document.write( "Question 130665: Steven invests $2000 into a GIC which will double every five years, wheras Dana invests $1000 into a GIC which will triple evry three years. How many years will pass before they have the same amount of money? \n" ); document.write( "
Algebra.Com's Answer #95437 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Steven invests $2000 into a GIC which will double every five years, wheras Dana invests $1000 into a GIC which will triple evry three years. How many years will pass before they have the same amount of money?
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document.write( "The rule is:\r\n" );
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document.write( "\"P+=+A%281%2Br%2Fn%29%5E%28nt%29\"\r\n" );
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document.write( "We will assume the interest accrues once yearly, so n = 1\r\n" );
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document.write( "and the formula becomes\r\n" );
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document.write( "\"A+=+P%281%2Br%2F1%29%5E%281t%29\"\r\n" );
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document.write( "\"A+=+P%281%2Br%29%5Et\"\r\n" );
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document.write( "So for Steven, P = 2000, so the amount A, in t years, is\r\n" );
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document.write( "\"A+=+2000%281%2Br%29%5Et\"\r\n" );
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document.write( "Steven's GIC doubles every five years,\r\n" );
document.write( "So we substitute t = 5, and A = 4000 (double 2000),\r\n" );
document.write( "so we can solve for r:\r\n" );
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document.write( "\"4000+=+2000%281%2Br%29%5E5\"\r\n" );
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document.write( "Divide both sides by 2000\r\n" );
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document.write( "\"2=%281%2Br%29%5E5\"\r\n" );
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document.write( "Take the 5th root of both sides:\r\n" );
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document.write( "\"root%285%2C2%29+=+root%285%2C%281%2Br%29%5E5%29+\"\r\n" );
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document.write( "Simplifying the right side:\r\n" );
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document.write( "\"root%285%2C2%29+=+1%2Br+\"\r\n" );
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document.write( "Solving for r:\r\n" );
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document.write( "\"root%285%2C2%29+-+1+=+r\"\r\n" );
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document.write( "Substituting that into\r\n" );
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document.write( "\"A+=+2000%281%2Br%29%5Et\"\r\n" );
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document.write( "\"A+=+2000%281%2Broot%285%2C2%29+-+1%29%5Et\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"A+=+2000%28root%285%2C2%29%29%5Et\"\r\n" );
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document.write( "This is the formula for Steven's amount A in t years.\r\n" );
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document.write( "---------------------------------------------\r\n" );
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document.write( "Now we do exactly the same thing with Dana's\r\n" );
document.write( "GIC.\r\n" );
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document.write( "\"A+=+P%281%2Br%29%5Et\"\r\n" );
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document.write( "Now for Dana, P = 1000, so the amount A, in t years, is\r\n" );
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document.write( "\"A+=+1000%281%2Br%29%5Et\"\r\n" );
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document.write( "DANA's GIC triples every three years,\r\n" );
document.write( "So we substitute t = 3, and A = 3000 (triple 1000),\r\n" );
document.write( "so we can solve for r:\r\n" );
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document.write( "\"3000+=+1000%281%2Br%29%5E3\"\r\n" );
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document.write( "Divide both sides by 1000\r\n" );
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document.write( "\"3=%281%2Br%29%5E3\"\r\n" );
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document.write( "Take the cube root of both sides:\r\n" );
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document.write( "\"root%283%2C3%29+=+root%283%2C%281%2Br%29%5E3%29+\"\r\n" );
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document.write( "Simplifying the right side:\r\n" );
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document.write( "\"root%283%2C3%29+=+1%2Br+\"\r\n" );
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document.write( "Solving for r:\r\n" );
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document.write( "\"+root%283%2C3%29+-+1+=+r\"\r\n" );
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document.write( "Substituting that into\r\n" );
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document.write( "\"A+=+1000%281%2Br%29%5Et\"\r\n" );
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document.write( "\"A+=+1000%281%2Broot%283%2C3%29+-+1%29%5Et\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"+A+=+1000%28+root%283%2C3%29+%29%5Et\"\r\n" );
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document.write( "This is the formula for Dana's amount A in t years.\r\n" );
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document.write( "-------------------------------\r\n" );
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document.write( "So to find out when they have an equal amount, we\r\n" );
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document.write( "set their two A's equal, and solve for t:\r\n" );
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document.write( "\"+A+=+2000%28+root%285%2C2%29+%29%5Et+\" and \"A+=+1000%28+root%283%2C3%29+%29%5Et\"\r\n" );
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document.write( "\"2000%28root%285%2C2%29%29%5Et+=+1000%28root%283%2C3%29%29%5Et\"\r\n" );
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document.write( "Divide both sides by 1000\r\n" );
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document.write( "\"2%28root%285%2C2%29%29%5Et+=+%28root%283%2C3%29%29%5Et\"\r\n" );
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document.write( "Change the roots to fractional exponentials of their radicands:\r\n" );
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document.write( "\"+2+%28+2%5E%281%2F5%29+%29%5Et+\" = \"+%283%5E%281%2F3%29%29%5Et++\"\r\n" );
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document.write( "Multiply inner exponents by outer exponents\r\n" );
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document.write( "\"2%282%5E%28t%2F5%29+%29+\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "Write the first \"2\" as \"2%5E1\"\r\n" );
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document.write( "\"2%5E1%282%5E%28t%2F5%29+%29+\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "Change the exponent \"1\" to \"5%2F5\" so you \r\n" );
document.write( "can add exponents on the left:\r\n" );
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document.write( "\"2%5E%285%2F5%29%282%5E%28t%2F5%29+%29+\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "Add exponents:\r\n" );
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document.write( "\"+2%5E%285%2F5%2Bt%2F5%29++\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "\"2%5E%28%285%2Bt%29%2F5%29++\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "\"2%5E%28%285%2Bt%29%2F5+%29++\" = \"3%5E%28t%2F3%29\"\r\n" );
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document.write( "Raise both sides to the 15th power\r\n" );
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document.write( "\"++2%5E%28+%285%2Bt%29%2F5+%29%5E15+%29++++\" = \"+%28+3%5E%28t%2F3%29+%29%5E15++\"\r\n" );
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document.write( "Multiply inner exponents by outer exponents:\r\n" );
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document.write( "\"++2%5E%28++15%285%2Bt%29+%29%2F5+%29\" = \"+3%5E%28%2815t%29%2F3%29\"\r\n" );
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document.write( "Cancel 5 into 15 on left and 3 into 15 on the right:\r\n" );
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document.write( "\"2%5E%283%285%2Bt%29%29\" = \"3%5E%285t%29\"\r\n" );
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document.write( "\"2%5E%2815%2B3t%29\" = \"3%5E%285t%29\"\r\n" );
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document.write( "Take the log (either log10 or ln) of both sides\r\n" );
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document.write( "\"log%28%282%5E%2815%2B3t%29%29%29\" = \"log%28%283%5E%285t%29%29%29\"\r\n" );
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document.write( "Use the rule of logs \"+log%28b%2Cx%5EN%29+=+N%2Alog%28b%2Cx%29\"\r\n" );
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document.write( "\"+%2815%2B3t%29log%282%29+\" = \"+%285t%29log%283%29+\"\r\n" );
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document.write( "Replace \"log%282%29\" by \"A\" and replace \"log%283%29\" by \"B\"\r\n" );
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document.write( "\"+%2815%2B3t%29A+\" = \"+%285t%29B+\"\r\n" );
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document.write( "\"+A%2815%2B3t%29\" = \"B%285t%29\"\r\n" );
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document.write( "\"15A%2B3At\" = \"5Bt%29\"\r\n" );
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document.write( "Isolate the terms in t on the right\r\n" );
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document.write( "\"15A\" = \"5Bt-3At\"\r\n" );
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document.write( "Factor out t on the right:\r\n" );
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document.write( "\"15A\" = \"t%285B-3A%29\"\r\n" );
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document.write( "Divide both sides by \"5B-3A\"\r\n" );
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document.write( "\"%2815A%29%2F%285B-3A%29\" = \"t\"\r\n" );
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document.write( "Now replace \"A\" by \"log%282%29\" \r\n" );
document.write( "and replace \"B\" by \"log%283%29\"\r\n" );
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document.write( "\"t\" = \"%2815log%282%29%29%2F%285log%283%29-3log%282%29%29\"  \r\n" );
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document.write( "Use calculator to get right side\r\n" );
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document.write( "\"t\" = \"3.045801234\"\r\n" );
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document.write( "So the answer is just after 3 years.\r\n" );
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document.write( "Edwin
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