Solve 500*[1-1/(1+0.055/12)^60/0.085/12]+1000*[1-1/(1+0.055/12)^36/0.055112]

$$500 \times [ \frac { 1 - \frac { 1 } { ( 1 + 0.055 / 12 ) ^ { 60 } } } { 0.085 / 12 } ] + 1000 \times [ \frac { 1 - \frac { 1 } { ( 1 + 0.055 / 12 ) ^ { 36 } } } { 0.055112 } ]$$

Evaluate

$$\frac{11595110332669863673944835929911142441764194394393942321976308929052377138511388632797358977448070998257313362724215591347398050425526024249705483329681915701740712943612188689528523833279015288189576994545400000}{588828479922289429843004132121070520940088708506766975517534355127097258824204535613576128845596298259174295126273522081564293428406436351542154224406651224386339240070560851247787102493004611082367436879289}\approx 19691.830011687$$

Solution Steps

Expand \frac{0.055}{12} by multiplying both numerator and the denominator by 1000.

$$500\times \frac{1-\frac{1}{\left(1+\frac{55}{12000}\right)^{60}}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Reduce the fraction \frac{55}{12000} to lowest terms by extracting and canceling out 5.

$$500\times \frac{1-\frac{1}{\left(1+\frac{11}{2400}\right)^{60}}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Add 1 and \frac{11}{2400} to get \frac{2411}{2400}.

$$500\times \frac{1-\frac{1}{\left(\frac{2411}{2400}\right)^{60}}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Calculate \frac{2411}{2400} to the power of 60 and get \frac{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.

$$500\times \frac{1-\frac{1}{\frac{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Divide 1 by \frac{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} by multiplying 1 by the reciprocal of \frac{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.

$$500\times \frac{1-1\times \frac{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Multiply 1 and \frac{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} to get \frac{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}.

$$500\times \frac{1-\frac{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Subtract \frac{64964261028908956064803498836830371684746041356377185527988338666957694562390245376000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} from 1 to get \frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}.

$$500\times \frac{\frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}}{\frac{0.085}{12}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Expand \frac{0.085}{12} by multiplying both numerator and the denominator by 1000.

$$500\times \frac{\frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}}{\frac{85}{12000}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Reduce the fraction \frac{85}{12000} to lowest terms by extracting and canceling out 5.

$$500\times \frac{\frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}}{\frac{17}{2400}}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Divide \frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} by \frac{17}{2400} by multiplying \frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} by the reciprocal of \frac{17}{2400}.

$$500\times \frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}\times \frac{2400}{17}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Multiply \frac{20509462286853771449059780611721017622858648512510457891598590513924473941631315897563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} and \frac{2400}{17} to get \frac{2895453499320532439867263145184143664403573907648535231755095131377572791759715185538318625852220640082811543578052419421877197477440141091946838852868280536978936379132513410079914663472125781063433167200}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}.

$$500\times \frac{2895453499320532439867263145184143664403573907648535231755095131377572791759715185538318625852220640082811543578052419421877197477440141091946838852868280536978936379132513410079914663472125781063433167200}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Multiply 500 and \frac{2895453499320532439867263145184143664403573907648535231755095131377572791759715185538318625852220640082811543578052419421877197477440141091946838852868280536978936379132513410079914663472125781063433167200}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} to get \frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\left(1+\frac{0.055}{12}\right)^{36}}}{0.055112}$$

Expand \frac{0.055}{12} by multiplying both numerator and the denominator by 1000.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\left(1+\frac{55}{12000}\right)^{36}}}{0.055112}$$

Reduce the fraction \frac{55}{12000} to lowest terms by extracting and canceling out 5.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\left(1+\frac{11}{2400}\right)^{36}}}{0.055112}$$

Add 1 and \frac{11}{2400} to get \frac{2411}{2400}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\left(\frac{2411}{2400}\right)^{36}}}{0.055112}$$

Calculate \frac{2411}{2400} to the power of 36 and get \frac{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{1}{\frac{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}}}{0.055112}$$

Divide 1 by \frac{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000} by multiplying 1 by the reciprocal of \frac{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-1\times \frac{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}}{0.055112}$$

Multiply 1 and \frac{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761} to get \frac{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1-\frac{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}}{0.055112}$$

Subtract \frac{48708493958471199415506599153950129703565945470976000000000000000000000000000000000000000000000000000000000000000000000000}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761} from 1 to get \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{\frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}}{0.055112}$$

Express \frac{\frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761}}{0.055112} as a single fraction.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761\times 0.055112}$$

Multiply 57424810880348845348455904536383566588657615182076285951468335068998659652385616301948641497455056715642094334539527379761 and 0.055112 to get 3164796177237785564844101810809171121834098487914588271357322882322654130762276085632993530207743085712467102965142432953.388232.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{3164796177237785564844101810809171121834098487914588271357322882322654130762276085632993530207743085712467102965142432953.388232}$$

Expand \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761}{3164796177237785564844101810809171121834098487914588271357322882322654130762276085632993530207743085712467102965142432953.388232} by multiplying both numerator and the denominator by 1000000.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761000000}{3164796177237785564844101810809171121834098487914588271357322882322654130762276085632993530207743085712467102965142432953388232}$$

Reduce the fraction \frac{8716316921877645932949305382433436885091669711100285951468335068998659652385616301948641497455056715642094334539527379761000000}{3164796177237785564844101810809171121834098487914588271357322882322654130762276085632993530207743085712467102965142432953388232} to lowest terms by extracting and canceling out 8.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+1000\times \frac{1089539615234705741618663172804179610636458713887535743933541883624832456548202037743580187181882089455261791817440922470125000}{395599522154723195605512726351146390229262310989323533919665360290331766345284510704124191275967885714058387870642804119173529}$$

Multiply 1000 and \frac{1089539615234705741618663172804179610636458713887535743933541883624832456548202037743580187181882089455261791817440922470125000}{395599522154723195605512726351146390229262310989323533919665360290331766345284510704124191275967885714058387870642804119173529} to get \frac{1089539615234705741618663172804179610636458713887535743933541883624832456548202037743580187181882089455261791817440922470125000000}{395599522154723195605512726351146390229262310989323533919665360290331766345284510704124191275967885714058387870642804119173529}.

$$\frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601}+\frac{1089539615234705741618663172804179610636458713887535743933541883624832456548202037743580187181882089455261791817440922470125000000}{395599522154723195605512726351146390229262310989323533919665360290331766345284510704124191275967885714058387870642804119173529}$$

Add \frac{1447726749660266219933631572592071832201786953824267615877547565688786395879857592769159312926110320041405771789026209710938598738720070545973419426434140268489468189566256705039957331736062890531716583600000}{85473723315762727513863279448551389307604689868887643419586929180882168504021561273563090266453229533919915100344537970904963482131867666067956775207816987136934132685521969988066062199594224282532651601} and \frac{1089539615234705741618663172804179610636458713887535743933541883624832456548202037743580187181882089455261791817440922470125000000}{395599522154723195605512726351146390229262310989323533919665360290331766345284510704124191275967885714058387870642804119173529} to get \frac{11595110332669863673944835929911142441764194394393942321976308929052377138511388632797358977448070998257313362724215591347398050425526024249705483329681915701740712943612188689528523833279015288189576994545400000}{588828479922289429843004132121070520940088708506766975517534355127097258824204535613576128845596298259174295126273522081564293428406436351542154224406651224386339240070560851247787102493004611082367436879289}.

$$\frac{11595110332669863673944835929911142441764194394393942321976308929052377138511388632797358977448070998257313362724215591347398050425526024249705483329681915701740712943612188689528523833279015288189576994545400000}{588828479922289429843004132121070520940088708506766975517534355127097258824204535613576128845596298259174295126273522081564293428406436351542154224406651224386339240070560851247787102493004611082367436879289}$$