Question Number 165404 by MWSuSon last updated on 01/Feb/22
$$\mathrm{given}\:\mathrm{a}\:\mathrm{loan}\:\mathrm{from}\:\mathrm{a}\:\mathrm{bank}\:\mathrm{on}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{interest}\:\mathrm{rate}\:\mathrm{5\%} \\ $$$$\mathrm{suppose}\:\mathrm{i}\:\mathrm{want}\:\mathrm{to}\:\mathrm{make}\:\mathrm{monthly}\:\mathrm{payments}\:\mathrm{for}\:\mathrm{14months} \\ $$$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{amount}\:\mathrm{i}\:\mathrm{should}\:\mathrm{pay}\:\mathrm{monthly}\:\mathrm{and} \\ $$$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{know}\:\mathrm{how}\:\mathrm{much}\:\mathrm{i}\:\mathrm{will}\:\mathrm{pay}\:\mathrm{back}\:\mathrm{to}\:\mathrm{the}\:\mathrm{bank}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{14months}? \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{explanations}. \\ $$
Commented by mr W last updated on 01/Feb/22
$${not}\:{only}\:{the}\:{interest}\:{rate}\:{but}\:{also}\:{the}\: \\ $$$${monthly}\:{payment}\:{must}\:{be}\:{arranged} \\ $$$${with}\:{the}\:{bank}. \\ $$
Commented by MWSuSon last updated on 01/Feb/22
$$\mathrm{So}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{way}\:\mathrm{to}\:\mathrm{know}\:\mathrm{how}\:\mathrm{much}\:\mathrm{money}\:\mathrm{you}'\mathrm{ll}\:\mathrm{pay} \\ $$$$\mathrm{monthly}\:\mathrm{from}\:\mathrm{the}\:\mathrm{interest}\:\mathrm{rate},\:\mathrm{loan}\:\mathrm{money}\:\mathrm{and}\:\mathrm{Time}\:\mathrm{to} \\ $$$$\mathrm{complete}\:\mathrm{payment}? \\ $$
Commented by mr W last updated on 01/Feb/22
$${if}\:{the}\:{time}\:{to}\:{pay}\:{back}\:{the}\:{complete}\: \\ $$$${money}\:{incl}.\:{interest}\:{is}\:{arranged}, \\ $$$${then}\:{the}\:{monthly}\:{payment}\:\left({usually}\right. \\ $$$$\left.{a}\:{fixed}\:{amount}\right)\:{is}\:{also}\:{determined}. \\ $$
Commented by mr W last updated on 01/Feb/22
$${A}={your}\:{loan} \\ $$$${p}={annual}\:{interest}\:{rate} \\ $$$$\frac{{p}}{\mathrm{12}}={p}'={monthly}\:{interest}\:{rate} \\ $$$${let}\:{q}=\mathrm{1}+\frac{{p}}{\mathrm{12}}=\mathrm{1}+{p}' \\ $$$${M}={monthly}\:{payment}\:\left({fixed}\:{amount}\right) \\ $$$$ \\ $$$${after}\:\mathrm{1}\:{month}: \\ $$$${interest}:\:{A}×\frac{{p}}{\mathrm{12}} \\ $$$${you}\:{owed}\:{the}\:{bank}:\:{A}+{A}×\frac{{p}}{\mathrm{12}}={qA} \\ $$$${your}\:{payment}:\:{M} \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}:\:{qA}−{M} \\ $$$$ \\ $$$${after}\:\mathrm{2}\:{monthes}: \\ $$$${interest}:\:\left({qA}−{M}\right)×\frac{{p}}{\mathrm{12}} \\ $$$${you}\:{owed}\:{the}\:{bank}:\:\left({qA}−{M}\right){q} \\ $$$${your}\:{payment}:\:{M} \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}:\:\left({qA}−{M}\right){q}−{M} \\ $$$${you}\:{can}\:{countinue}\:{further}. \\ $$$$ \\ $$$${generally}\:{after}\:{n}\:{monthes}: \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}: \\ $$$${Aq}^{{n}} −{M}\left(\mathrm{1}+{q}+{q}^{\mathrm{2}} +..+{q}^{{n}−\mathrm{1}} \right) \\ $$$$={Aq}^{{n}} −\frac{{M}\left({q}^{{n}} −\mathrm{1}\right)}{{q}−\mathrm{1}} \\ $$$$ \\ $$$${such}\:{that}\:{you}\:{don}'{t}\:{owe}\:{the}\:{bank}\:{any} \\ $$$${money}\:{after}\:{n}\:{monthes}, \\ $$$${Aq}^{{n}} −\frac{{M}\left({q}^{{n}} −\mathrm{1}\right)}{{q}−\mathrm{1}}=\mathrm{0} \\ $$$$\Rightarrow{M}=\frac{\left({q}−\mathrm{1}\right){q}^{{n}} {A}}{{q}^{{n}} −\mathrm{1}} \\ $$$$ \\ $$$${example}:\: \\ $$$${you}\:{loan}\:\mathrm{1000\$}\:{from}\:{the}\:{bank}\:{with} \\ $$$${an}\:{annual}\:{interest}\:{rate}\:\mathrm{6\%}. \\ $$$${you}\:{must}\:{pay}\:{back}\:{completely}\:{in} \\ $$$$\mathrm{20}\:{monthes}. \\ $$$${A}=\mathrm{1000\$} \\ $$$${p}=\mathrm{6\%} \\ $$$${q}=\mathrm{1}+\frac{{p}}{\mathrm{12}}=\mathrm{1}.\mathrm{005} \\ $$$${the}\:{monthly}\:{payment}\:{is}\:{then} \\ $$$${M}=\frac{\mathrm{0}.\mathrm{005}×\mathrm{1}.\mathrm{005}^{\mathrm{20}} ×\mathrm{1000}}{\mathrm{1}.\mathrm{005}^{\mathrm{20}} −\mathrm{1}}\:\approx\mathrm{52}.\mathrm{67\$} \\ $$$$ \\ $$$${you}\:{payed}\:{the}\:{bank}\:{totally}\: \\ $$$$\mathrm{20}×\mathrm{52}.\mathrm{67}=\mathrm{1053}.\mathrm{40\$} \\ $$$${for}\:{a}\:{loan}\:{of}\:\mathrm{1000\$}. \\ $$
Commented by mr W last updated on 01/Feb/22
$${M}=\frac{\left({q}−\mathrm{1}\right){q}^{{n}} {A}}{{q}^{{n}} −\mathrm{1}}=\frac{{p}'\left(\mathrm{1}+{p}'\right)^{{n}} {A}}{\left(\mathrm{1}+{p}'\right)^{{n}} −\mathrm{1}} \\ $$$$\approx\frac{{p}'\left(\mathrm{1}+{np}'\right){A}}{{np}'}=\left(\frac{\mathrm{1}}{{n}}+{p}'\right){A} \\ $$
Commented by MWSuSon last updated on 03/Feb/22
$$\mathrm{Sorry}\:\mathrm{sir}\:\mathrm{that}\:\mathrm{it}\:\mathrm{took}\:\mathrm{me}\:\mathrm{this}\:\mathrm{long}\:\mathrm{to}\:\mathrm{reply},\:\mathrm{i}\:\mathrm{do}\:\mathrm{not}\:\mathrm{know}\:\mathrm{why} \\ $$$$\mathrm{i}\:\mathrm{do}\:\mathrm{not}\:\mathrm{recieve}\:\mathrm{notifications}. \\ $$$$\mathrm{I}\:\mathrm{just}\:\mathrm{have}\:\mathrm{one}\:\mathrm{question}\:\mathrm{sir},\:\mathrm{what}\:\mathrm{does}\:\mathrm{q}\:\mathrm{stand}\:\mathrm{for}\:\mathrm{and}\:\mathrm{why} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{1}+\frac{\mathrm{p}}{\mathrm{12}}? \\ $$
Commented by mr W last updated on 03/Feb/22
$${all}\:{is}\:{explained}: \\ $$$${p}={annual}\:{interest}\:{rate} \\ $$$$\frac{{p}}{\mathrm{12}}={monthly}\:{interest}\:{rate} \\ $$$${let}\:{q}=\mathrm{1}+\frac{{p}}{\mathrm{12}},\: \\ $$$${that}\:{means}\:{q}\:{stands}\:{for}\:\mathrm{1}+\frac{{p}}{\mathrm{12}}. \\ $$$${the}\:{meaning}\:{of}\:{q}\:{is}\:{clear}.\:{it}\:{is}\:{the}\: \\ $$$${common}\:{ratio}\:{of}\:{the}\:{geometric} \\ $$$${progression}.\:{when}\:{you}\:{owe}\:{the} \\ $$$${bank}\:{money}\:{of}\:{amount}\:{A},\:{one} \\ $$$${month}\:{later}\:{you}\:{owe}\:{the}\:{bank}\:{qA}, \\ $$$${two}\:{monthes}\:{later}\:{you}\:{owe}\:{the} \\ $$$${bank}\:{q}^{\mathrm{2}} {A},\:{three}\:{monthes}\:{later}\:{you} \\ $$$${owe}\:{q}^{\mathrm{3}} {A},\:{etc}. \\ $$
Commented by MWSuSon last updated on 07/Feb/22
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{you}\:\mathrm{really}\:\mathrm{helped}\:\mathrm{me}. \\ $$
Commented by mr W last updated on 07/Feb/22
$${nice}\:{to}\:{know}\:{that}! \\ $$