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Solve for the unknown number of years in each of the following (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.) Present Value Years $ 600 850 8,800 21,900 Interest Rate Future Value 8% $ 1,393 2,330 367,247 382,983 12 18 14
Expert Answer
| Solution: | ||||
| 1. | Present Value | Years | Interest Rate | Future Value |
| $600 | 10.94 | 8% | $1,393 | |
| 850 | 8.90 | 12 | 2,330 | |
| 18,800 | 17.96 | 18 | 367,247 | |
| 21,900 | 21.84 | 14 | 382,983 | |
| Working Notes: | ||||
| Future value = Present Value (1+r)^t | ||||
| For | ||||
| $600 | 10.94 | 8% | $1,393 | |
| Future value = Present Value (1+r)^t | ||||
| 1,393 = 600 (1+0.08)^t | ||||
| (1.08)^t = (1,393/600) | ||||
| taking log on both side | ||||
| (1.08)^t =2.32166 | ||||
| Log(1.08)^t = Log(2.32166) | ||||
| t x Log(1.08) = Log(2.32166) | ||||
| t=Log(2.32166)/Log(1.08) | ||||
| t= 0.3657986/0.0334237 | ||||
| t= 10.94 | ||||
| For | ||||
| 850 | 8.90 | 12 | 2,330 | |
| Future value = Present Value (1+r)^t | ||||
| 2,330 = 850(1+0.12)^t | ||||
| (1.12)^t = (2,330/850) | ||||
| taking log on both side | ||||
| (1.12)^t =2.741176 | ||||
| Log(1.12)^t = Log(2.741176) | ||||
| t x Log(1.12) = Log(2.741176) | ||||
| t=Log(2.741176)/Log(1.12) | ||||
| t= 0.4379369/0.0492180 | ||||
| t= 8.8979 | ||||
| t=8.90 | ||||
| For | ||||
| 18,800 | 17.96 | 18 | 367,247 | |
| Future value = Present Value (1+r)^t | ||||
| 367,247 = 18,800(1+0.18)^t | ||||
| (1.18)^t = (367,247/18,800) | ||||
| taking log on both side | ||||
| (1.18)^t =19.53441489 | ||||
| Log(1.18)^t = Log(19.53441489) | ||||
| t x Log(1.18) = Log(19.53441489) | ||||
| t=Log(19.53441489)/Log(1.18) | ||||
| t= 1.2908004074/0.071882007 | ||||
| t= 17.95721156 | ||||
| t=17.96 | ||||
| For | ||||
| 21,900 | 21.84 | 14 | 382,983 | |
| Future value = Present Value (1+r)^t | ||||
| 382,983 = 21,900(1+0.14)^t | ||||
| (1.14)^t = (382,983/21,900) | ||||
| taking log on both side | ||||
| (1.14)^t =17.48780822 | ||||
| Log(1.14)^t = Log(17.48780822) | ||||
| t x Log(1.14) = Log(17.48780822) | ||||
| t=Log(17.48780822)/Log(1.14) | ||||
| t= 1.2427353819/0.05690485 | ||||
| t= 21.83883064273 | ||||
| t=21.84 | ||||
| Notes: | Exact value of t is calculation showing t value having more than two decimal , as per demand of question t value is round off to two decimal | |||
| Log value is calculate from online calculator | ||||
| 2. | Annual increase in selling price = 2.97% | |||
| Working Notes | ||||
| Using formula | ||||
| Future value = Present Value (1+r)^t | ||||
| $280,400=$197,300(1+r)^12 | ||||
| (1+r)^12 = 280,400/197,300 | ||||
| (1+r)^12 = 1.421186011 | ||||
| (1+r) = (1.421186011)^(1/12) | ||||
| (1+r) = 1.029724178 | ||||
| r=1.029724178 – 1 | ||||
| r=0.029724178 | ||||
| r=2.97% | ||||
| Exact value of r is calculation showing r value having more than two decimal , as per demand of question r value is round off to two decimal | ||||
