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Question Number 165404 by MWSuSon last updated on 01/Feb/22

given a loan from a bank on a fixed interest rate 5% suppose i want to make monthly payments for 14months how do i calculate the amount i should pay monthly and how do i know how much i will pay back to the bank at the end of the 14months? please i need explanations.

$$\mathrm{given}\:\mathrm{a}\:\mathrm{loan}\:\mathrm{from}\:\mathrm{a}\:\mathrm{bank}\:\mathrm{on}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{interest}\:\mathrm{rate}\:\mathrm{5\%} \\ $$$$\mathrm{suppose}\:\mathrm{i}\:\mathrm{want}\:\mathrm{to}\:\mathrm{make}\:\mathrm{monthly}\:\mathrm{payments}\:\mathrm{for}\:\mathrm{14months} \\ $$$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{amount}\:\mathrm{i}\:\mathrm{should}\:\mathrm{pay}\:\mathrm{monthly}\:\mathrm{and} \\ $$$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{know}\:\mathrm{how}\:\mathrm{much}\:\mathrm{i}\:\mathrm{will}\:\mathrm{pay}\:\mathrm{back}\:\mathrm{to}\:\mathrm{the}\:\mathrm{bank}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{14months}? \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{need}\:\mathrm{explanations}. \\ $$

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.

$${not}\:{only}\:{the}\:{interest}\:{rate}\:{but}\:{also}\:{the}\: \\ $$$${monthly}\:{payment}\:{must}\:{be}\:{arranged} \\ $$$${with}\:{the}\:{bank}. \\ $$

Commented by MWSuSon last updated on 01/Feb/22

So there′s no way to know how much money you′ll pay monthly from the interest rate, loan money and Time to complete payment?

$$\mathrm{So}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{way}\:\mathrm{to}\:\mathrm{know}\:\mathrm{how}\:\mathrm{much}\:\mathrm{money}\:\mathrm{you}'\mathrm{ll}\:\mathrm{pay} \\ $$$$\mathrm{monthly}\:\mathrm{from}\:\mathrm{the}\:\mathrm{interest}\:\mathrm{rate},\:\mathrm{loan}\:\mathrm{money}\:\mathrm{and}\:\mathrm{Time}\:\mathrm{to} \\ $$$$\mathrm{complete}\:\mathrm{payment}? \\ $$

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 (usually a fixed amount) is also determined.

$${if}\:{the}\:{time}\:{to}\:{pay}\:{back}\:{the}\:{complete}\: \\ $$$${money}\:{incl}.\:{interest}\:{is}\:{arranged}, \\ $$$${then}\:{the}\:{monthly}\:{payment}\:\left({usually}\right. \\ $$$$\left.{a}\:{fixed}\:{amount}\right)\:{is}\:{also}\:{determined}. \\ $$

Commented by mr W last updated on 01/Feb/22

A=your loan p=annual interest rate (p/(12))=p′=monthly interest rate let q=1+(p/(12))=1+p′ M=monthly payment (fixed amount) after 1 month: interest: A×(p/(12)) you owed the bank: A+A×(p/(12))=qA your payment: M you still owe the bank: qA−M after 2 monthes: interest: (qA−M)×(p/(12)) you owed the bank: (qA−M)q your payment: M you still owe the bank: (qA−M)q−M you can countinue further. generally after n monthes: you still owe the bank: Aq^n −M(1+q+q^2 +..+q^(n−1) ) =Aq^n −((M(q^n −1))/(q−1)) such that you don′t owe the bank any money after n monthes, Aq^n −((M(q^n −1))/(q−1))=0 ⇒M=(((q−1)q^n A)/(q^n −1)) example: you loan 1000$ from the bank with an annual interest rate 6%. you must pay back completely in 20 monthes. A=1000$ p=6% q=1+(p/(12))=1.005 the monthly payment is then M=((0.005×1.005^(20) ×1000)/(1.005^(20) −1)) ≈52.67$ you payed the bank totally 20×52.67=1053.40$ for a loan of 1000$.

$${A}={your}\:{loan} \\ $$$${p}={annual}\:{interest}\:{rate} \\ $$$$\frac{{p}}{\mathrm{12}}={p}'={monthly}\:{interest}\:{rate} \\ $$$${let}\:{q}=\mathrm{1}+\frac{{p}}{\mathrm{12}}=\mathrm{1}+{p}' \\ $$$${M}={monthly}\:{payment}\:\left({fixed}\:{amount}\right) \\ $$$$ \\ $$$${after}\:\mathrm{1}\:{month}: \\ $$$${interest}:\:{A}×\frac{{p}}{\mathrm{12}} \\ $$$${you}\:{owed}\:{the}\:{bank}:\:{A}+{A}×\frac{{p}}{\mathrm{12}}={qA} \\ $$$${your}\:{payment}:\:{M} \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}:\:{qA}−{M} \\ $$$$ \\ $$$${after}\:\mathrm{2}\:{monthes}: \\ $$$${interest}:\:\left({qA}−{M}\right)×\frac{{p}}{\mathrm{12}} \\ $$$${you}\:{owed}\:{the}\:{bank}:\:\left({qA}−{M}\right){q} \\ $$$${your}\:{payment}:\:{M} \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}:\:\left({qA}−{M}\right){q}−{M} \\ $$$${you}\:{can}\:{countinue}\:{further}. \\ $$$$ \\ $$$${generally}\:{after}\:{n}\:{monthes}: \\ $$$${you}\:{still}\:{owe}\:{the}\:{bank}: \\ $$$${Aq}^{{n}} −{M}\left(\mathrm{1}+{q}+{q}^{\mathrm{2}} +..+{q}^{{n}−\mathrm{1}} \right) \\ $$$$={Aq}^{{n}} −\frac{{M}\left({q}^{{n}} −\mathrm{1}\right)}{{q}−\mathrm{1}} \\ $$$$ \\ $$$${such}\:{that}\:{you}\:{don}'{t}\:{owe}\:{the}\:{bank}\:{any} \\ $$$${money}\:{after}\:{n}\:{monthes}, \\ $$$${Aq}^{{n}} −\frac{{M}\left({q}^{{n}} −\mathrm{1}\right)}{{q}−\mathrm{1}}=\mathrm{0} \\ $$$$\Rightarrow{M}=\frac{\left({q}−\mathrm{1}\right){q}^{{n}} {A}}{{q}^{{n}} −\mathrm{1}} \\ $$$$ \\ $$$${example}:\: \\ $$$${you}\:{loan}\:\mathrm{1000\$}\:{from}\:{the}\:{bank}\:{with} \\ $$$${an}\:{annual}\:{interest}\:{rate}\:\mathrm{6\%}. \\ $$$${you}\:{must}\:{pay}\:{back}\:{completely}\:{in} \\ $$$$\mathrm{20}\:{monthes}. \\ $$$${A}=\mathrm{1000\$} \\ $$$${p}=\mathrm{6\%} \\ $$$${q}=\mathrm{1}+\frac{{p}}{\mathrm{12}}=\mathrm{1}.\mathrm{005} \\ $$$${the}\:{monthly}\:{payment}\:{is}\:{then} \\ $$$${M}=\frac{\mathrm{0}.\mathrm{005}×\mathrm{1}.\mathrm{005}^{\mathrm{20}} ×\mathrm{1000}}{\mathrm{1}.\mathrm{005}^{\mathrm{20}} −\mathrm{1}}\:\approx\mathrm{52}.\mathrm{67\$} \\ $$$$ \\ $$$${you}\:{payed}\:{the}\:{bank}\:{totally}\: \\ $$$$\mathrm{20}×\mathrm{52}.\mathrm{67}=\mathrm{1053}.\mathrm{40\$} \\ $$$${for}\:{a}\:{loan}\:{of}\:\mathrm{1000\$}. \\ $$

Commented by mr W last updated on 01/Feb/22

M=(((q−1)q^n A)/(q^n −1))=((p′(1+p′)^n A)/((1+p′)^n −1)) ≈((p′(1+np′)A)/(np′))=((1/n)+p′)A

$${M}=\frac{\left({q}−\mathrm{1}\right){q}^{{n}} {A}}{{q}^{{n}} −\mathrm{1}}=\frac{{p}'\left(\mathrm{1}+{p}'\right)^{{n}} {A}}{\left(\mathrm{1}+{p}'\right)^{{n}} −\mathrm{1}} \\ $$$$\approx\frac{{p}'\left(\mathrm{1}+{np}'\right){A}}{{np}'}=\left(\frac{\mathrm{1}}{{n}}+{p}'\right){A} \\ $$

Commented by MWSuSon last updated on 03/Feb/22

Sorry sir that it took me this long to reply, i do not know why i do not recieve notifications. I just have one question sir, what does q stand for and why is it 1+(p/(12))?

$$\mathrm{Sorry}\:\mathrm{sir}\:\mathrm{that}\:\mathrm{it}\:\mathrm{took}\:\mathrm{me}\:\mathrm{this}\:\mathrm{long}\:\mathrm{to}\:\mathrm{reply},\:\mathrm{i}\:\mathrm{do}\:\mathrm{not}\:\mathrm{know}\:\mathrm{why} \\ $$$$\mathrm{i}\:\mathrm{do}\:\mathrm{not}\:\mathrm{recieve}\:\mathrm{notifications}. \\ $$$$\mathrm{I}\:\mathrm{just}\:\mathrm{have}\:\mathrm{one}\:\mathrm{question}\:\mathrm{sir},\:\mathrm{what}\:\mathrm{does}\:\mathrm{q}\:\mathrm{stand}\:\mathrm{for}\:\mathrm{and}\:\mathrm{why} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{1}+\frac{\mathrm{p}}{\mathrm{12}}? \\ $$

Commented by mr W last updated on 03/Feb/22

all is explained: p=annual interest rate (p/(12))=monthly interest rate let q=1+(p/(12)), that means q stands for 1+(p/(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^2 A, three monthes later you owe q^3 A, etc.

$${all}\:{is}\:{explained}: \\ $$$${p}={annual}\:{interest}\:{rate} \\ $$$$\frac{{p}}{\mathrm{12}}={monthly}\:{interest}\:{rate} \\ $$$${let}\:{q}=\mathrm{1}+\frac{{p}}{\mathrm{12}},\: \\ $$$${that}\:{means}\:{q}\:{stands}\:{for}\:\mathrm{1}+\frac{{p}}{\mathrm{12}}. \\ $$$${the}\:{meaning}\:{of}\:{q}\:{is}\:{clear}.\:{it}\:{is}\:{the}\: \\ $$$${common}\:{ratio}\:{of}\:{the}\:{geometric} \\ $$$${progression}.\:{when}\:{you}\:{owe}\:{the} \\ $$$${bank}\:{money}\:{of}\:{amount}\:{A},\:{one} \\ $$$${month}\:{later}\:{you}\:{owe}\:{the}\:{bank}\:{qA}, \\ $$$${two}\:{monthes}\:{later}\:{you}\:{owe}\:{the} \\ $$$${bank}\:{q}^{\mathrm{2}} {A},\:{three}\:{monthes}\:{later}\:{you} \\ $$$${owe}\:{q}^{\mathrm{3}} {A},\:{etc}. \\ $$

Commented by MWSuSon last updated on 07/Feb/22

Thank you sir. you really helped me.

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{you}\:\mathrm{really}\:\mathrm{helped}\:\mathrm{me}. \\ $$

Commented by mr W last updated on 07/Feb/22

nice to know that!

$${nice}\:{to}\:{know}\:{that}! \\ $$

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