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Question & Answer: Solve for the unknown number of years in each of the following (Do not round intermediate calculations and round your answer…..

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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 |