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The difference between the amount and the money borrowed is called the compound interest for given period of time

A = P*[1+ (r/100)]^{n};

CI = {P*[1+ (r/100)]^{n} -1}

A= P*[1+ (r/2*100)]^{2n}

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

Pr^{2}/100^{2}or P(r/100)2

P(r/100)^{2}*{(r/100)+3}

P ((1 + R^{1})/100) ((1 + R^{2})/100) ((1 + R^{2})/100)

x/(1+R/100)

Find the compound interest on Rs. 20,000 in 2 years at 4 % per annum, the interest being compounded half-yearly.

Principal = Rs. 20000, Rate = 2 % per half-year, Time = 2 years = 4 half- years

Amount=Rs.21648.64

Compound Interest = Total amount – Principal

= 21648.64 – 20000

= Rs. 1648.64

Find compound interest on Rs. 8500 at 4 % per annum for 2 years, compounded annually.

Solution:

We are given:

Principal = Rs. 8500, Rate = 4 % per annum, Time = 2 years

Principal = Rs. 8500, Rate = 4 % per annum, Time = 2 years

= Rs. 9193.6

Compound Interest = Total amount – Principal

= 9193.6 – 8500

= 693.6

Compound Interest = Rs. 693.6

Compound Interest = Total amount – Principal

= 9193.6 – 8500

= 693.6

Compound Interest = Rs. 693.6

Find the compound interest on a principal amount of Rs.5000 after 2 years, if the rate of interest for the 1st year is 2% and for the 2nd year is 4%.

Here R1 = 2% R2 = 4% and p = Rs.5000, we have to find CI (compound interest).

CI = 5000(1 + 2/100)(1 + 4/100) – 5000

= 5000 x (102/100)(104/100) – 5000

= 5000 x (51/50) x (52/50) – 5000

= 5000 x (51 x 52/2500) – 5000

= 5000 x (2652 / 2500) – 5000

= 5304 – 5000 = 304

Hence the required compound interest is Rs.304.

CI = 5000(1 + 2/100)(1 + 4/100) – 5000

= 5000 x (102/100)(104/100) – 5000

= 5000 x (51/50) x (52/50) – 5000

= 5000 x (51 x 52/2500) – 5000

= 5000 x (2652 / 2500) – 5000

= 5304 – 5000 = 304

Hence the required compound interest is Rs.304.

If the annual rate is r% per annum and is to be calculated for n years, then in this case, rate = (n/2%) half-yearly and time = (2n) half-yearly.

Form the above we get

Sam investment Rs.15,000 @ 10% per annum for one year. If the interest is compounded half-yearly, then the amount received by Sam at the end of the year will be.

P = Rs. 15000; R = 10% p.a = 5% half-year, T = 1 year = 2 half year

Amount = Rs

= Rs.16537.50

If the simple interest for certain sum for 2yrs at the annual rate of interest R% is SI. Then,

**Compound interest (CI) = SI (1+r/200) (no. of years =2)**

**5).** If the simple interest for a certain sum for 2 yr at 5%pa is 200, then what will be the compound interest for same sum for same period and the same rate of interest?

Sol:

Si =200 r=5%

Ci =200(1+5/200) =200*(205/200) =205

If a certain sum at compound interest becomes x times n_{1}^yr and y times n_{2}^yr then,

**X ^{1/N1} = Y^{1/N2}**

**6).** If an amount at compound interest becomes twice in 5yr, then in how many years, it will be 16 times at the same rate of interest?

2^{1/5 }= 16 ^{1/x2}

=2^{4*1/x2}

1/5 = 4/x_{2}

X_{2} = 5*4 =20yrs

If a certain sum at compound interest amounts to A_{1} in n yrs and A_{2} in (n+1) yrs,

then

**Rate of compound interest =(A _{2} – A_{1})/A_{1} *100%**

**7).** A sum of money invested at compound interest amounts to 800 in 2yr and 840 in 3yrs .Find the rate of interest and the sum.

A_{1} =800 ; A_{2 }=840,

Rate of interest = (840-800)/800 *100% =40/8 =5%

Sum = 800 *(800/840)^{2} =320000/441 = Rs.725.62

If the populations of a city P and it increases with the rate of R% per annum, then

**Populations after n yr = p(1+R/100)**^{n}**Populations n yr ago = p / (1+R/100)**^{n}

**8).** The population of a city A is 5000. It increases by 10% in 1^{st} year. It decreases by 20% in the 2^{nd} yr because of some reason. In the 3^{rd} yr, the population increases by 30%. What will be the [population of area A at the end of 3yrs?

=5000(1+10/100)(1-20/100)(1+30/100)

= 500*(11/10)*(4/5)*(13/10) = 5720

**Difference between ci and si 2yr =pr ^{2} /100^{2 } **

**9).** The difference between c.i and s.i for 2yr at the rate of 5% per annum is 5 .then the sum

5 = p (5/100)^{2} = Rs.2000

**Rate of interest (no .of years =2)**

**(for only ci)**

2% = 4.04%

3% = 6.09%

4% = 8. 16%

5% = 10.25%

6% = 12.36%

7% = 14.49%

8% = 16.64%

9% = 18.81%

10%= 20.00

+ 1.00 =21%

**10).** What is the Compound interest for Rs. 1500 at 5% rate of interest for 2 years?

1500*(10.25/100) =153.75

Example:

If the difference between the compound interest and the simple interest on a certain sum of money at 5% per annum for 3 years is Rs. 1220. What is the sum?

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