Aptitude - Compound Interest - Discussion

Let principal, P be Rs.100

Amount after 1 year on Rs.100 at 6% per annum when interest is compounded half-yearly

=P(1+(R/2)/100)^2T,
=100(1+(6/2)/100)^2*1,
=100(1+3/100)^2,
=100(103/100)^2,
=100*103*103/100*100,
=103*103/100,
=106.09.

This means, if interest is compounded half-yearly at 6%, Rs.100 becomes Rs.106.09 after 1 year

Now, we need to find out the rate of interest on which Rs.100 becomes Rs.106.09 after 1 year.

when the interest is compounded annually

P(1+R/100)^T=106.09
100(1+R/100)^1=106.09
100(1+R/100)=106.09
100+R=106.09
R=106.09/100=6.09%.

i.e, effective annual rate of interest is 6.09%

The amount for a sum of 'p' for 'n' years with effective rate 'r' compounded yearly should be equal to the amount for the same sum 'p' for 'n' years with rate 6% p.a compounded half-yearly.

p(1+r/100)^n = p(1+(6/2)/100)^2n
'p' and 'n' on both sides gets cancelled and remaining EQ becomes;
(1+r/100) = (1+3/100)^2.
1+r/100 = 1.0609 (on simplication),
r/100 = 0.0609 ==> r= 6.09,
Therefore, 'r' becomes 6.09% which is the required answer.

Effective rate of Interest means,
the simple interest rate that produces the same amount of interest as the compound interest rate.

i.e assume the principal amount be 'x'
Then according to definition,

p*r*t = (p(1+r/100)^n - p),

For S.I., as per question, t= 1 year;
For C.I., as per question, we use formula C.I= (P(1+(r/2)/100)^2 - P);
Therefore,
x*r*1 = ( x(1+(6/2)/100)^2 - x),

r = 0.0609;
or, rate percent = 6.09.

As per formula "n" account for compounding period which is taken in years.

But when we count for the half year we get 2 periods in a year. So, we just multiply no. of years (n) with 2 in original formula to obtain the exact compounding periods.

How ever the rate of interest are (R.P.A) will be reduced to half in considered for half year.

Hope you understood!

@All.

Since,it says rate to be 6% per annumn and we need to find the effective rate for half yearly
So, rate=6/100=0.06% (mandatory step).

E.F(effective rate)=(1+r/n)^n-1 (n=time; i.e-if time is 1 year ,annually=1,half .yearly=2,quarterly=4).

And then,
(E.F * 100 ) will give you an actual effective rate.

I still don't understand what is n=? In this formula for calculating compound interest half year.

A = P[1+R/200]^2n.
Is it the no of slots like n=2 for 1 year, because Two six months slots(6 months+ 6 months).
Or is it n=1/2 because it is half of 1 year.

Please explain it clearly.

The effective rate is the actual rate that you earn on an investment or pay on a loan after the effects of compounding frequency are considered.

Formula for Effective Rate = (1 + (i / n))^n -1


i = Nominal or stated interest rate
n = Number of compounding periods per year

Effective rate is actual rate for full year corresponding to rate for a year half yearly.

Eg. Rs.100 A = 100(1+6/100) = 106 for full year.

A = 100(1+3/100)^2 = 106.09 for a year half yearly. So effective rate is 106.09 = 100(1+R/100) is equal to 6.09. That's it.

Half yearly calculation is not correct.

As for half yearly, Amount = P[1+R/100]2N; so for this context, it seems to be 1, if half yearly (2*1/2=1).

How is it possible to give the time still as 2 years?

Please explain.

Thank you.

r=6.
t=1.
Let sum=100.

CI = (p(1+r/2*100)^t)-p.
=100(1+6/200)^2*1-100 //as we have to solve for 1 year
=100(1+3/100)^2-100,
=100(103/100)^2-100,
=100*103/100*103/100-100,
=10609/100-100,
=106.09-100,
=6.09 Ans.