Aptitude - Compound Interest - Discussion

Calculate compound interest when interest is compounded yearly:

Principal (P) = Rs. 5000
Rate of interest (r) = 4% per annum (0.04)
Time (t) = 1.5 years.

C.I., when interest compounded yearly = P * (1 + r/n)^(n*t), where n is the number of times interest, is compounded per year.

Plugging in the values:

C.I. when interest compounded yearly = 5000 * (1 + (4/100)) * (1 + (4/200)).
= 5000 * (26/25) * (51/50).
≈ Rs. 5304.

Calculate compound interest when interest is compounded half-yearly:
Principal (P) = Rs. 5000.
Rate of interest (r) = 4% per annum (0.04).
Time (t) = 1.5 years.

C.I. when interest compounded half-yearly = P * (1 + r/(2100))^(2t), as the interest is compounded twice a year.

Plugging in the values:
C.I. when interest compounded half-yearly = 5000 * (1 + (2/100))^3.
= 5000 * (51/50) * (51/50) * (51/50).
≈ Rs. 5306.04.

Calculate the difference in compound interest:
Difference = C.I. when interest compounded half-yearly - C.I. when interest compounded yearly
= Rs. (5306.04 - 5304),
= Rs. 2.04.

As we know that,

The formula for annual compound interest, including principal sum, is:

A = P (1 + r/n) (nt).

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

C.I. when interest compounded yearly = Rs. 5000 * (1 + 4/100) * (1 + (4/2) /100)

= Rs.(5000 * 26/25 * 51/50) = Rs. 5304.

C.I. when interest is compounded half-yearly = Rs. 5000 * (1 + 2/100) 3
= Rs.(5000 * 51/50 * 51/50 * 51/50) = Rs. 5306.04,

Difference = Rs.(5306.04 - 5304) = Rs. 2.04.

Compounded yearly:
P = 5000,
R = 4% -----> 200,
So, 5000 + 200 = 5200,
R = 2% -----> 104,
Then total = 5200 + 104 = 5304.

Compounded half-yearly:
P = 5000,
R = 2% -----> 100,
Then, 5000 + 100 = 5100,
R = 2% -----> 102,
Then, 5100 + 102 = 5202,
R = 2% -----> 104.04,
Total = 5202 + 104.04 = 5306.04.

And finally, 5304 - 5306.04 = 2.04.

Sum = 5000.
Interest rate = 4%.

yearly (1+1/2).
=>Interest gained in 1 year = 200.
=>Interest gained in another 1/2 year on 5000 and on 200(interest of 1st year) = 104 = >(100+4)
=>Total interest yearly = 200+104 = 304.

Now,
Half-yearly (1/2+1/2+1/2),
=>Interest gained in 1st half year = 100.
=>Interest gained in 2nd half year= 100 + 2.
=>Interest gained in 3rd half year = 100 + 2 + 2 + 0.04.
=>Total interest gained half yearly = 306.04

So, the difference is 306.04-304 = 2.04.

It's actually simple if you look at this in a different way.
The first CI on 4% per annum on 5000 is 200.
And for the next 6 months, it is 2% of 5200 so that's 102.
So,
Total is 200+104 = 304.

On the second case, it's taken half yearly so 2% every 6 months then put the value;
100 in the first 6 months.
102 in the next 6 months.
On the third 6 month, it becomes
2% of 5222.
ie 52.2 + 52.2 = 104.4,
100 + 102 + 104.4 = 306.4,
304 - 306.4 = 2.04.

Short cut method:
Rate given - 4%.
CI for yearly,
4%=1/25(first year).
2%=1/50(6 month)(rate divided by 2).

25:26.
50:51.
1250: 1326 //(25*50)and (26*51).
1250x=5000.

So x = 4,
CI = 1326*4 = 5304.

CI for half-yearly,
2% = 1/50(rate divided by 2).
50:51
50:51
50:51
125000:132651 //(50*50*50) and (51*51*51).

125000x = 5000.
x = 0.04.

So 0.04*132651=5306.04

So difference is 5306.04 - 5304=2.04(answer).

C.I ON PER YEAR.

N=1,1/2 r=4%. P=5000.
1st year-- 5000 in 4% is = 200,
Next we need 1/2 year interest so r=4/2. R=2%,
Hence 5200 in 2% is = 204.
Hence interest is 200+100=304.

Then find half year interest.
N=3 i.e ::-:'(1*1/2 in 3 half year),
R=2 because (4%is full year but we need half yr),

5000 in 2% is =100,
5100 in 2%is =102,
5202 in 2%is =104.04,
Total = 306.04.

Hence(half year - 1year) = 306.04 - 304.
Ans = 2.04.

If the calculator is not provided.

Then short cut (same as Given Solution).

For yearly (Amount+compound interest) = A (1 + [r/100]) ^ 3/2 A = 5000; are = 4.

= A (1 + [r/100]) ^ (1+1/2).

= A (1 + [r/100]) * (1 +[r/100] ^ (1/2)).

= A (1 + [r/100]) * (1 + 1/2[r/100]).

For half yearly (Amount +compound interest) =A (1 +[(r/2) /100]) ^ (3/2*2).

Direct formula for finding the amount of compound interest.

A = P( 1 + R*t/100)^n/t.

Where,
A : Amount ( principle + interest).
P : principle amount.
R : rate of interest in %.
t : number of years for interested is compounded.
n : number of years.

In above example,
For the half year compounded t=1/2.

Try this method.
Thanks.

Explanation for the amount when compounded yearly is as below:

We learn from our formulas that when interest is compounded, annually but time is in fraction then it should be:

Amount = P*(1+r/100)^1*(1+(1/2*4)/100).

Please check general formula section 5.