1MS.3 MATEMÀTICA FINANCERA/1MS.3.3 EquivalènciaTaxes d'interès 1MS.3.3.10DT EQUIVALÈNCIA DE TAXES

Equivalència de taxes d'interès

Dues taxes d'interès són equivalents si, i només si, produeixen els mateixos interessos, en un temps igual, amb un capital inicial idèntic.

Interès simple

Si f és la freqüència anual amb què es paguen els interessos, l'interès equivalent pel període f és de:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

Interès compost

Si f és la freqüència anual amb què es paguen els interessos, l'interès equivalent pel període f és de:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mroot mathcolor=¨#003300¨»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»r«/mi»«/mrow»«mi mathvariant=¨bold¨»f«/mi»«/mroot»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/math»

]]> 0.0000000 0.0000000 0 1MS.3.3.11Q TaxaEquivalent Quina és la taxa d'interès #f equivalent a una taxa d'interès anual del #i1 %?

a) a interès simple?
b) a interès compost?

Format de la resposta: tant per cent arrodonit als centèsims.

]]> 1.0000000 0.5000000 0 0 a) =#i_1b) =#i_2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>bimestral</mi><mo>,</mo><mo> </mo><mi>trimestral</mi><mo>,</mo><mi>quadrimestral</mi><mo>,</mo><mi>semestral</mi></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>39</mn><mo>)</mo><mo>*</mo><mn>0</mn><mo>.</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</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>C_0</mi><mo>=</mo><mn>1000</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>bimestral</mi></mrow><mrow><mi>f_1</mi><mo>=</mo><mn>6</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>trimestral</mi></mrow><mrow><mi>f_1</mi><mo>=</mo><mn>4</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>quadrimestral</mi></mrow><mrow><mi>f_1</mi><mo>=</mo><mn>3</mn></mrow><mrow><mi>f_1</mi><mo>=</mo><mn>2</mn></mrow></apply></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>=</mo><mn>100</mn><mo>*</mo><mfrac><mi>r</mi><mi>f_1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>=</mo><mn>100</mn><mo>*</mo><mfenced><mrow><mroot><mrow><mn>1</mn><mo>+</mo><mi>r</mi></mrow><mi>f_1</mi></mroot><mo>-</mo><mn>1</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mi>r1</mi></mrow></mfenced></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>i_2</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mi>r2</mi></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>bimestral</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>,</mo><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.43333</mn><mo>,</mo><mn>0.43</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>,</mo><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.42871</mn><mo>,</mo><mn>0.43</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">i</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">i</mi><mi>_</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="float_format">mr</option><option name="tolerance">10^(-5)</option><option name="relative_tolerance">false</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="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MS.3.3.21 ComparaTaxesEquivalents Dues entitats bancàries ens ofereixen el següent per un dipòsit de #C_0 €:

l'entitat A paga els interessos #f a un interès compost anual del #i_1 %
l'entitat B paga els interessos #g a un interès compost anual del #i_2 %

a) Quant ens paga d'interessos l'entitat A cada #f_2?

b) Quant ens paga d'interessos l'entitat B cada #g_2?

c) Quina és la que ens dona més interessos al cap de l'any?

Format de la resposta: arrodonida als centims.

]]> 1.0000000 0.5000000 0 0 a) =#M1Ab) =#M1Bc) =#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>C_0</mi><mo>=</mo><mn>10000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd/></mtr><mtr><mtd><mi>d_11</mi><mo>=</mo><mi>bimestralment</mi></mtd></mtr><mtr><mtd><mi>d_12</mi><mo>=</mo><mi>trimestralment</mi></mtd></mtr><mtr><mtd><mi>d_13</mi><mo>=</mo><mi>quadrimestralment</mi></mtd></mtr><mtr><mtd><mi>d_14</mi><mo>=</mo><mi>semestralment</mi></mtd></mtr><mtr><mtd><mi>f</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>d_11</mi><mo>,</mo><mi>d_12</mi><mo>,</mo><mi>d_13</mi><mo>,</mo><mi>d_14</mi></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>g</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>d_11</mi><mo>,</mo><mi>d_12</mi><mo>,</mo><mi>d_13</mi><mo>,</mo><mi>d_14</mi></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>i_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>39</mn><mo>)</mo><mo>*</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>j_1</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>i_2</mi><mo>=</mo><mi>i_1</mi><mo>+</mo><mi>j_1</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mfenced><mrow><mi>f</mi><mo>≠</mo><mi>g</mi></mrow></mfenced></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>bimestralment</mi></mrow><mtable><mtr><mtd><mi>f_1</mi><mo>=</mo><mn>6</mn></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>=</mo><mi>bimestre</mi></mtd></mtr><mtr><mtd><mi>i11</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>600</mn></mfrac></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>trimestralment</mi></mrow><mtable><mtr><mtd><mi>f_1</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>=</mo><mi>trimestre</mi></mtd></mtr><mtr><mtd><mi>i11</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>400</mn></mfrac></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f</mi><mo>=</mo><mi>quadrimestralment</mi></mrow><mtable><mtr><mtd><mi>f_1</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>=</mo><mi>quadrimestre</mi></mtd></mtr><mtr><mtd><mi>i11</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>300</mn></mfrac></mtd></mtr></mtable><mtable><mtr><mtd><mi>f_1</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>=</mo><mi>semestre</mi></mtd></mtr><mtr><mtd><mi>i11</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>200</mn></mfrac></mtd></mtr></mtable></apply></apply></apply></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>N1</mi><mo>></mo><mi>N2</mi></mrow><mrow><mi>sol</mi><mo>=</mo><mi>A</mi></mrow><mrow><mi>sol</mi><mo>=</mo><mi>B</mi></mrow></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C0</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>,</mo><mi>i_1</mi><mo>,</mo><mi>i11</mi><mo>,</mo><mi>M1A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>bimestralment</mi><mo>,</mo><mn>1.7</mn><mo>,</mo><mn>0.0028333</mn><mo>,</mo><mn>170.</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>,</mo><mi>i_2</mi><mo>,</mo><mi>i12</mi><mo>,</mo><mi>M1B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>semestralment</mi><mo>,</mo><mn>2.1</mn><mo>,</mo><mn>0.0105</mn><mo>,</mo><mn>630.</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N1</mi><mo>,</mo><mi>N2</mi><mo>,</mo><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60010.</mn><mo>,</mo><mn>60013.</mn><mo>,</mo><mi>B</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>M</mi><mn>1</mn><mi>A</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>M</mi><mn>1</mn><mi>B</mi><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">10</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="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> a) Si es fa el pagament #f, cal dividir la taxa d'interès per #f_1 i és #i11. Atenció, no es demana el capital final, sinó l'interès generat.

b) Si es fa el pagament #g, cal dividir la taxa d'interès per #g1i és #i12. Atenció, no es demana el capital final, sinó l'interès generat.

c) Amb l'expressió: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«mrow»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msub»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«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¨»f«/mi»«/msup»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=92a1861a115e6ef0ec8c411f8764c7b2&cw=102&ch=31&cb=20" style="vertical-align: -11px;"/>" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=92a1861a115e6ef0ec8c411f8764c7b2&cw=102&ch=31&cb=20" style="vertical-align: -11px;"/> es pot calcular el capital final anual i comparar.

]]> 1MS.3.3.30DT InteresCapitalitzacioInferiorAny

Taxes d'interès anual
amb període de capitalització inferior a l'any

Si invertim un capital C₀ a una taxa d'interès anual r amb una freqüència de capitalització f, durant t anys, el capital final és de:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»C«/mi»«mo»=«/mo»«msub»«mi mathvariant=¨bold¨»C«/mi»«mn»0«/mn»«/msub»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mi mathvariant=¨bold¨»t«/mi»«/mrow»«/msup»«/math»

]]> 0.0000000 0.0000000 0 1MS.3.3.31Q CPeriodeInfAny Si invertim un capital #C_0 a una taxa d'interès anual del #i_1 %, quina quantitat s'obté al cap de #t anys si el període de conversió és:

anual?
semestral?
trimestral?

Resposta arrodonida als cèntims.

]]> 1.0000000 0.5000000 0 0 a) =#a_1b) =#s_1c) =#t_1]]> <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>C_0</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</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>i</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>100</mn></mfrac></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>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><msup><mo>)</mo><mi>t</mi></msup></mrow></mfenced></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>s_1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><msup><mo>)</mo><mrow><mn>2</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></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>t_1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>4</mn></mfrac><msup><mo>)</mo><mrow><mn>4</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>a</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>t</mi><mi>_</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</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="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Només cal calcular la taxa d'interès que correspon al període i aplicar-la al nombre total de períodes.
Si el període és anual, l'interès és #i % i el nombre total de períodes #t.
Si el període és semestral, l'interès és de #i %/2 i el nombre total de períodes és de 2 · #t.
Si el període és trimestral, l'interès és de de #i %/4 i el nombre total de períodes és de 4 · #t.]]> 1MS.3.3.35Q iPeriòdeInferiorAny Si invertim un capital #C_0 a una taxa d'interès anual del #i_1 %, quants interessos s'obtenen al cap de #t anys si el període de conversió és:

a) #f_1?
b) #f_2?

Resposta arrodonida als cèntims.

]]> 1.0000000 0.5000000 0 0 a) =#r_1b) =#r_2]]> <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>C_0</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</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>i</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>100</mn></mfrac></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>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>semestral</mi><mo>,</mo><mo> </mo><mi>trimestral</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>quadrimestral</mi><mo>,</mo><mo> </mo><mi>bimestral</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_1</mi><mo>=</mo><mi>semestral</mi></mrow><mrow><mi>r_1</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><msup><mo>)</mo><mrow><mn>2</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mi>C_0</mi></mrow><mrow><mi>r_1</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>4</mn></mfrac><msup><mo>)</mo><mrow><mn>4</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mi>C_0</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_2</mi><mo>=</mo><mi>quadrimestral</mi></mrow><mrow><mi>r_2</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>3</mn></mfrac><msup><mo>)</mo><mrow><mn>3</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mi>C_0</mi></mrow><mrow><mi>r_2</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>6</mn></mfrac><msup><mo>)</mo><mrow><mn>6</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mi>C_0</mi></mrow></apply></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>r</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>r</mi><mi>_</mi><mn>2</mn></math>]]></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="inputCompound">true</data><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Només cal calcular la taxa d'interès que correspon al període i aplicar-la al nombre total de períodes per trobar el Capital Final. Els interessos seran la diferència entre el Capital Inicial i el Capital Final.
Si el període és semestral, l'interès és de #i %/2 i el nombre total de períodes és de 2 · #t.
Si el període és quadrimestral, l'interès és de de #i %/3 i el nombre total de períodes és de 3 · #t.
Si el període és semestral, l'interès és de #i %/2 i el nombre total de períodes és de 2 · #t.
Si el període és bimestral, l'interès és de de #i %/64 i el nombre total de períodes és de 6 · #t.]]> 1MS.3.3.41Q C0PeríodeInfAny Quin capital hem invertit a una taxa d'interès anual del #i_1 %, si obtenim #C € al cap de #t anys si el període de conversió és #f_1?

Resposta arrodonida al miller d'euros.

]]> 1.0000000 0.5000000 0 0 #C_0]]> <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>C_0</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</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>i</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>100</mn></mfrac></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>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>semestral</mi><mo>,</mo><mo> </mo><mi>trimestral</mi><mo>,</mo><mi>quadrimestral</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_1</mi><mo>=</mo><mi>semestral</mi></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><msup><mo>)</mo><mrow><mn>2</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow><mtable><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_1</mi><mo>=</mo><mi>trimestral</mi></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>4</mn></mfrac><msup><mo>)</mo><mrow><mn>4</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>3</mn></mfrac><msup><mo>)</mo><mrow><mn>3</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow></apply></mtd></mtr><mtr><mtd/></mtr></mtable></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>C</mi><mi>_</mi><mn>0</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">10</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> Només cal aïllar el Capital Inicial a partir de la fórmula, amb les dades del problema:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«msub»«mi mathvariant=¨bold¨»C«/mi»«mi mathvariant=¨bold¨»t«/mi»«/msub»«msup»«mfenced»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»ft«/mi»«/msup»«/mfrac»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=fc5dc23f256a6193748de88fffa83752&cw=90&ch=39&cb=18" width="90" height="39" style="vertical-align: -21px;"/>
]]> 1MS.3.3.45Q tPeríodeInfAny Durant quant de temps hem invertit #C_0 € a una taxa d'interès anual del #i_1 %, si obtenim #C € si el període de conversió és #f_1?

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>C_0</mi><mo>=</mo><mn>1000</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</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>i</mi><mo>=</mo><mfrac><mi>i_1</mi><mn>100</mn></mfrac></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>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>semestral</mi><mo>,</mo><mo> </mo><mi>trimestral</mi><mo>,</mo><mi>quadrimestral</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_1</mi><mo>=</mo><mi>semestral</mi></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><msup><mo>)</mo><mrow><mn>2</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow><mtable><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>f_1</mi><mo>=</mo><mi>trimestral</mi></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>4</mn></mfrac><msup><mo>)</mo><mrow><mn>4</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow><mrow><mi>C</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>100</mn><mo>*</mo><mfenced><mrow><mi>C_0</mi><mo>*</mo><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>i</mi><mn>3</mn></mfrac><msup><mo>)</mo><mrow><mn>3</mn><mo>*</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></mrow></mfenced></mrow></apply></mtd></mtr><mtr><mtd/></mtr></mtable></apply></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer>#t</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">8</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> Per aïllar el temps, recordeu que cal emprar logaritmes:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»log«/mi»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/msub»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»ft«/mi»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»log«/mi»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/msub»«mfrac mathcolor=¨#0000FF¨»«msub»«mi mathvariant=¨bold¨»C«/mi»«mi mathvariant=¨bold¨»t«/mi»«/msub»«msub»«mi mathvariant=¨bold¨»C«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»ft«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»log«/mi»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/msub»«mfrac mathcolor=¨#0000FF¨»«msub»«mi mathvariant=¨bold¨»C«/mi»«mi mathvariant=¨bold¨»t«/mi»«/msub»«msub»«mi mathvariant=¨bold¨»C«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»ft«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»log«/mi»«mfrac»«msub»«mi mathvariant=¨bold¨»C«/mi»«mi mathvariant=¨bold¨»t«/mi»«/msub»«msub»«mi mathvariant=¨bold¨»C«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«/mfrac»«/mrow»«mrow»«mi mathvariant=¨bold¨»log«/mi»«mfenced»«mrow»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mi mathvariant=¨bold¨»i«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=c988d45be59b2d8e93822b176b659b0b&cw=201&ch=132&cb=65" width="201" height="132" style="vertical-align: -67px;"/>" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=c988d45be59b2d8e93822b176b659b0b&cw=201&ch=132&cb=65" width="201" height="132" style="vertical-align: -67px;"/>
]]> 1MS.3.3.60 DT TAE

Taxa anual equivalent TAE

Si el període de capitalització és inferior a l'any, la taxa d'interès anual real augmenta i es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»TAE«/mi»«mo»=«/mo»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mfrac»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»f«/mi»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨bold¨»f«/mi»«/msup»«mo»-«/mo»«mn»1«/mn»«/math»

]]> 0.0000000 0.0000000 0 1MS.3.3.61Q Nominal → TAE Calculeu les TAE que corresponen a:

a) un #i_1 % anual i un període de conversió mensual
b) un #i_2 % anual i un període de conversió quadrimestral
c) un #i_3 % anual i un període de conversió trimestral

Resposta: tant per cent arrodonit als centèsims i sense el símbol %.

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]]> 1MS.3.3.65Q TAE→Nominal Calcula les taxes nominals que corresponen a:

a) una TAE de #j_1 % i un període de conversió mensual
b) una TAE de #j_2 %i un període de conversió quadrimestral
c) una TAE de #j_3 % i un període de conversió trimestral

Resposta: tant per cent arrodonit als centèsims sense el símbol %

]]> 1.0000000 1.0000000 0 0 a) =#i_1b) =#i_2c) =#i_3]]> <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_1</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>01</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>250</mn><mo>,</mo><mn>490</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>01</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>250</mn><mo>,</mo><mn>490</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_3</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>01</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>250</mn><mo>,</mo><mn>490</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi><mo>=</mo><mfenced><mfrac><mi>i_1</mi><mn>100</mn></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi><mo>=</mo><mfenced><mfrac><mi>i_2</mi><mn>100</mn></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_3</mi><mo>=</mo><mfenced><mfrac><mi>i_3</mi><mn>100</mn></mfrac></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t_1</mi><mn>12</mn></mfrac></mrow></mfenced><mn>12</mn></msup><mo>-</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t_2</mi><mn>3</mn></mfrac></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_3</mi><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t_3</mi><mn>4</mn></mfrac></mrow></mfenced><mn>4</mn></msup><mo>-</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j_1</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>10000</mn><mo>*</mo><mi>r_1</mi></mrow></mfenced></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>j_2</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>10000</mn><mo>*</mo><mi>r_2</mi></mrow></mfenced></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>j_3</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mfenced><mrow><mn>10000</mn><mo>*</mo><mi>r_3</mi></mrow></mfenced></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">i</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">i</mi><mi>_</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">i</mi><mi>_</mi><mn>3</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</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"/><data name="inputCompound">true</data></localData></question> Aïlla i en la fórmula de la TAE

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