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:

`A= P(1+i)^n`     and 
`A=P(1+ i n)`

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.

1. Compound
   interest: `A = P (1 + 0.08/2)^(1times 2)`

Remember that the
interest is a percentage so always divide by 100 (`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: `A= P(1+0.08 times 1)`

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:

`P (1.0816) -
P(1.08) = 16`

`therefore 0,0016P = 16`

`therefore P = 16 /
0.0016`

`therefore P = 10 000`

Therefore the
amount that was lent out is Rs 10 000