Aptitude - Compound Interest - Discussion

To calculate the interest earned by the customer, we can use the formula for compound interest:

A = P left( 1 + frac{r}{n} right)^{nt}

Where:
- (A) = Final amount.
- (P) = Principal (initial deposit).
- (r) = Annual interest rate (in decimal).
- (n) = Number of times interest is compounded per year.
- (t) = time the money is invested (in years).

For the first deposit of Rs. 1600 made on January 1:
- (P_1 = 1600)
- (r = 5% = 0.05)
- (n = 2) (compounded half-yearly)
- (t = 1) year for this deposit.

The amount after 1 year:

A1 = 1600 left( 1 + frac{0.05}{2} right)^{2 times 1}
= 160 times left( 1 + 0.025 right)^2
= 1600 times (1.025)^2
= 1600 times 1.050625 = 1681

So, the amount from the first deposit is Rs. 1681, and the interest earned from this deposit is Rs. 1681 - Rs. 1600 = Rs. 81.

For the second deposit of Rs. 1600 made on July 1:
- (t = 0.5) years (since only half a year has passed).

The amount after 0.5 years:

A_2 = 1600 left( 1 + frac{0.05}{2} right)^{2 times 0.5}
= 1600 times (1.025)
= 1600 times 1.025 = 1640.

So, the amount from the second deposit is Rs. 1640, and the interest earned from this deposit is Rs. 1640 - Rs. 1600 = Rs. 40.

Now, adding the interest from both deposits:
{Total interest} = 81 + 40 = 121.

Thus, the customer would have gained Rs. 121 by way of interest at the end of the year.

So in the question, they have given that compounded half-yearly (1 year = 12 months) so we have to calculate for every 6 months.

First, let us understand the difference b/w simple interest and compound interest.

In Simple interest, the interest after a year will not be added to the principal (sum) amount.

Whereas compound interest. The interest after a year will be added to the principal (sum) amount.

So calculating for the first 6 months using formula S. I = PTR/100. Here 6 months to covert in to years just divide by 12. (in first 6 months TIME in years = 6/12).

S.I = (1600* 6/12*5) / 100.

S.I = 40.

AS I have mentioned this interest will be added to PRINCIPAL since we are calculating for Compound Interest.

So, 1600 + 40 = 1640 this is for 6 Months.

GIVEN ====> in July he again invest 1600 so total will be 1640+1600 = 3240.

We have to calculate for another 6 months.

Here the PRINCIPAL WILL BECOME NEW PRINCIPAL ====== 3240.
S.I = PTR/100 ==> (3240*6/12*5) /100 ==> 81.

This will be again added to the last Principal to obtain ANOTHER NEW PRINCIPAL SO.

3240+81 = 3321.
The total amount deposited is 3200.
3321-3200 = 121.

I hope this info will be helpful.

Thank you.

@ALL.

You all are facing problem because you are obsessed with "n" to substitute as "year" always.
"n" is basically how many "times" rate is applied on amount.

Here, it is clearly given that the interest is calculated on "half-yearly" basis. Now because of half yearly basis.

Here "n" will be "1" for 6 months (and r=r/2).
and "n" will be "2" for 1 year ( and r=r/2).

Suppose if it was given quarter-yearly,
"n" would be "1" for 3 months ( and r=r/4).
"n" would be "2" for 6 months (and r=r/4).
"n" would be "3" for 9 months (and r=r/4).
"n" would be "4" for 1 year (and r=r/4).

So, this guy kept his first 1600Rs amount for " two period" time.
that is, from 1st jan to 30th June (1st period) and from 1st July to 31st Dec (2nd Period).
he also deposited another amount on 1st July, this amount is kept for "one period".

That is, from 1st July to 31st Dec.

Therefore there is square(n=2) in a first Formula and no square(n=1) in a second formula.

Hope I helped little bit.

Formula for compound interest: A=P(1+r/m)^m(t).
where:
A=future value.
P=principal.
r=rate.
m=no. of compounding period a yr.
t= no. of years.

In the above problem, we have to solve January and July.

FOR: on January 1st, the customer deposits 1600 which is logically an end of the year.

GIVEN:
P=1600,
r=5% or 0.05,
m=2 (half-yearly basis).
t=1 (one year, as you read a while ago, it's also an end of the year deposit so it's one year).

Using the formula:
A= 1600(1+(0.05/2))^2(1),
= 1681.

FOR: on July 1st he again deposits 1600. 1st of July is the 7th month of the year which is the starting of half of year (1/2).

GIVEN:
P=1600.
r=5% or 0.05,
m=2 (half-yearly basis),
t=1/2 (half of the year),

Using the formula:
A= 1600(1+(0.05/2))^2(1/2).
= 1640.

Now that we're done solving January and July's future value we got:
customer's total deposit is 1600+1600 = 3200.
customer's total deposit WITH interest is 1681+1640 = 3321.

Finally, the question is " what is the amount he would have gained BY WAY OF INTEREST?"

From the formula:
I= Amount-Principal.
= 3321-3200.
= 121 (amount OF INTEREST).

Hope this helps.

My gosh. So many comments. I know it is bit confusing and I too.
Let me try to make you understand.

So here we start. you might have familiar with the formula,

C.I = p[1+(R/100)]^n.

Here p = principal amount.
R = rate.
n = no.of years.

But in the problem we are dealing with half year.

Means we are getting C.I on 6 months
* we have given annual rate of 5%.

So for half year it would be R/2
* As we are calculating C.I over every 6 months, so for a year n become 2 (as two half year is equal to one year).

So here n = 2.
So our formula becomes,

C.I = P[1+(R/2*100)]^2.

Here p is given as 1600 Rs. Now after 6 months, on date 1 July another amount of 1600 Rs got deposited.

So again we have to calculate the C.I for this amount for a 6 months only(upto 31 Dec) so that we can get the C.I from Jan 1 to July 1 and from July 1 to Dec 31.

So as to complete one year. as We are asked about C.I over total one year.

So,
For a second amount formula for C.I becomes,

C.I = P[1+(R/2*100)]^1.

Combining two we have,

C.I = P[1+(R/2*100)]^2 +P[1+(R/2*100)]^1.

@ALL.

P (1+ R/2 /100) ^ 2n for half yearly.

P (1+R/100) ^n for yearly.

Don't think about the second 1600 deposited on july 1st, forget this completly to avoid confusion.

1600 first deposit becomes 1640 at the June end (use half yearly formula).

Since this is not given back on June end it will remain in the bank.

So what we think is, 1600 is kept for 6 +6 months = 1 year and we use yearly formula. (completely mistaken).

How can we use yearly interest formula when question clearly says interest calculated half yearly.

If we use yearly formula we get 1600 (1+ 5/100) ^1 =1680.

Now see when half yearly formula is used.

1600 (1+ (5/200) ) check half yearly formula.

Which gives 1640.

As I said this 1640 not given back, again apply half year formula.

1640 (1+5/200) = 1681.

So interest is 81.

Compare the interest you got using yearly formula (mistaken formula) and actual formula (correct formula). We got 80 (mistaken) and we got 81 (correct) see the difference.

So it is used p (1+ R/2 /100) ^2n = 1600 (1+ 5/200) ^2 = 1681 correct n= 1/2 +1/2 =1.

PRINCIPLE AMT p=1600RS. THIS AMOUNT IS COMPOUNDED HALF YEARLY SO r=5/2%.

Time= 2t this is theory based on the amount calculated half yearly.

For the first six months he received an interest we calculate as follow.

= p (1 + r/100) ^2t.

Here t = 6 month since we have to put it in the formula as a year so 6 month means 1/2 year.

So compound interest for 6 month = 1600 (1 + 5/2 * 100) ^2 * 1/2.

= 1600 (1 + 5/200) =1640 this is the amount he has in his account after six months.

Again bank gives him six-month interest on this amount after the completion of first six months now understand clearly after the six-month interest will be given on 1640 so this is principle amount for the bank again follow the above procedure.

= 1640 (1 + 5/2 * 100) ^2 * 1/2.
= 1640 (1= 5/200) =1681.

The total amount he received at the end of year = 1640 + 1681 = 3321.

Total amount he paid = amount paid for first six month + amount paid for next six month = 1600 + 1600 = 3200.

Interest he received at the end of 12 month = 3321 - 3200 = 121.

Still in confusion, Check out these:

According to question Rs. 1600 is invested twice. i.e. from 1st Jan to till the end of the year, and from 1st July to the end of the year.

So we have -.

P= Rs. 1600; R = 5%; N1= 1yr; N2= 1/2yr.

So, Amount->1 = p{1+ (R/2) /100}^2 * N1, ie CI calculated half yearly.

= 1600 (1 + 5/200) ^2 * 1.
= 1600 (205/200) ^2.
= 1600 (41/40) ^2.
= 1600 * 41 * 41/40 * 40.
= 41 * 41 = 1681 --->(i).

And,
Amount->2 = p{1 + (R/2) /100}^2 * N2.

= 1600 (1 + 5/200) ^2 * 1/2.
= 1600 (1 + 5/200).
= 1600 (41/40).
= 40 * 41.
= 1640 ----> (ii).

Therefore Total Amount:

(i) + (ii) = 1681 + 1640 = 3321 Rs.

As the same amount, 1600 is invested twice so Total principal = 1600 * 2 =3200.

Therefore, C.I = Amount - Principal.

= 3321 - 3200.
= Rs. 121.

Let me try to make you understand.

>> First 6 months .

* Deposit is 1600.

* Interest is 5% half yearly.

So, Interest is 5/100 for 6 months,

i.e., one year is represented as 1 (one).

Half yearly means 1/2 (six months), {quarterly 1/4 (4 months each) for example}.

To calculate interest on deposit.

>> deposit * interest * years.
>> 1600 * 5/100 * 1/2.
>> 40.

Interest on 1600 is 40,
So 1600 + 40 = Rs. 1640.

After first six months, the total amount available is 1640.

>>> Next Six months.
>> Deposit is 1600.

So, Total available balance is 1600 + 1640(from first 6 months).

>> 3240 * (5/100) * (1/2).
>> 81.

Finally, So total interest is 40 + 81 = 121.

Hello all,

My solution:

First calculate Amount for money deposited in 1st Jan for half year.

n=1/2 R=5 P= 1600.
Amount = P(1+(R/2)/100)^2n ------> (Formula when interest compound half-yearly)
=1600(1+(5/2)/100)^2/2,
=1600(1+5/200),
=1640.

Now we will add the amount 1640 into money deposited in 1st July.
1640+1600 = 3240.
Now P=3240 and will calculate amount for the remaining half year.
P=3240 R=5 n=1/2.
Amount = P(1+(R/2)/100)^2n.
= 3240(1+(5/2)/100)^2/2
= 3240(1+5/200)
= 3321.

Total amount deposited =1600 + 1600 = 3200.
Amount gained = 3321 - 3200 = 121.

Hope this helps