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.

If time period is T and the rate R.
for half yearly basis calculation;
T`=2T. And. R`=R/2.

Here total period=12 month= 1 year,
Rate= 5%.
T`=2. R`= 2.5.

For the first half,
1600 × 2.5/100 = 40,
CI = 40 × 2 = 80,

For second half
P = 1640 × 2.5/100 = 41.
CI = 41×1 = 41,
Total CI = 121.

Note.
For 2 year CI calculation
Multiple is 2,1.

For 3-year CI calculation
Multiple are 3,3,1.
For 4 year CI calculation
Multiple are 4,6,4,1.

@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.

Anyone, please give a clear explanation about the given formula.

Why is it used? Please elobarate.

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.

But why they didn't take 2t for next half year? please explain to me.

BANK INTEREST - 5%.

1st jan deposit = 1600.
1st jul deposit = 1600.

1st Jan = 1600 * 5/100 = 80.
For 6months 80/2v= 40.
40 + 1600 = 1640.
1st july = 1640(jan) + 1600(jul) = 3240.
3240*5/100v= 81.

= 40+81 = 121.

First, Deposit
Principal Amount (P1): 1600.
Rate of Interest (R): 5.
Frequency of Compounding(N): 2.
Tenure Period(T): 0.5.
Compound Interest (CI): 40.
Compound Amount (CA): 1640.

CA = 1600*(1 + 0.05/2)^1.
CA = 1640.

Second Deposit, 1600
Now, for the rest of the tenure we will be investing.. CA of first deposit + second deposit.

1640 + 1600 = 3240 (P2).
CA = 3240*(1 + 0.05/2)^1.
CA = 3321.

CI = CA - P.
CI = 3321 - (1600+1600).
CI = 121.

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.

@All.

Guys we can do this in an easy way listen;

Firstly, from 1st Jan to July we give 1600 to the bank
1600*5/100*1/2(1/2 because it's on half yearly based)
So, the answer is 40.

After that from July to December the process will be the same but now the ci will be on 1640
1640*5/100*1/2 answer is 41.

Now we have 1600 in July so the amount will be;
1600*5/100*1/2.
40.

So, the total interest is 40+41+40 is equal to 121.