On a certain sum of money, the difference between the compound interest for a year, payable half yearly, and the simple interest for a year is Rs...

Tuesday, July 5, 2011

On a certain sum of money, the difference between the compound interest for a year, payable half yearly, and the simple interest for a year is Rs...

The
formulae for compound interest and simple interest:

$\displaystyle{A}={P}{{\left({1}+{i}\right)}}^{{n}}$ and
$\displaystyle{A}={P}{\left({1}+{i}{n}\right)}$

A is the future amount. P is the present amount (which is what
we need) i = interest and n= months/ years.

By using a form of simultaneous
equations, we will be able to deduce the amount borrowed.

Compound
interest: $\displaystyle{A}={P}{{\left({1}+\frac{{0.08}}{{2}}\right)}}^{{{1}\times{2}}}$

Remember that the
interest is a percentage so always divide by 100 ( $\displaystyle\frac{{8}}{{100}}={0.08}$ ).We have divided by 2 because
the interest is compunded half yearly (ie twice a year) and we have multiplied n (1 year) by the
same 2 that we divided the interest by.

(In other words had it been
compounded monthly we would have divided the interest by 12 and would have multiplied n by 12)

2. Simple interest: $\displaystyle{A}={P}{\left({1}+{0.08}\times{1}\right)}$

we are
working with only 1 year and there is now compounding so n=1

Now we know
that the difference is Rs 16. So if we subtract the simple interest from the compound interest
formula (ie 1. - 2.) we can deduce the amount originally saved:

$\displaystyle{P}{\left({1.0816}\right)}-$
P(1.08) = 16