**Compound Interest**

**Compound Interest: **sometimes it so happens that the borrower and the lender agree to fix up a certain unit of time, say *yearly or half-yearly or quarterly* to settle the previous amount.

In such case, the amount after first unit of time becomes the principal for the second unit, the amount after second unit becomes the principal for the third unit and so on.

After a specified period, *the difference between the amount and the money borrowed is called the ***compound Interest*** (abbreviated as C.I.) for that period.*

**FORMULAE:**

** **Let

**Principal** =p,

**Rate** = R% per annum,

**Time**=n years

**I. When interest is compounded annually: **

Amount = P[ 1 + (R/100)]^{n}^{ }

**II. When the interest is compounded half-yearly : **

** **Amount = p[ 1 + (R/2)/100]^{2n}^{ }

**III. When the interest is compounded quarterly : **

** **Amount = P[ 1 + (R/4)/100 ]^{4n }

**IV.** When the interest is compounded annually but time is in fraction, say**3 2/5 Years. **

** **Amount = P{ [1 + (R/100)]^{3 }x [ 1 + (2R/5)/100] }

**V. ** When rates are different for different years, say R_{1}%, R_{2}%, R_{3}% for 1^{st}, ** 2**^{nd} and 3^{rd} year respectively.

** **Then, Amount= P [ 1 + R_{1}/100][ 1 + R_{2}/100][ 1+ R_{3}/100]

**VI.** Present worth of Rs. X Due n years hence is given by :

** ** Present worth = X/ (1 + R/100)^{n }

^{ }

**Solved Problem**

**Ex.1. ***Find compound interest on Rs.6250 at 16% per annum for 2 years, compounded annually. *

** Sol. **Amount = Rs. [ 6250 x ( 1 + 16/100)^{2} ]

** = Rs.**[ 6250 x 29/25 x 29/25 ] = Rs.8410.

** .**. C.I = Rs.(8410 – 6250) = Rs. 2160.

**Ex.2. ***Find compound interest on Rs. 5000 at 12% per annum for 1 year, **compounded half-yearly.*

**Sol***.* Principal = Rs.5000, Rate=6% per half year

** **Time = 1 year = 2 half-*years*

Amount = Rs. {5000 x ( 1 + 6/100)^{2}]* *= Rs. [ 5000 x 53/50 x 53/50 ]

= Rs. 5618

.. C.I. = Rs. (5618 – 5000) = Rs. 618

**Ex.3. ***Find compound interest on Rs. 16000 at 20% per annum for 9 months,**compounded quarterly.*

**Sol. **Principal = Rs. 16000, Time = 9 months = 3 quarters

** **Rate = 20% per annum = 5% per quarter.

.. Amount = Rs. [ 16000 x { 1 + 5/100 )^{3 }]

= Rs. [ 16000 x 21/20 x 21/20 x 21/20 ] = Rs. 18522.

.. C.I. = Rs. (18522 – 16000) = Rs. 2522.

**Ex.4. *** The difference between the compounded interest and simple interest on a certain sum at 10% per annum for 2 years is Rs.631. Find the sum. *

**Sol. **Let the sum be Rs. X .Then,

** C.**I.. = *x*[ 1 + 10/100 ]^{2 } - *x *] = 21*x*/100

** S.**I. = [ (*X*x10x2)/100] = *X*/5

** .**. (C.I.) – (S.I.) = [ (21x)/100 – (x/5) ] = x/100

** .**. x/100 = 631 x= 63100

** **Hence, the sum is Rs. 63100

**Ex.5. ***If the compound interest on a certain sum for 2 years at 12% per annum is * *Rs. 1590, what would be the simple interest? *

**Sol. **Let the sum be Rs. X. Then

** ** X{ 1 + 12/100 }^{2} – X = 1590 or (784X/625) – X = 1590

Or 159X/625 = 1590 or x = {1590 x 625}/159 =6250

.. Sum = Rs. 6250

.. S.I. = Rs. [ (6250 x 12 x 2)/100] = Rs. 1500

**Ex.6. ***A sun of money amounts to Rs.6690 after 3 years and to Rs 10035 after 6 * *years on compound interest. Find the sum. *

* * P{ 1 + R/100 }^{3} = 6690 ….(i) and P{ 1 + R/100 }^{6 } = 10035 … (ii)

On Dividing, we get [ 1 + R/100 ]^{3 }= 10035/6690 = 3/2

Substituting this value in (i), we get;

P x 3/2 =6690 or P=[6690 x 2/3] = 4460

Hence, the sum is Rs. 4460

**Ex.7. ***A sum of money doubles itself at compound interest in 15 years. In how many * *years will it become eight times? *

**Sol. ** P { 1 + R/100 }^{15 }= 2P ie 2P= [ 1 + R/100]^{15} = 2 … (i

** **Let P[ 1 + R/100]n = 8P [ 1 + R/100]^{n} =8 = 2^{3} = { [1 + R/100]^{15} }^{3}

[ 1 + R/100]^{n} = [ 1 + R/100]^{45}

i.e n=45

Thus required time = 45 years

**Ex.8. ***A certain sum amounts to Rs.7350 in 2 years and to Rs.8575 in 3 years . Find *** ***the sum and rate per cent. *

**Sol. **S.I. on Rs.7350 for 1 year = Rs.(8575 – 7350) = Rs.1225.

** .***. *Rate = [(100 x 1225)/(7350 x 1)]% = 16 2/3%

** **Let the sum be Rs. X .Then

X[1 + 50/(3 x100)]2 = 7350

i.e X x 7/6 x 7/6 =7350

X= {7350 x 36/49]

.. Sum = Rs. 5400