A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600.<\/p>\n\n\n\n
An alternative machine costs N4,000, also with a lifespan of 8 years, but its resale value at the end of its life is N1,900.<\/p>\n\n\n\n
At what annual discount rate (compounded annually) would it be equally economical to choose either machine? (Ignore taxes.)<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To find the annual discount rate<\/strong> at which both machines are equally economical<\/strong>, we need to compare the present value (PV) of their costs<\/strong> over their 8-year lifespans.<\/p>\n\n\n\n Let\u2019s define:<\/p>\n\n\n\n The present cost<\/strong> of each machine is:<\/p>\n\n\n\n $$ Where:<\/p>\n\n\n\n Set the net present cost of both machines equal<\/strong>:<\/p>\n\n\n\n $$ Move all terms to one side:<\/p>\n\n\n\n $$ $$ Multiply both sides by $(1 + r)^8$:<\/p>\n\n\n\n $$ Divide both sides by -1000:<\/p>\n\n\n\n $$ Take the 8th root of both sides:<\/p>\n\n\n\n $$ Using a calculator:<\/p>\n\n\n\n $$ $$ To determine which machine is more economical, we use the present value<\/strong> approach, which helps compare money spent or saved at different points in time. Since both machines last 8 years and provide similar services, the main difference lies in their initial cost<\/strong> and salvage (resale) value<\/strong>.<\/p>\n\n\n\n The concept of discounting<\/strong> helps us find how much future money (like resale value) is worth in today’s terms. A higher discount rate makes future money less valuable today.<\/p>\n\n\n\n We calculate each machine\u2019s net present cost<\/strong>, which is the initial cost minus the present value of the resale value. This is because we “recover” some money at the end by selling the machine.<\/p>\n\n\n\n For both machines to be equally economical, their net present costs<\/strong> must be the same. We set up an equation that expresses this equality and solve it to find the discount rate.<\/p>\n\n\n\n The solution involves solving an equation where the only unknown is the discount rate $r$<\/strong>. Using algebra and exponentiation, we find that the value of $r$ that satisfies the condition is approximately 3.32% per year<\/strong>, compounded annually.<\/p>\n\n\n\n This means that if the time value of money (or cost of capital) is exactly 3.32%, both machines are financially equivalent over their lifespan. If the actual discount rate is higher than 3.32%<\/strong>, the cheaper machine (N3,000) is preferable. If the rate is lower<\/strong>, the more expensive machine with the higher salvage value (N4,000) becomes more economical in the long run.<\/p>\n","protected":false},"excerpt":{"rendered":" A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600. An alternative machine costs N4,000, also with a lifespan of 8 years, but its resale value at the end of its life is N1,900. At what annual discount rate (compounded annually) would it be equally economical […]<\/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-215299","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215299","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=215299"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215299\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=215299"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=215299"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=215299"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n\n\n\nStep 1: Net Present Cost Formula<\/h3>\n\n\n\n
\\text{Net Present Cost} = \\text{Initial Cost} – \\frac{\\text{Salvage Value}}{(1 + r)^n}
$$<\/p>\n\n\n\n\n
3000 – \\frac{600}{(1 + r)^8} = 4000 – \\frac{1900}{(1 + r)^8}
$$<\/p>\n\n\n\n
\n\n\n\nStep 2: Solve the Equation<\/h3>\n\n\n\n
3000 – 4000 = \\frac{600 – 1900}{(1 + r)^8}
$$<\/p>\n\n\n\n
-1000 = \\frac{-1300}{(1 + r)^8}
$$<\/p>\n\n\n\n
-1000(1 + r)^8 = -1300
$$<\/p>\n\n\n\n
(1 + r)^8 = \\frac{1300}{1000} = 1.3
$$<\/p>\n\n\n\n
1 + r = (1.3)^{1\/8}
$$<\/p>\n\n\n\n
(1.3)^{1\/8} \\approx 1.0332
$$<\/p>\n\n\n\n
r = 1.0332 – 1 = 0.0332 \\text{ or } 3.32\\%
$$<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer: 3.32% annual discount rate<\/strong><\/h3>\n\n\n\n
\n\n\n\n\ud83d\udd0d Explanation (300+ words)<\/strong><\/h3>\n\n\n\n