Aptitude - Simple Interest - Discussion

The rate is always annual(Not necessary to mention).

If it is not mentioned then we have to understand it is 10 p. a.

But in this question interest is calculating at every six months so rate (10/2 =5%).
So, Rate is 5 % for six months,
Now, financier includes interest after 6 months that means He will give interest on the interest that means he will give interest 5% on 5.

So, it is 5*5/100=0.25,
one year interest =5+5+=10%,
But effective interest =10+0.25=10.25,
(here 0.25% is interest on interest)

Doesn't matter if you are taking p=100 or p=200 or p=10.
But interest is always on 100.

If it is given 5% then it means we got an interest of rupees 5 on rupees 100.

If you are taking p=200 and rate is unchanged (5%) then we got 10 rupees interest on 200.
In this case, we are also getting 5 rupees on every 100 rupees.
So in this question, you take principle 100 /200/10 but we get effective interest =10.25.

Let the principal amount be x.
i.e p=x and rate of interest r=10.
and time = 1/2 (half year).

For first six months simple interest i = ptr/100.
=> i =((x )(1/2)(10))/100,

By solving above equation we get i= x/20,
Now new principal amount =x+(x/20) =>21x/20,
=>p=21x/20.

For second six months simple interest i= ptr/100.
=> i=( (21x/20)(10)(1/2) )/100.
=>i=21x/400.

Now total amount TA is = principal amount +interest on first six months +interest second six months.
=>TA=(x)+(x/20)+(21x/400).
=>TA=441x/400,

Now interest for 1 year is= total amount TA -actual principal amount.
ie. i= (441x/400)-x.
i= 41x/400.
Now for an effective rate of interest r.
i=ptr/100.
p=x,r=?,t=1(for complete 1 year),i=41x/400(interest for complete one year).
=> 41x/400=( (x)(1)(r) )/100.
=> r=41/4.
=> r=10.25.

As amount is not given so we will have to consider the amount to be 100
hence let P=100.

Now the lender add interest after every six month so we will have to calculate SI for every six month add then add this SI toh the P.

Hence
SI=P*R*T/100
=100*10*1/100*2(6 monh=1/2 year)
=5

Now P becomes
P=100+5
=105, this is SI for 1st 6 month
Now calculate SI for last 6 motnh
SI=105*10*1/100*2
=5.25
P=5+5.25+100
=110.25, this is the amount at the end of 1 year.

Now SI for I year=110.25-100
=10.25
Now we have to calculate rate
SI=P*R*T/100
10.25 =100*R*1/100
R=10.25 //

It is said as SI. So even for the second half of the year the principal should be taken as 100. Here it is taken as 105 which is in case of CI.

My answer:

For first 6 months :

ASSUME p = 100 AND r = 10%.

SO INTEREST IS 10 RS AND AMOUNT = 110.

FOR NEXT 6 MONTHS :

SAME P = 100 AND R = 10% AND WE WILL GET THE SAME INTEREST WHICH IS 10.

NET INTEREST = 20.

Net amount = 120.

So answer should be 20%. option D.

Answer will be 10.25 if its CI.

Kindly correct me if I am wrong.

Let the sum be Rs. 100. Then, for six month time will be (6/12) year which become 1/2 year.

S.I. for first 6 months = Rs. (100 x 10 x 1)/100*2 = Rs. 5.

Now Principal for next 6 months will 100+5 = 105.

S.I. for next 6 months = Rs. (105 x 10 x 1)100*2 = Rs. 5.25.

Total interest in a year 10.25.

Apply formula:

S.I. = P*r*t/100.

Put value for a year now.

10.25 = 100*r*1/100.

R = 10.25*100/100*1.

R = 10.25%.

We can find effective interest rate by the following formula.

EAR (effective annual rate) = [(1+r/m)^m]-1.
Where r means interest rate.
Here, m means the number of compounding period.
Here our require ans is [(1+.10/2)^2]-1.
= .1025.

where r = 0.10.
m= 2 ( it is 2 because every six months come two time in a year).
= 10.25 %.

eff. interest = ( (1 + r%/(100*t) ) ^ t - 1 )* 100.
r% = rate of interest (10).
t = time divided (in this case = 2).

Soln; ( (1 + 10/(100*2) ) ^ 2 - 1 )* 100,
= ( (1 + 10/200 ) ^ 2 - 1 )* 100,
= ( (1 + 0.05 ) ^ 2 - 1 )* 100,
= ( (1.05)^2 - 1 )*100.
= ( 1.1025 - 1 )*100.
= ( 0.1025)*100,
= 10.25%.

Jadhav, it is written in the question he adds interest obtained, after every six months.. So interest obtained in 6 months was 5rs
He added this to principal after 6 months and calculated SI for the rest 6 of months using this added principal amount (using 105)..he now gets principal as 5.25
So, for first six months , SI=5
Remaining six months, SI=5.25
Total SI =10.25
Hope this helps

The question said at the interest rate of 10% which probably means for 1year per annum, but he includes every 6 months interest. That means an interest rate of 6 months will become (5% half of 1 year) + another 6 months (5%half of 1year).

Now, using the effective rate of interest formula.
i,e (a + b + a * b/100),
(5+5 + 5 * 5/100)
(10 + 25/100),
(10 + 0.25),
(10.25) answer.

Say X is the Principal amount
then SI for first the 6 months= X*10/100*1/2 =X/20

SI for the next 6 months =21X/20*10/100*1/2= 21X/400
[Here Principal amount is taken as X+ X/20= 21X/20]

SI for one year =X/20+ 21X/400 = 41X/400

Now With Principal amount X and SI 41X/400 for one year we have to find the rate of interest, say y
y= 100*41X/400*1/X = 10.25//