Aptitude - Compound Interest - Discussion

The formula is amount=P+C. I.
Then if we have to find C. I mean, it should be like C. I=amount-P, right?
But here they didn't put 1, can someone teach me this part of thought?

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.

How come 51/50? It must be 52/50, Right?

Anyone, explain to me.

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.

Yearly
5000+4%+2% = 5304.
Half-yearly
5000 + 2% + 2% + 2% = 5306.04.

Therefore, the difference is 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.

Very good explanation, Thank you.

The issue is for 1 year.
I can understand the time n =1 but isn't it suppose to be 2x1 (2xn) for a half year but instead, they are taking 1 (1/2), which means they are taking a period of 1 and half years for calculating half-year compound interest.

Since compound interest (CI= p(1+(1/2)/100)}] ^2n for half year

Thus:

CI= 5000[1+{(4/2)/(100)}]^(2*(3/2)]
= 5000[1+{2/100}]^(3).
= 5306.04.

Can we take time (n) in fraction as 3/2 in the yearly formula?

Please explain me.