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 determine the annual discount rate at which both machines are equally economical, we compare their Net Present Cost (NPC)<\/strong> over the 8-year lifespan.<\/p>\n\n\n\n Let:<\/p>\n\n\n\n Let the annual discount rate be r<\/strong>.<\/p>\n\n\n\n The Net Present Cost (NPC)<\/strong> of a machine is: NPC=Initial Cost\u2212Resale Value(1+r)n\\text{NPC} = \\text{Initial Cost} – \\frac{\\text{Resale Value}}{(1 + r)^n}<\/p>\n\n\n\n where:<\/p>\n\n\n\n Set the NPCs equal to each other for the machines: 3000\u2212600(1+r)8=4000\u22121900(1+r)83000 – \\frac{600}{(1 + r)^8} = 4000 – \\frac{1900}{(1 + r)^8}<\/p>\n\n\n\n Move all terms to one side: 3000\u22124000=600\u22121900(1+r)83000 – 4000 = \\frac{600 – 1900}{(1 + r)^8} \u22121000=\u22121300(1+r)8-1000 = \\frac{-1300}{(1 + r)^8}<\/p>\n\n\n\n Multiply both sides by (1+r)8(1 + r)^8: \u22121000(1+r)8=\u22121300-1000(1 + r)^8 = -1300 (1+r)8=13001000=1.3(1 + r)^8 = \\frac{1300}{1000} = 1.3<\/p>\n\n\n\n Now take the 8th root of both sides: 1+r=(1.3)1\/81 + r = (1.3)^{1\/8} 1+r\u22481.033431 + r \\approx 1.03343 r\u22480.03343=3.343%r \\approx 0.03343 = 3.343\\%<\/p>\n\n\n\n Annual discount rate \u2248 3.34%<\/strong><\/p>\n\n\n\n When businesses or individuals choose between two capital investments with different initial costs and resale values, they often use Net Present Cost (NPC)<\/strong> to compare options. The NPC helps us account for the time value of money<\/strong> \u2014 the concept that a naira today is worth more than a naira in the future due to potential earning capacity.<\/p>\n\n\n\n In this scenario, both machines serve the same purpose for the same lifespan (8 years). The goal is to find a discount rate (interest rate) at which the present value of the total cost<\/strong> of owning either machine is the same \u2014 this is when it’s equally economical<\/strong> to choose either option.<\/p>\n\n\n\n Machine A has a lower initial cost (N3,000), but also a lower resale value (N600). Machine B is more expensive up front (N4,000), but has a higher resale value (N1,900). To properly compare these, we calculate each machine’s net present cost<\/strong> by subtracting the discounted resale value from the initial cost.<\/p>\n\n\n\n Setting both NPCs equal and solving gives us a discount rate where the buyer is indifferent between machines. Mathematically, we solve: 3000\u2212600(1+r)8=4000\u22121900(1+r)83000 – \\frac{600}{(1 + r)^8} = 4000 – \\frac{1900}{(1 + r)^8}<\/p>\n\n\n\n This simplifies to: (1+r)8=1.3(1 + r)^8 = 1.3<\/p>\n\n\n\n Solving gives r\u22483.34%r \\approx 3.34\\%. This is the break-even discount rate: below this rate, the cheaper Machine A is better; above it, the higher-resale Machine B becomes more cost-effective.<\/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. 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Initial cost = N3,000
Resale value = N600<\/li>\n\n\n\n
Initial cost = N4,000
Resale value = N1,900<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nStep 1: Net Present Cost Formula<\/h3>\n\n\n\n
\n
\n\n\n\nStep 2: Solve the Equation<\/h3>\n\n\n\n
\n\n\n\n\u2705 Final Answer:<\/h3>\n\n\n\n
\n\n\n\n\ud83d\udd0d Explanation (300+ words):<\/h3>\n\n\n\n