Question 372469


{{{A=P(1+r/n)^(n*t)}}} Start with the compound interest formula



{{{1000=500(1+0.12/4)^(4*t)}}} Plug in {{{A=1000}}} (double the initial investment of $500), {{{P=500}}}, {{{r=0.12}}} (the decimal equivalent of 12%), and {{{n=4}}}.



{{{1000=500(1+0.03)^(4*t)}}} Evaluate {{{0.12/4}}} to get {{{0.03}}}



{{{1000=500(1.03)^(4*t)}}} Add {{{1}}} to {{{0.03}}} to get {{{1.03}}}



{{{1000/500=(1.03)^(4*t)}}} Divide both sides by {{{500}}}.



{{{2=(1.03)^(4*t)}}} Evaluate {{{1000/500}}} to get {{{2}}}.



{{{ln(2)=ln((1.03)^(4*t))}}} Take the natural log of both sides.



{{{ln(2)=4*t*ln(1.03)}}} Pull down the exponent using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}.



{{{ln(2)/ln(1.03)=4*t}}} Divide both sides by {{{ln(1.03)}}}.



{{{0.693147180559945/ln(1.03)=4*t}}} Evaluate the natural log of {{{2}}} to get {{{0.693147180559945}}}.



{{{0.693147180559945/0.0295588022415444=4*t}}} Evaluate the natural log of {{{1.03}}} to get {{{0.0295588022415444}}}.



{{{23.4497722504377=4*t}}} Divide.



{{{23.4497722504377/4=t}}} Divide both sides by 4 to isolate "t".



{{{5.86244306260943=t}}} Divide.



{{{t=5.86244306260943}}} Rearrange the equation.



{{{t=6}}} Round to the nearest whole number (ie to the nearest whole year).



So it will take about 6 years for the investment to double.



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