Question 601674
Part 1)

If $7000 is invested at 8% compounded monthly, how much will this amount be at the end of 15 years?




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



{{{A=7000(1+0.08/12)^(12*15)}}} Plug in {{{P=7000}}}, {{{r=0.08}}} (the decimal equivalent of 8%), {{{n=12}}} (since we're compounding 12 times a year) and {{{t=15}}}.



{{{A=7000(1+0.00666666666666667)^(12*15)}}} Evaluate {{{0.08/12}}} to get {{{0.00666666666666667}}}



{{{A=7000(1.00666666666667)^(12*15)}}} Add {{{1}}} to {{{0.00666666666666667}}} to get {{{1.00666666666667}}}



{{{A=7000(1.00666666666667)^(180)}}} Multiply {{{12}}} and {{{15}}} to get {{{180}}}.



{{{A=7000(3.30692147741001)}}} Evaluate {{{(1.00666666666667)^(180)}}} to get {{{3.30692147741001}}}.



{{{A=23148.45034187}}} Multiply {{{7000}}} and {{{3.30692147741001}}} to get {{{23148.45034187}}}.



{{{A=23148.45}}} Round to the nearest hundredth (ie to the nearest penny).



So at the end of 15 years, there will be <font color="red">$23,148.45</font> in the account.


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Part 2)


How long will it be until it is worth $12000? 





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



{{{12000=7000(1+0.08/12)^(12*t)}}} Plug in {{{A=12000}}}, {{{P=7000}}}, {{{r=0.08}}} (the decimal equivalent of 8%), and {{{n=12}}}.



{{{12000=7000(1+0.00666666666666667)^(12*t)}}} Evaluate {{{0.08/12}}} to get {{{0.00666666666666667}}}



{{{12000=7000(1.00666666666667)^(12*t)}}} Add {{{1}}} to {{{0.00666666666666667}}} to get {{{1.00666666666667}}}



{{{12000/7000=(1.00666666666667)^(12*t)}}} Divide both sides by {{{7000}}}.



{{{1.71428571428571=(1.00666666666667)^(12*t)}}} Evaluate {{{12000/7000}}} to get {{{1.71428571428571}}}.



{{{ln(1.71428571428571)=ln((1.00666666666667)^(12*t))}}} Take the natural log of both sides.



{{{ln(1.71428571428571)=12*t*ln(1.00666666666667)}}} Pull down the exponent using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}.



{{{ln(1.71428571428571)/ln(1.00666666666667)=12*t}}} Divide both sides by {{{ln(1.00666666666667)}}}.



{{{0.538996500732687/ln(1.00666666666667)=12*t}}} Evaluate the natural log of {{{1.71428571428571}}} to get {{{0.538996500732687}}}.



{{{0.538996500732687/0.00664454271866851=12*t}}} Evaluate the natural log of {{{1.00666666666667}}} to get {{{0.00664454271866851}}}.



{{{81.118674911717=12*t}}} Divide.



{{{81.118674911717/12=t}}} Divide both sides by 12 to isolate "t".



{{{6.75988957597641=t}}} Divide.



{{{t=6.75988957597641}}} Rearrange the equation.



{{{t=7}}} Round to the nearest whole number (ie to the nearest whole year).



So it will take about <font color="red">7</font> years for the account to be worth $12,000