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Question Number 168189 by mathls last updated on 05/Apr/22

Commented by mathls last updated on 05/Apr/22

please help me!

$${please}\:{help}\:{me}! \\ $$

Commented by mr W last updated on 05/Apr/22

≈534%

$$\approx\mathrm{534\%} \\ $$

Commented by mr W last updated on 06/Apr/22

two years=24 monthes 1+q_(2y) =(1+q_m )^(24) ⇒q_(2y) =(1+q_m )^(24) −1 ⇒q_m =(1+q_(2y) )^(1/(24)) −1

$${two}\:{years}=\mathrm{24}\:{monthes} \\ $$$$\mathrm{1}+{q}_{\mathrm{2}{y}} =\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} \\ $$$$\Rightarrow{q}_{\mathrm{2}{y}} =\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} −\mathrm{1} \\ $$$$\Rightarrow{q}_{{m}} =\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right)^{\frac{\mathrm{1}}{\mathrm{24}}} −\mathrm{1} \\ $$

Commented by mathls last updated on 06/Apr/22

please describe the formula and calculate the last ans by formula.

$${please}\:{describe}\:{the}\:{formula}\:{and}\: \\ $$$${calculate}\:{the}\:{last}\:{ans}\:{by}\:{formula}. \\ $$

Commented by MJS_new last updated on 06/Apr/22

8% → factor 1+(8/(100))=1.08 1.08^(24) =6.3411... =1+((534)/(100)) → 534%

$$\mathrm{8\%}\:\rightarrow\:\mathrm{factor}\:\mathrm{1}+\frac{\mathrm{8}}{\mathrm{100}}=\mathrm{1}.\mathrm{08} \\ $$$$\mathrm{1}.\mathrm{08}^{\mathrm{24}} =\mathrm{6}.\mathrm{3411}...\:=\mathrm{1}+\frac{\mathrm{534}}{\mathrm{100}}\:\rightarrow\:\mathrm{534\%} \\ $$

Commented by mathls last updated on 06/Apr/22

what means 534% ?

$${what}\:{means}\:\mathrm{534\%}\:? \\ $$

Commented by MJS_new last updated on 06/Apr/22

534 percent

$$\mathrm{534}\:\mathrm{percent} \\ $$

Commented by mathls last updated on 07/Apr/22

percentage must 100% but it is 534%?

$${percentage}\:{must}\:\mathrm{100\%}\:{but}\:{it}\:{is}\:\mathrm{534\%}? \\ $$

Commented by MJS_new last updated on 07/Apr/22

percentage can be more than 100% if you need 120$ to buy something but you only have 60$ than you have 100×((60)/(120))% = =50% of the price. but if you have 180$ you have 100×((180)/(120))% = 150% of it. if you put all your money on your bank account you put 100% of your money there. if you get 5% per year (tell me the name of the bank) you have 105% of what you had before after the first year (=100×1.05) if you leave it there another year you have 100×1.05×1.05=110.25% of what you had before. after n years you have 100×1.05^n % of what you had in the beginning. after 48 years you have 1040%

$$\mathrm{percentage}\:\mathrm{can}\:\mathrm{be}\:\mathrm{more}\:\mathrm{than}\:\mathrm{100\%} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{need}\:\mathrm{120\$}\:\mathrm{to}\:\mathrm{buy}\:\mathrm{something}\:\mathrm{but}\:\mathrm{you} \\ $$$$\mathrm{only}\:\mathrm{have}\:\mathrm{60\$}\:\mathrm{than}\:\mathrm{you}\:\mathrm{have}\:\mathrm{100}×\frac{\mathrm{60}}{\mathrm{120}}\%\:= \\ $$$$=\mathrm{50\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{price}.\:\mathrm{but}\:\mathrm{if}\:\mathrm{you}\:\mathrm{have}\:\mathrm{180\$}\:\mathrm{you} \\ $$$$\mathrm{have}\:\mathrm{100}×\frac{\mathrm{180}}{\mathrm{120}}\%\:=\:\mathrm{150\%}\:\mathrm{of}\:\mathrm{it}. \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{put}\:\mathrm{all}\:\mathrm{your}\:\mathrm{money}\:\mathrm{on}\:\mathrm{your}\:\mathrm{bank} \\ $$$$\mathrm{account}\:\mathrm{you}\:\mathrm{put}\:\mathrm{100\%}\:\mathrm{of}\:\mathrm{your}\:\mathrm{money}\:\mathrm{there}. \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{get}\:\mathrm{5\%}\:\mathrm{per}\:\mathrm{year}\:\left(\mathrm{tell}\:\mathrm{me}\:\mathrm{the}\:\mathrm{name}\:\mathrm{of}\right. \\ $$$$\left.\mathrm{the}\:\mathrm{bank}\right)\:\mathrm{you}\:\mathrm{have}\:\mathrm{105\%}\:\mathrm{of}\:\mathrm{what}\:\mathrm{you}\:\mathrm{had} \\ $$$$\mathrm{before}\:\mathrm{after}\:\mathrm{the}\:\mathrm{first}\:\mathrm{year}\:\left(=\mathrm{100}×\mathrm{1}.\mathrm{05}\right) \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{leave}\:\mathrm{it}\:\mathrm{there}\:\mathrm{another}\:\mathrm{year}\:\mathrm{you}\:\mathrm{have} \\ $$$$\mathrm{100}×\mathrm{1}.\mathrm{05}×\mathrm{1}.\mathrm{05}=\mathrm{110}.\mathrm{25\%}\:\mathrm{of}\:\mathrm{what}\:\mathrm{you}\:\mathrm{had} \\ $$$$\mathrm{before}.\:\mathrm{after}\:{n}\:\mathrm{years}\:\mathrm{you}\:\mathrm{have}\:\mathrm{100}×\mathrm{1}.\mathrm{05}^{{n}} \% \\ $$$$\mathrm{of}\:\mathrm{what}\:\mathrm{you}\:\mathrm{had}\:\mathrm{in}\:\mathrm{the}\:\mathrm{beginning}.\:\mathrm{after}\:\mathrm{48} \\ $$$$\mathrm{years}\:\mathrm{you}\:\mathrm{have}\:\mathrm{1040\%} \\ $$

Commented by mathls last updated on 07/Apr/22

thanks

$${thanks} \\ $$

Commented by mathls last updated on 08/Apr/22

please prove me both formula.

$${please}\:{prove}\:{me}\:{both}\:{formula}. \\ $$

Answered by mr W last updated on 08/Apr/22

q_m =rate per month that means if you have money of amount A in this month, next month it becomes q_m A more, i.e. next month you totally have money of amount A+q_m A=(1+q_m )A. similarly after two monthes you′ll have (1+q_m )^2 A, and after n monthes you′ll have (1+q_m )^n A. after two years, i.e. after 24 monthes, you′ll have (1+q_m )^(24) A. q_(2y) =rate per 2 years that means if you have money of amount A in this month, after 2 years it becomes q_(2y) A more, i.e. after 2 years you′ll have totally A+q_(2y) A=(1+q_(2y) )A. we see (1+q_(2y) )A=(1+q_m )^(24) A 1+q_(2y) =(1+q_m )^(24) ⇒q_(2y) =(1+q_m )^(24) −1 ...(I) or 1+q_m =(1+q_(2y) )^(1/(24)) ⇒q_m =(1+q_(2y) )^(1/(24)) −1 ...(II) examples: given: monthly rate q_m =8%=0.08 rate per 2 years: q_(2y) =(1+q_m )^(24) −1 =(1+0.08)^(24) −1 ≈5.34=534% given: rate per 2 years q_(2y) =18%=0.18 monthly rate: q_m =(1+q_(2y) )^(1/(24)) −1 =(1+0.18)^(1/(24)) −1 ≈0.0069=0.69%

$$\boldsymbol{{q}}_{\boldsymbol{{m}}} =\boldsymbol{{rate}}\:\boldsymbol{{per}}\:\boldsymbol{{month}} \\ $$$${that}\:{means}\:{if}\:{you}\:{have}\:{money}\:{of}\: \\ $$$${amount}\:{A}\:{in}\:{this}\:{month},\:{next}\:{month} \\ $$$${it}\:{becomes}\:{q}_{{m}} {A}\:\underline{{more}},\:{i}.{e}.\:{next}\:{month} \\ $$$${you}\:{totally}\:{have}\:{money}\:{of}\:{amount}\: \\ $$$${A}+{q}_{{m}} {A}=\left(\mathrm{1}+{q}_{{m}} \right){A}. \\ $$$${similarly}\:{after}\:{two}\:{monthes}\:{you}'{ll}\: \\ $$$${have}\:\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{2}} {A},\:{and}\:{after}\:{n}\:{monthes} \\ $$$${you}'{ll}\:{have}\:\left(\mathrm{1}+{q}_{{m}} \right)^{{n}} {A}. \\ $$$${after}\:{two}\:{years},\:{i}.{e}.\:{after}\:\mathrm{24}\:{monthes}, \\ $$$${you}'{ll}\:{have}\:\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} {A}. \\ $$$$ \\ $$$$\boldsymbol{{q}}_{\mathrm{2}\boldsymbol{{y}}} =\boldsymbol{{rate}}\:\boldsymbol{{per}}\:\mathrm{2}\:\boldsymbol{{years}} \\ $$$${that}\:{means}\:{if}\:{you}\:{have}\:{money}\:{of}\: \\ $$$${amount}\:{A}\:{in}\:{this}\:{month},\:{after}\:\mathrm{2}\:{years} \\ $$$${it}\:{becomes}\:{q}_{\mathrm{2}{y}} {A}\:{more},\:{i}.{e}.\:{after}\:\mathrm{2}\:{years} \\ $$$${you}'{ll}\:{have}\:{totally}\:{A}+{q}_{\mathrm{2}{y}} {A}=\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right){A}. \\ $$$$ \\ $$$${we}\:{see} \\ $$$$\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right){A}=\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} {A} \\ $$$$\mathrm{1}+\boldsymbol{{q}}_{\mathrm{2}\boldsymbol{{y}}} =\left(\mathrm{1}+\boldsymbol{{q}}_{\boldsymbol{{m}}} \right)^{\mathrm{24}} \:\:\:\: \\ $$$$\Rightarrow{q}_{\mathrm{2}{y}} =\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} −\mathrm{1}\:\:\:\:\:...\left({I}\right) \\ $$$${or} \\ $$$$\mathrm{1}+{q}_{{m}} =\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right)^{\frac{\mathrm{1}}{\mathrm{24}}} \\ $$$$\Rightarrow{q}_{{m}} =\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right)^{\frac{\mathrm{1}}{\mathrm{24}}} −\mathrm{1}\:\:\:...\left({II}\right) \\ $$$$ \\ $$$${examples}: \\ $$$${given}:\:{monthly}\:{rate}\:{q}_{{m}} =\mathrm{8\%}=\mathrm{0}.\mathrm{08} \\ $$$${rate}\:{per}\:\mathrm{2}\:{years}: \\ $$$${q}_{\mathrm{2}{y}} =\left(\mathrm{1}+{q}_{{m}} \right)^{\mathrm{24}} −\mathrm{1} \\ $$$$\:\:\:\:\:\:=\left(\mathrm{1}+\mathrm{0}.\mathrm{08}\right)^{\mathrm{24}} −\mathrm{1} \\ $$$$\:\:\:\:\:\:\approx\mathrm{5}.\mathrm{34}=\mathrm{534\%} \\ $$$$ \\ $$$${given}:\:{rate}\:{per}\:\mathrm{2}\:{years}\:{q}_{\mathrm{2}{y}} =\mathrm{18\%}=\mathrm{0}.\mathrm{18} \\ $$$${monthly}\:{rate}: \\ $$$${q}_{{m}} =\left(\mathrm{1}+{q}_{\mathrm{2}{y}} \right)^{\frac{\mathrm{1}}{\mathrm{24}}} −\mathrm{1} \\ $$$$\:\:\:\:\:=\left(\mathrm{1}+\mathrm{0}.\mathrm{18}\right)^{\frac{\mathrm{1}}{\mathrm{24}}} −\mathrm{1} \\ $$$$\:\:\:\:\:\approx\mathrm{0}.\mathrm{0069}=\mathrm{0}.\mathrm{69\%} \\ $$

Commented by mathls last updated on 08/Apr/22

what a great solution! u are a star of mathematics.

$${what}\:{a}\:{great}\:{solution}!\:{u}\:{are}\:{a}\:{star}\:{of}\: \\ $$$${mathematics}. \\ $$

Commented by peter frank last updated on 08/Apr/22

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

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