1MS.3 MATEMÀTICA FINANCERA/1MS.3.5 Anualitat d'amortització 1MS.3.5.10DT AMORTITZACIÓ  

Anualitat d'amortització
Si tenim un deute amb una entitat financera, el podem saldar amb pagaments periòdics, com anualitats d'amortització, de manera que al final del procés la capitalització d'aquestes anualitats sigui igual a l'evolució del deute sotmès al mateix interès.

Així doncs, la quantitat prestada, D es transforma al cap de t anys en D (1+r)t .
Per altra banda, les anualitats aportades es transformen en
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»A«/mi»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mi mathvariant=¨bold¨»r«/mi»«/mfrac»«/mstyle»«/math»
Igualant, es pot calcular A:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»D«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»r«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mstyle»«/math»

En cas d'aportacions amb periodicitat inferior a l'any, aquesta expressió s'escriu:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»D«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mfenced»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»ft«/mi»«/msup»«/mrow»«mrow»«msup»«mfenced»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»ft«/mi»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mstyle»«/math»
]]> 0.0000000 0.0000000 0 1MS.3.5.11Q Aïlla t Aïlla t en l'expressió 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨ mathsize=¨12px¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨ mathsize=¨12px¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨ mathsize=¨12px¨»D«/mi»«mo mathvariant=¨bold¨ mathsize=¨12px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«mo mathvariant=¨bold¨ mathsize=¨12px¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathsize=¨12px¨»(«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«mo mathvariant=¨bold¨ mathsize=¨12px¨»+«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»)«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»t«/mi»«/msup»«/mrow»«mrow»«mo mathvariant=¨bold¨ mathsize=¨12px¨»(«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«mo mathvariant=¨bold¨ mathsize=¨12px¨»+«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»)«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»-«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«/mrow»«/mfrac»«/math»
 

Resposta: t = loga b 

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>49</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><mfrac><mi>i1</mi><mn>100</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D1</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>80</mn><mo>,</mo><mn>250</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>30</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfrac><mrow><mi>D1</mi><mo>*</mo><mi>s1</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>s1</mi><msup><mo>)</mo><mi>t1</mi></msup></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>s1</mi></mrow></mfenced><mi>t1</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>sol1</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>t</mi><mo>=</mo><msub><mi>log</mi><mrow><mn>1</mn><mo>+</mo><mi>r</mi></mrow></msub><mfrac><mi>A</mi><mrow><mi>A</mi><mo>-</mo><mi>D</mi><mo>·</mo><mi>r</mi></mrow></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mfenced><mfrac><mi>A</mi><mrow><mi>A</mi><mo>-</mo><mi>D</mi><mo>*</mo><mi>r</mi></mrow></mfrac></mfenced></mrow><mrow><mi>ln</mi><mfenced><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MS.3.5.12Q CàlculA (anual) Quant haurem de pagar cada any, si hem comprat un immoble de #D €, la taxa d'interès és del #i % i el paguem durant #t anys?

Resposta arrodonida als cèntims.

]]> 1.0000000 0.5000000 0 0 #A <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>49</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mi>i</mi><mn>100</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>80</mn><mo>,</mo><mn>250</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>30</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>D</mi><mo>*</mo><mi>s</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>s</mi><msup><mo>)</mo><mi>t</mi></msup></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>s</mi></mrow></mfenced><mi>t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>a</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#A</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Has d'aplicar la fórmula per calcular l'anualitat A

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨12px¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨12px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨ mathsize=¨12px¨»D«/mi»«mo mathvariant=¨bold¨ mathsize=¨12px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«mo mathvariant=¨bold¨ mathsize=¨12px¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathsize=¨12px¨»(«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«mo mathvariant=¨bold¨ mathsize=¨12px¨»+«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»)«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»t«/mi»«/msup»«/mrow»«mrow»«mo mathvariant=¨bold¨ mathsize=¨12px¨»(«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«mo mathvariant=¨bold¨ mathsize=¨12px¨»+«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»r«/mi»«msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»)«/mo»«mi mathvariant=¨bold¨ mathsize=¨12px¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathsize=¨12px¨»-«/mo»«mn mathvariant=¨bold¨ mathsize=¨12px¨»1«/mn»«/mrow»«/mfrac»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=ea87400d39a2b9e30dece985154348b1&cw=111&ch=43&cb=25" width="111" height="43" style="vertical-align: -18px;"/>

]]> 1MS.3.5.21Q CàlculA (crèdit mensual) Una moto val #P_1 €. Donem una entrada de #P_2 € i fem un crèdit pel que falta a tornar en #t_1 anys amb un interès de #i %. Quin és l'import de la mensualitat?

Resposta arrodonida als cèntims.

]]> 1.0000000 0.5000000 0 0 #m <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>39</mn><mo>,</mo><mn>82</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mi>i</mi><mn>1200</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_1</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>8</mn><mo>,</mo><mn>20</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_2</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>P_1</mi><mo>-</mo><mi>P_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>12</mn><mo>*</mo><mi>t_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>D</mi><mo>*</mo><mi>s</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>s</mi><msup><mo>)</mo><mi>t</mi></msup></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>s</mi></mrow></mfenced><mi>t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>a</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#m</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer calcula quan falta per pagar. 

Aplica la fórmula per calcular l'anualitat però com que el pagament és mensual, cal substituir r per r/12 i el temps t per 12·t.

]]> 1MS.3.5.31Q CàlculA (hipoteca mensual) Volem comprar un pis que val #P_1 € però només disposem de #P_2 €. Pel pagar els diners que ens falten, el banc ens concedeix un préstec hipotecari a tornar en #t_1 anys amb un interès de #i %. Quin és l'import de la mensualitat?

Resposta arrodonida als cèntims.

]]> 1.0000000 0.5000000 0 0 #m <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>49</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mi>i</mi><mn>1200</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_1</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>80</mn><mo>,</mo><mn>250</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_2</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>30</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>P_1</mi><mo>-</mo><mi>P_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>30</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>12</mn><mo>*</mo><mi>t_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>D</mi><mo>*</mo><mi>s</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>s</mi><msup><mo>)</mo><mi>t</mi></msup></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>s</mi></mrow></mfenced><mi>t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>a</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#m</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer calcula el deute, o sigui quants diners et falten per comprar el pis.

Aplica la fórmula per calcular l'anualitat però com que el pagament és mensual, i es vol la mensualitat cal substituir r per r/12 i el temps t per 12·t.

]]> 1MS.3.5.41Q CàlculTemps (mensual) Quants mesos trigarem per a tornar un crèdit de #D € amb un interès de #i % si cada mes paguem #m €?

Resposta arrodonida a la unitat.

]]> 1.0000000 0.5000000 0 0 #t <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>39</mn><mo>,</mo><mn>82</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mi>i</mi><mn>1200</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_1</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>8</mn><mo>,</mo><mn>20</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P_2</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>P_1</mi><mo>-</mo><mi>P_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>12</mn><mo>*</mo><mi>t_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>D</mi><mo>*</mo><mi>s</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>s</mi><msup><mo>)</mo><mi>t</mi></msup></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>s</mi></mrow></mfenced><mi>t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><mi>a</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi>m</mi><mrow><mi>m</mi><mo>-</mo><mi>D</mi><mo>*</mo><mi>s</mi></mrow></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>304.71</mn><mo>,</mo><mn>5.9</mn><mo>,</mo><mn>48</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13000</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.2654</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>48</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#t</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Cal aïllar el temps a partir de la fórmula, emprant logaritmes.
Si m és la mensualitat, r l'interès mensual i t el temps en mesos
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»D«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»r«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»D«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»D«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«msup mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»t«/mi»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»m«/mi»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»D«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»r«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Un cop trobat (1+i)t= #k cal emprar logaritmes per a calcular t:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»t«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»log«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»k«/mi»«/mrow»«mrow»«mi mathvariant=¨bold¨»log«/mi»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
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