<p>The present value of all these future payments must equal the amount borrowed. This relationship is captured by the annuity formula, which equates a stream of equal periodic payments to a lump-sum amount based on an interest rate and number of periods.<\/p>\n\n\n\n<p>To solve for the quarterly interest rate, a numerical approach or financial calculator is applied. After determining the quarterly rate to be approximately 0.6126%, it must then be converted to a nominal annual interest rate compounded monthly, as requested.<\/p>\n\n\n\n<p>This involves first finding the equivalent monthly rate using the compound interest relationship: a quarterly interest rate is equivalent to compounding the monthly rate three times. Solving for the monthly rate gives approximately 0.2037%. This is then scaled to a nominal annual rate by multiplying by 12, resulting in approximately 2.44%.<\/p>\n\n\n\n<p>This rate represents the nominal interest rate compounded monthly that equates a $40,000 present value to 100 quarterly payments of $1,000. The process shows the difference between payment frequency (quarterly) and compounding frequency (monthly), which often occurs in financial contracts.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-635.jpeg\" alt=\"\" class=\"wp-image-236085\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A property worth 50,000 can be purchased for 20% down and quarterly mortgage payments of $ 1000 for 25 years. What nominal rate of interest compounded monthly is charged? The Correct Answer and Explanation is: To find the nominal rate of interest compounded monthly, first identify key values from the problem: Step 1: Set Up [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-236084","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236084","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=236084"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236084\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236084"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236084"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236084"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}