SI And CI
- What is the meaning of interest?
- Suppose you are in need of some money and you went to your friend for help. He gave you 100
rupees and said to give it back in 1 year, but when you return it, you need to give an
additional 10 rupees (total 110). This extra money that you have to pay is known as interest.
- The extra money paid is interest.
- The money we borrowed, 100 rupees, is the principal.
- The duration after which we have to give it back is known as time, here 1 year.
- The interest in this case is 10 rupees.
- The total money we return, 110 rupees, is known as the amount.
Formula of amount: principal + interest.
- Rate of interest is the interest we pay in one year upon the principal amount multiplied by 100.
Rate = (Interest of 1 year / Principal) * 100
Rate = (10 / 100) * 100 = 10%
- There are two types of interest:
- Simple Interest: It is calculated only on the principal amount every year.
- Compound Interest: It is calculated on the principal amount as well as on the interest received
every year.
Example Questions
P = 1000
R = 10% per annum
T = 2 years
Find Simple Interest (SI) and Compound Interest (CI)?
- Simple Interest (SI):
1st year → 10% of 1000 = 100
2nd year → 10% of 1000 = 100
Total = 200, which is the Simple Interest (SI)
- Compound Interest (CI):
1st year → 10% of 1000 = 100
2nd year → 10% of 1000 = 100 and 10% of 1st year interest (100) = 10
Total for the 2nd year = 100 (principal interest) + 10 (interest on interest) = 110
Total = 100 (1st year) + 110 (2nd year) = 210, which is the Compound Interest (CI)
P = 1200
R = 10% pa
T = 2 year
Si and CI?
- Simple Interest (SI):
1st year → 10% of 1200 = 120
2nd year → 10% of 1000 = 120
Total = 240, which is the Simple Interest (SI)
- Compound Interest (CI):
1st year → 10% of 1200 = 120
2nd year → 10% of 1200 = 120 and 10% of 1st year interest (120) = 12
Total for the 2nd year = 120 (principal interest) + 12 (interest on interest) = 132
Total = 120 (1st year) + 132 (2nd year) = 252, which is the Compound Interest (CI)
P = 1000
R = 10% pa
T = 3 year
SI, CI?
- Simple Interest (SI):
1st year → 10% of 1000 = 100
2nd year → 10% of 1000 = 100
3rd year → 10% of 1000 = 100
Total = 300, which is the Simple Interest (SI)
- Compound Interest (CI):
1st year → 10% of 1000 = 100
2nd year → 10% of 1000 = 100 and 10% of 1st year interest (100) = 10, total = 110
3rd year → 10% of 1000 = 100 and 10% of 1st year interest (100) = 10 and also 10% of
2nd
year interest (110) = 11, total = 121
Total = 100 (1st year) + 110 (2nd year) + 121 (3rd year) = 331 , which is the Compound
Interest
(CI)
P = 3430
R = 14(2/7)%
T = 3 years
Find Compound Interest (CI)?
- Compound Interest (CI) Calculation:
14(2/7)% can be simplified to 1/7th
1st year → 1/7th of 3430 = 490
2nd year → 1/7th of 3430 = 490 and 1/7th of 1st year interest (490) = 70, total = 560
3rd year → 1/7th of 3430 = 490 and 1/7th of 1st year interest (490) = 70 and also 1/7th
of
2nd year interest (560) = 80, total = 640
Total = 490 (1st year) + 560 (2nd year) + 640 (3rd year) = 1690, which is the Compound
Interest
(CI)
- Another method - cube method:
R = 14(2/7)% in fraction = 1/7
Rate = Interest / Principal
Here, we pay 1 unit of interest for every 7 units of principal amount.
For 3 years:
CI = Principal * (1 + Rate)^n - Principal, where n is the number of years
CI = 3430 * (1 + 1/7)^3 - 3430
CI = 5120 - 3430
CI = 1690
P = 2160
R = 16(2/3)%
T = 3 year
CI?
- CI:
16(2/3)% = 1/6
1st year → 1/6th of 2160 = 360
2nd year → 1/6th of 2160 = 360 and 1/6th of 1st year interest (360) = 60, total = 420
3rd year → 1/6th of 2160 = 360 and 1/6th of 1st year interest (360) = 60 and also 1/6th
of
2nd year interest (420) = 70, total = 490
Total = 360 (1st year) + 420 (2nd year) + 490 (3rd year) = 1270, which is the Compound
Interest
(CI)
- CI: using cube method
Rate = 16(2/3)% which in fraction = 1/6
Rate = Interest / Principal
so here Interest = 1 unit and principal = 6 unit
CI = principal * (1+Rate)^3 - Principal
CI = 2160 * (1+1/6)^3 - Principal
CI = 3430 - 2160
CI = 1270
- CI: using cube method
R = 16(2/3)% in fraction = 1/6
Rate = Interest / Principal
Here, we pay 1 unit of interest for every 6 units of principal amount.
For 3 years:
CI = Principal * (1 + Rate)^n - Principal, where n is the number of years
CI = 2160 * (1 + 1/6)^3 - 2160
CI = 3430 - 2160
CI = 1270
- CI: using ratio method
Rate = 1/6
Here, 1 represents the interest and 6 represents the principal, so the amount (principal +
interest) is 7.
Now, for 3 years:
Principal ratio^3 = 6^3 = 216
Amount ratio^3 = 7^3 = 343
This is all in ratio.
This means 216 in ratio corresponds to 2160 rupees (given principal amount).
So, 1 ratio unit = 2160 / 216 = 10 rupees.
We can also find the interest using the ratio:
Interest ratio = Amount ratio - Principal ratio = 343 - 216 = 127
Therefore, Compound Interest (CI) = 127 * 10 (value of one ratio unit)
CI = 1270 rupees
P = 2000
T = 2(1/5) years
R = 10%
CI?
-
We will try to find the CI for 2 and 3 years using the ratio method:
11 is the amount and 10 is the principal in terms of ratio.
For 2 years, 11^2 = 121 (amount ratio units) and 10^2 = 100 (principal ratio units).
So, 1 ratio unit = 2000 / 100 = 20
Interest ratio for 2 years = 121 - 100 = 21
CI for 2 years = 21 * 20 = 420
So, this was CI for 2 years.
Now for 3 years:
Amount ratio for 3 years = 11^3 = 1331
Principal ratio for 3 years = 10^3 = 1000
Interest ratio for 3 years = 1331 - 1000 = 331
CI for 3 years = 331 * 20 = 662
We need CI for 2 years and 1/5 of the 3rd year:
CI for 2 years = 420
Additional CI for 1/5 of the 3rd year = 1/5 * (662 - 420) = 1/5 * 242 = 48.4
Total CI for 2(1/5) years = 420 + 48.4 = 468.4
- We can also do it like this:
On 1st year, CI is 10% of 2000 = 200
For the 2nd year, the amount becomes 2200, so CI will be 10% of 2200 = 220
For the 3rd year, the amount is 2420, so CI will be 10% of 2420 = 242
CI for 2 years = 200 + 220 = 420
1/5 of the 3rd year's CI = 1/5 * 242 = 48.4
Total CI for 2(1/5) years = 420 + 48.4 = 468.4
R = 14(2/7)%
T = 3 years
Amount = 5120
P?
-
The rate 14(2/7)% can be written as a fraction: 14(2/7)% = 1/7.
This means that the amount ratio after 1 year is 8 (principal + interest).
For 3 years, the amount ratio is 8^3 = 512.
If 512 ratio units equal 5120 in actual currency,
then 1 ratio unit = 5120 / 512 = 10.
Now, the principal ratio for 3 years is 7^3 = 343.
Therefore, the principal amount = 343 * 10 (one ratio unit value) = 3430.
So, the principal (P) is 3430.
P = 5000
R of 1st year = 20%
R of 2nd year = 10%
CI?
-
For the 1st year:
20% of 5000 = 0.20 * 5000 = 1000
For the 2nd year:
Amount at the end of 1st year = Principal + 1st year interest = 5000 + 1000 = 6000
10% of 6000 = 0.10 * 6000 = 600
Therefore, the compound interest for 2 years = 1st year interest + 2nd year interest
CI = 1000 + 600 = 1600
Effective Rate Of Interest
- If we have two different interest rates, for example, a% and b%, then we can combine them using the
formula: a + b + (ab/100)
P = 5000
R of 1st year = 20%
R of 2nd year = 10%
CI?
-
Effective Rate (R) = 20 + 10 + (20 * 10 / 100)
= 30 + 200 / 100
= 30 + 2
Effective R = 32%
Now, we can calculate the compound interest directly using the effective rate:
32% of 5000 = 0.32 * 5000 = 1600
Therefore, the compound interest (CI) for 2 years using the effective rate is 1600.
Another thing about effective interest rate
-
For Simple Interest (SI), if T = 2 years and R = 10%, then the effective R will be just double
of R, which will be 20%.
- But for Compound Interest (CI), if T = 2 years and R = 10%, then the effective R will be
calculated using the formula:
10 + 10 + (10 * 10 / 100) = 21%
- Now, if T = 3 years and R = 10%, we first calculate the effective rate for the first two years,
and then use that result with the rate for the last year:
10 + 10 + (10 * 10 / 100) = 21%
21 + 10 + (21 * 10 / 100) = 31 + 2.1 = 33.1%
P = 1200
R = 10%
T = 2 years
CI - SI?
-
SI = 20% of 1200
SI = 240
CI = 21% of 1200
CI = 252
So, CI - SI = 12
-
We also have a formula for calculating the difference between CI and SI, which only works when T
is 2 years:
Difference = (PR^2) / 100^2
Difference = (1200 * 10 * 10) / 10000
Difference = 12
Half Yearly and Quarterly Concept
-
If we have R = 20% and T = 2 years, then:
Half yearly:
R = 10% (half)
T = 4 times (double)
Quarterly:
R = 5% (quarter)
T = 8 times
P = 3000
R = 20% per annum
T = 1 year
CI? Compounded half yearly
-
We have half yearly compounding, so:
R = 10% (half of 20%)
T = 2 periods (double the time for half-yearly)
Using the short method:
CI for 1st period (half-year) = 10% of 3000 = 300
Amount after 1st period = 3000 + 300 = 3300
CI for 2nd period (half-year) = 10% of 3300 = 330
Total CI for 1 year = 300 (1st period) + 330 (2nd period) = 630
If the compound interest accrued on an amount of Rs. 15000 in two years is Rs. 2,496, what is the
rate of interest?
- Note:
Amount of Denotes Principal
and Amount to Denotes Amount
- P = 15000
T = 2 years
CI = 2496
Amount = 15000 + 2496 = 17496
First, we will find the ratio between Principal and Amount:
15000 : 17496
625 : 729
Now, we use the square root or cube root of the ratio:
(625)^1/2 = 25
(729)^1/2 = 27
Interest = 27 - 25 = 2
Rate = Interest / Principal
So rate = 2/25 * 100 = 8%