Question

What is the formula for the following ?

Answer 1

\(\large\tt \begin{align} \color{black}{\text{1.) each annual installment (x) upto (n) years }\hspace{.33em}\\~\\
\text{for a sum (P) at the rate (r%) at simple interest }\hspace{.33em}\\~\\
\text{2.) each annual installment (x) upto (n) }\hspace{.33em}\\~\\
\text{ years for a sum (P) at the rate (r%) at }\hspace{.33em}\\~\\
\text{ compound interest }\hspace{.33em}\\~\\ }\end{align}\)

Answer 2

is it about paying off the loan or saving money in bank account ?

Answer 3

That reminds me, I have to deposit my check in the bank tomorrow, haha :(

Answer 4

it is equivalent to loan

Answer 5

lol tomoro is half day for banks i think.. make sure u reach in time ;)

Answer 6

yes upto 2:PM ^

Answer 7

so the situation is like, you want to find the annual payment given below info :
1) loan amount
2) interest rate
3) number of years to pay off the loan

Answer 8

*annual payment

Answer 9

yes

Answer 10

like for sample question


what annual payment will discharge a debt of Rs 1092 due in 3 years at 12% simple interest ?


solution :
\(\large\tt \begin{align} \color{black}{x+\left(x+\dfrac{x\times 12\times 1}{100}\right)+\left(x+\dfrac{x\times 12\times 2}{100}\right)=1092 \hspace{.33em}\\~\\ }\end{align}\)

Answer 11

yeah that can be interpreted like this :

3 annual payments + interest = 1092

Answer 12

notice you're making payments at different times so the interest accumulated will also be different :

Answer 13

the payment made in the end of year1 stays in bank for two years

the payment made in the end of year2 stays in bank for one year

the payment made in the end of year3 stays in bank for 0 year

By this time the total amount in the bank equals 1092

Answer 14

so the first payment earns an interst for 2 years
second payment earns an interest for 1 year
last payment doesnt earn any interest

so total amount in the bank would be :
(x + interest for 2 years) + (x + interest for 1 year) + (x + interest for 0 years)

Answer 15

if you want a general formula for #1 : \[P = \sum \limits_{k=0}^{n-1} \left(x+x\frac{r}{100}k\right)\]

Answer 16

solve \(x\) ^

Answer 17

you can pull \(x\) out of summation because it is not dependent on the index variable k :

\[P = x\sum \limits_{k=0}^{n-1} \left(1+\frac{r}{100}k\right)\]

Answer 18

\[\dfrac{P}{\sum \limits_{k=0}^{n-1} \left(1+\frac{r}{100}k\right)} = x\]

Answer 19

thats right!
we're assuming payments are made at the end of each year

Answer 20

looks better :)

Answer 21

sry this one

\(\large\tt \begin{align} \color{black}{x+\left(x+\dfrac{x\times r\%\times 1}{100}\right)+\left(x+\dfrac{x\times r\%\times 2}{100}\right)\\\cdot \cdot \cdot \cdot \cdot \left(x+\dfrac{x\times r\%\times (n-1)}{100}\right) =P\hspace{.33em}\\~\\ }\end{align}\)

Answer 22

since you're making the first payment at the END of first year, it stays in bank for (n-1) years

Answer 23

ok so thats correct
but what about the compound interest

Answer 24

we can use the same trick

Answer 25

let me post a sample problem for this one too

Answer 26

for #2 :

first payment stays in bank for (n-1) years and becomes : \(\large x\left(1+\frac{r}{100}\right)^{n-1}\)

yes ?

Answer 27

a sum of 8000 is borrowed at 5% p.a. compound interest and paid back in 3 equal annual instalments .what is the amount of each installment?
solution :
\(\large\tt \begin{align} \color{black}{\dfrac{x}{(1+\dfrac{5}{100})^1}+\dfrac{x}{(1+\dfrac{5}{100})^2}+\dfrac{x}{(1+\dfrac{5}{100})^3}=8000 \hspace{.33em}\\~\\ }\end{align}\)

Answer 28

Notice this is a different situation, you took the loan already here.

Answer 29

ok

Answer 30

can i conclude for such problem

\(\large\tt \begin{align} \color{black}{\dfrac{x}{(1+\dfrac{r}{100})^1}+\dfrac{x}{(1+\dfrac{r}{100})^2}\\+\cdot \cdot \cdot \cdot \cdot \cdot \dfrac{x}{(1+\dfrac{r}{100})^n}=P \hspace{.33em}\\~\\ }\end{align}\)

Answer 31

looks good!

Answer 32

that works because the present value of a sum of "x" amount in "n" years is given by :

\[PV = \dfrac{x}{(1+\frac{r}{100})^n}\]

Answer 33

oh yes the present worth formula

Answer 34

otherwise you can use the EMI formula directly
http://www.financeformulas.net/Formula%20Images/Loan%20Payment%20Formula%201.gif

Answer 35

yes money now is more valuable than the money in future

Answer 36

thats for compund interest ? ^^^

Answer 37

yes interest is always compound in general
nobody uses simple interest anymore

Answer 38

100 rupees now is worth same as 1000 rupees after 10 years

Answer 39

ohk thnks very much