# 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 16.  Find the sum lent out, if the rate of...

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 16.  Find the sum lent out, if the rate of interest in both cases is 8 %?

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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

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