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# Find the time required for an investment of $5000 to grow to$7000 at an interest rate of 7.5% per year, compo

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Find the time required for an investment of $5000 to grow to$7000 at an interest rate of 7.5% per year, compounded quarterly.

Jul 10, 2014

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$$\\i=0.075/4=0.01875\\ n \;\;quarters\\ \begin{array}{rlll} FV&=&P(1+i)^n&\\\\ 7000&=&5000(1.01875)^n&\\\\ 7000/5000&=&(1.01875)^n&\\\\ 7/5&=&(1.01875)^n&\\\\ log(7/5)&=&log(1.01875)^n&\\\\ log(7/5)&=&nlog(1.01875)&\\\\ log(7/5)log(1.01875)&=&n&\\\\ n&\approx &18.11\;\;quarters&\\\\ n&\approx &18.11/4\;\;years\\\ n&\approx &4.528\;\;years\\\\ \end{array}$$

Since the payments are made in quarters, the best answer is probably 19 quaters which is 4 and 3/4 years.

Jul 12, 2014

$$\\i=0.075/4=0.01875\\ n \;\;quarters\\ \begin{array}{rlll} FV&=&P(1+i)^n&\\\\ 7000&=&5000(1.01875)^n&\\\\ 7000/5000&=&(1.01875)^n&\\\\ 7/5&=&(1.01875)^n&\\\\ log(7/5)&=&log(1.01875)^n&\\\\ log(7/5)&=&nlog(1.01875)&\\\\ log(7/5)log(1.01875)&=&n&\\\\ n&\approx &18.11\;\;quarters&\\\\ n&\approx &18.11/4\;\;years\\\ n&\approx &4.528\;\;years\\\\ \end{array}$$