Aptitude - Compound Interest - Discussion

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.

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.

1600* 5/100=80.
For 6 month : 80/2= 40
3200+40(including previous)=3240*5/100=162. For the next 6 month, the value will be 81.
Now we have to add 40 + 81.

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.

@All.

Here, is my explanation for the answer.
Basically in question, there is not mention the rate if for a year,

If the rate is given for the year then for half a year the rate becomes 2.5%,
Then 1600's 5 % is 80 then half is 40 means 2.5% of it,
Then next 6month 1600 + 40 '5% is 162 so 2.5 is 81 then total;
Interest he earn is 40 + 81 = 121.

Amount after 1 year on Rs 1600 (deposited on 1st. Jan)
= 1600(1+5/2100)^(2*1)=1600(41/40)^2,
= 1681.

Compound interest
= 1681 - 1600
= 81.

Amount after 12 year on Rs 1600(deposited on 1st Jul)
= 1600(1+5/2100)^(2*(1/2))
= 1600(41/40)
= 1640

Compound interest
=1640 - 1600 = 40.

Required gain;
= 81 + 40,
= 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

How it's come 5/2*100 because r is 5% only? Explain please.

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.

Simple trick: firstly for the half-year rate of interest would be half that is 2.5%.
2.5%of 1600 = 40.

This is compounded 1640 and again deposited another 1600 it becomes 3240.
Now 2.5%of 3240 = 81.
Finally, 81+ 40 = 121.