Aptitude - Compound Interest - Discussion

I think it will be as correct answer for everyone, the first deposit amt is 1600, after six month total amt is 1640 (Including 5% of one year interest for six months). We should assign this as A-case.

Next July 1st day second deposit amt is 1600, It should be as B-case. The end of the year amt is 1640 (Including 5% of one year interest of six months). But A-case deposited at one year, So total Mature amt is 1681 (Including 5% of one year interest of twelve months. Interest of Rs. 40 is Rs-1, So amt Rs-40 became as Rs-41).

Finally A-case + B-case are 1681 + 1640 = 3321. Deposit amt is Rs-3200 and Interest amt is Rs-121.

On 1st january customer deposit amount and receive end of year. It means first he deposit for 1 year.

When interest is compounded Half-yearly:

Amount = P * [ { 1 + (1/2) * (R/100) } ^ 2(n) ]
Amount = 1600 * [ { 1 + (1/2) * (5/100) } ^2(1) ] = 1681


But in 1st july he deposit same amount (1600 Rs. ) and receive it after the same date (after 6 month). It means second he deposit for 6 month(1/2 year).

When interest is compounded Annually but time is in fraction, say 1/2 years.

Amount = P * [ 1 + { (1/2)* R/100}]
Amount = 1600 * [ 1 + { (1/2)* 5/100}] = 1640

So Total = 1681 + 1640 = 3321

You people might confused with the formulae and there of three formulaes where n represents exactly year i.e n=1 year, n=1/2 half year so on.,

Then another thing the three formulaes were having a much difference look at it as individual dont miggle those all three formulaes 2gether .

Ex: to calculate half year compund intrest u shoud go with the second one
don't bother about the duration here we are calculating C.I for every half yearly what ever the term it be either 1 year or 2 years u shud use the half year formulae for calculating that... look at it as very abstract.

On 1st January he deposited 1600.

Using the formula of compound interest half yearly
A=P(1+(R/2/100))²n.

n is 1/2 here as he deposits for 6 months i.e, till July.
Now, A=1600(1+(5/200)^2*.5
A = 1600 * 1.025,
= 1640.

Therefore compound interest = 1640 - 1600,
= 40.

Now in July he deposits 1600.
So, the principal amount becomes 1640 + 1600 = 3240.

So, using the formula
A=3240*(1+5/200)^2*0.5
A=3240*1.025,
= 3321.

Therefore, compound interest is 3321 - 3240 = 81

Total compound interest is 40 + 81 = 121.

So, basically the person is depositing the amount of 1600 in 1st Jan and again the amount of 1600 in 1st July.

So, Customer deposit = 1600 + 1600 => 3200.

Now, using Simple interest formula cal the 1st jan deposit SI.
SI = 1600(5)(1)/100(2).
Si = 40.
AMOUNT = 1600 + 40 => 1640 ---> Eq 1
SI for 1st July with same process but this time the principal will be P = 1640.
SO, SI = 41 and;
AMOUNT will be = 1640+41 => 1681 ---> Eq 2.
ADD BOTH Eq 1 and Eq 2 we get =>3321.

CI = customer deposit - amount.
CI = 3321 - 3200.
CI = 121.
HOPE YU GET IT.

@Anom.

You are right but just once again read the 2nd line of the question.

It is being said that the amount is received in 1st of Jan and 1st of July which are basically half years in simple words - from 1st of Jan to 1st of July it's 6 months which is a half year and same way from 1st of July to 1st Jan is another 6 months which is another half year.

Therefore, the 5% of interest is calculated in Jan and July separately [ p- (1+r/100) ^t + p- (1+r/100) ^t] - p. Which calculated in half year basis only not for a year.

Given:

Interest rate = 5% per annum, compounded half-yearly, so every 6 months.

So, half-yearly interest rate = 2.5% (5% ÷ 2).

Two deposits:

Rs. 1600 on 1st January (for 1 year, or 2 half-years)
Rs. 1600 on 1st July (for 6 months, or 1 half-year)

Now, calculate the compound interest separately for both:

1st deposit: Rs. 1600 for 2 half-years
Interest = 1681 - 1600 = Rs. 81

2nd deposit: Rs. 1600 for 1 half-year,
Interest = 1640 - 1600 = Rs. 40.

Total interest gained:
81 + 40 = Rs. 121

Answer: Rs. 121.

At the time of first deposit i.e on January 1st

Amount= p[1+(R/2)/100]^2(n)
here n is 1 year beginning of the year
so amount = 1600[1+(5/2)/100]^2
=1600[1+(5/200)]^2
=1600[1+(1/40)]^2
=1600[41/40]^2
=1681.

Secondly when he deposited its July i.e after 6 months
so n=half of the year=1/2

Amount=p[1+(R/2)/100]^2n n=1/2 year
=1600[1+(5/200)]
=1600[41/40]
=1640.

now add both amounts
1681+1640=3321
1600 deposited 2 times by the customer in a year therefore 1600*2=3200

gain => 3321-3200=121.

Amount from 1st January to 1st July is
Amount= 1600[1+5/(2*100)]^2.
= 1681.

Note: The principal amount becomes 1681 instead of 1600 because the money is deposited in the bank. There is not provided with any information regarding the interest is withdrawn.

So, the new principal amount at 1st July is 1600+1681 = 3281.
The amount at the end of the year becomes;
=3281[1+5/(2*100)]^2,
=3447.1.
The gain is =3447.1-3200.
=247.1 Rs.

Correct me, whether it is wrong.

Can someone explain me?

If I don't go for formula.. so let's do with normally.

January to july,
Principal-1600
rate-5%
time-1 (i.e half year)

so CI= 1600*0.05*1
= 80.
Hence the amount is 1600+80=1680 which is the principal for the second time.

Now July to December,

His new principal=1680 and he deposit 1600 more so his new principal is 1680+1600=3280.
rate is same = 5%
time=1 (half yearly)

So CI = 3280*0.05*1.
= 164.

So its 164. How 121? Please correct me.