document.write( "Question 16752: How many years will it take $5,000 to grow to $7,500 if it is invested at 8% compounded semiannually?\r
\n" ); document.write( "\n" ); document.write( "Compounded continously? \n" ); document.write( "

Algebra.Com's Answer #8171 by bam878s(77) $\"\"$ $\"About$

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we are given annual interest rate i = 8% or .08
\n" ); document.write( "the semiannual interest rate is then $\"i%2F2\"$ = $\".08%2F2\"$ = .04
\n" ); document.write( "the per period accumulation function is then $\"%281+%2B+.04%29%5En\"$
\n" ); document.write( "so, we have 7500 = 5000 * $\"%281+%2B+.04%29%5En\"$ . First divide 5000 from both sides.
\n" ); document.write( "1.5 = $\"%281+%2B+.04%29%5En\"$
\n" ); document.write( "log 1.5 = n log 1.04
\n" ); document.write( "divide log 1.04 from both sides
\n" ); document.write( "n = 10.3380351 half years or 5.1690175 years.\r
\n" ); document.write( "\n" ); document.write( "For continous interest we need to find the force of interest (d) over the accumulation period.
\n" ); document.write( "we know $\"%281+%2B+i%29%5En\"$ = $\"e%5E%28nd%29\"$
\n" ); document.write( "so $\"%281+%2B+.08%29%5En\"$ = $\"e%5E%28nd%29\"$
\n" ); document.write( "n ln 1.08 = nd (take the natural log of both sides)
\n" ); document.write( "ln 1.08 = d
\n" ); document.write( "now the accumulation function for continuous interest is $\"e%5E%28nd%29\"$
\n" ); document.write( "so 5000 * $\"e%5E%28n+%2A+ln+1.08%29\"$ = 7500
\n" ); document.write( " $\"e%5E%28n+%2A+ln+1.08%29\"$ = 1.5
\n" ); document.write( " $\"n+%2A+ln+1.08\"$ = ln 1.5 (take the natural log of both sides)
\n" ); document.write( "n = $\"ln+1.5%2F+ln+1.08\"$
\n" ); document.write( "n = 5.2684462\r
\n" ); document.write( "\n" ); document.write( "Hope this helps, let me know if you have any more financial mathematics questions. I know them fairly well. \n" ); document.write( "

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