# Compound Interest Formula

Simple interest is calculated on the principal or on the original amount of the loan. If principle = p, rate of interest = r, time = t, Then **SI = (p * t * r)/100**. But Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. It is also known as “interest on interest”.

- When interest compounded annually, Amount A = P (1 + R/100)
^{n} - When interest compounded Half-yearly, Amount A = P {1 + (R/2)/100}
^{2n}

[Half-yearly: Calculating twice a year so rate(R) is divided by 2 and number of years (n) multiplied by 2]

Now Compound interest (C.I) = Amount – Principle

C.I = P[(1 + r/100)^{n}– 1]

**Some Key Points and Useful Formulas**

**Final Amount**

The Interest at end of a certain period is added to the original sum(P) to get the amount. Now This amount becomes the principle for the next period. This process will be repeated until the amount for the last period is found which is the Final Amount (A).

__Basic rules for the compound interest :__

**Rule 1:**

If rate of interest is R1% For first year , R2% for second year and R3% for the third year , then

A=P*(1+R1/100)(1+R2/100)(1+R3/100)

**Rule 2 :**

IF principle =P, rate=R% , AND time=T years then

**a) If the interest is compounded annually:**

A=P*(1+R/100)^T

**b) If the interest is compounded half yearly (two times in year)**

A=P*(1+R/100)^2T

**c) If the interest is compounded quarterly (four times in year):**

A=P*(1+R/100)^4T

**Rule 3: IF difference between simple interest and compound interest is given :**

- If the difference between compound interest and simple interest on a certain sum of money for 2 years at R% rate is given then

**sum = difference *(100/R)^2**

- If the difference between compound interest and simple interest on a certain sum of money for 2 years at R% rate is asked then

** difference = p (r/100)^2**

- If the difference between simple interest and compounded interest on a certain sum of money for 3 years at R% rate is given then

p(r/100)^2(300+r/100)

**RULE 4: if sum A becomes B in T1 years at compounded interest , then after T2 years **

sum = (b^t2/t1)/ A^t2/t1-1

**TRICK: **If a certain sum becomes ‘m’ in times in ‘t’ years the rate of compound interest r is equal to

=100[(m^1/t -1]

**Compound Interest Of Consecutive Years**

If we have the same sum and at the same rate of interest. The C.I of a particular year is always more than C.I of Previous Year. (C.I of 3rd year is greater than C.I of 2nd year). The difference between C.I for any two consecutive years is the interest of one year on C.I of the preceding year.

C.I of 3rd year – C.I of 2nd year = C.I of 2nd year * r * 1/100

[r = rate; t = 1 year]

The difference between the amounts of any two consecutive years is the interest of one year on the amount of the preceding year.

Amount of 3rd year – Amount of 2nd year = Amount of 2nd year * r * 1/100

[r = rate; t = 1 year]

**Key Results**

When we have the same sum and same rate,

C.I for nth year = C.I for (n – 1)th year + Interest for one year on C.I for (n – 1)th year.

C.I for 6th year = C.I for 5th year + Interest for one year on C.I for 5th year

For Amount,

The amount for 6th year = Amount for 5th year + Interest for one year on Amount for 5th year.

**Some Other Applications of Amount**

**Growth:** This is mainly used for growth if industries related.

Production after n years = initial production * (1 + r/100)^{n}

**Depreciation: **When the cost of a product depreciates by r% every year, then its value after n years is

Present value * (1 + r/100)^{n}

**Population Problems: **When the population of a town, city, village increases at a certain rate per year.

Population after n years = present population * (1 + r/100)^{n}

**Examples**

**Example 1: Find the Compound Interest when principal = Rs 6000, rate = 10% per annum and time = 2 years?**

**Solution: **

Interest for first year = (6000 * 10 * 1)/100 = 600

Amount at the end of first year = 6000 + 600 = 6600

Principal interest for second year = (6600 * 10 * 1) / 100 = 660

Amount at the end of second year = 6600 + 660 = 7260

Compound Interest = 7260 – 6000 = 1260

**Example 2: What will be the compound interest on Rs 8000 in two years when the rate of interest is 2% per annum?**

**Solution:**

Given principal P = 8000

rate r = 2%

time = 2years

by formula ,

A = P (1 + R/100)

^{n}= 8000 (1 + 2/100)

^{2}= 8000 (102/100)

^{2}= 8323

Compound interest = A – P

= 8323 – 8000

= Rs 323

**Example 3: Hari deposited Rs. 4000 with a finance company for 2 years at an interest of 5% per annum. What is the compound interest that Rohit gets after 2 years?**

**Solution: **

Given

principal P = 4000

rate r = 5%

time = 2years

By formula ,

A = P (1 + R/100)

^{n}= 4000 (1 + 5/100)

^{2}= 4000 (105/100)

^{2}= 4410

Compound Interest = A – P

= 4410 – 4000

= 410

**Example 4: Find the compound interest on Rs. 2000 at the rate of 4 % per annum for 1.5 years. When interest is compounded half-yearly?**

**Solution:**

Given,

principal p = 2000

rate r = 4%

time = 1.5 ( i.e 3 half years )

by formula ,

A = P (1 + R/200)

^{2n}= 2000 (1 + 4/200)

^{3}= 2000 (204/200)

^{3}= 2122

Compound Interest = A – P

= 2122 – 2000

= 122

**Example 5: What is the compound interest on 10000 for one year at the rate of 20% per annum, if the interest compounded quarterly?**

**Solution: **

Given,

Principal P = Rs 10000

rate R = 12% (12/4 = 3 % per quarter year)

Time = 1 year (1 * 4 = 4 quarters)

by formula,

A = P (1 + R/100)

^{n}= 10000 (1 + 3/100)

^{4}= 10000 (103/100)

^{4}= 11255

Compound Interest = A – P

= 11255 – 10000

= 1255

**Example 6: Find the Compound interest at the rate of 5% per annum for 2 years on that principal which in 2 years at the rate of 5% per annum given Rs. 400 as simple interest?**

**Solution: **

Given

Simple interest SI = 400

rate R = 5%

time T = 2years

by formula,

Simple interest = (P * T * R)/100

P = (SI * 100)/T * R

= (400 * 100)/2 * 5

= 40000/10

= Rs 4000

Rate of Compound Interest = 5%

time = 2 years

by formula ,

A = P (1 + R/100)

= 4000 (1 + 5/100)

= 4410

Compound Interest = A – P

= 4410 – 4000

= 410