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Parin Sir > Quantitative Aptitude (Maths) > Compound Interest > Formulae > Formulae

Formulae

June 15, 2020
Category: Formulae

☼ FORMULAE :

(i) When interest is compounded K times in a year, then

A = P $\left (1 + \frac{R}{100 \times K} \right ) ^{kN}$

$\therefore$ C.I. = A – P = P $\left (1 + \frac{R}{100 \times K} \right ) ^{kN}$ – P = P $\left [\left (1 + \frac{R}{100 \times K} \right ) ^{kN}-1 \right ]$

$\large \mapsto$ When interest is compound Annually, then

A = P ( $\because$ K = 1 )

e.g. Find amount and C. I. on Rs. 10000 at 15 % per annum for 2 years, compounded annually.

Soln. Here, K = 1 $\large \Rightarrow$ No. of years = N = 1 $\times$ 2 = 2

A = 10000 $\times \frac{115}{100}\times \frac{115}{100}$ $\left (\because k=1 \Rightarrow \textup{use } \frac{100+R}{100} \right )$

= Rs. 13225

$\therefore$ C.I. = 13225 – 10000 = Rs. 3225

$\large \mapsto$ When interest is compound Half–yearly, then

A = P $\left (1 + \frac{R}{100 \times 2} \right ) ^{2N}$ ( $\because$ K = 2 )

e.g. Find amount on Rs. 10000 at 16 % per annum for 1 year, compounded half–yearly.

Soln. Here, K = 2 $\large \Rightarrow$ No. of years = N = 2 $\times$ 1 = 2

A = 10000 $\times \frac{216}{200}\times \frac{216}{200}$ $\left (\because k=2 \Rightarrow \textup{use } \frac{200+R}{200} \right )$

= Rs. 11664

$\large \mapsto$ When interest is compound Quarterly, then

A = P $\left (1 + \frac{R}{100 \times 4} \right ) ^{4N}$ ( $\because$ K = 4 )

e.g. Find amount on Rs. 10000 at 16 % per annum for 6 months, compounded quarterly.

Soln. Here, K = 4 $\large \Rightarrow$ No. of years = N = 4 $\times \frac{6}{12}$ = 2

A = 10000 $\times \frac{416}{400}\times \frac{416}{400}$ $\left (\because k=4 \Rightarrow \textup{use } \frac{400+R}{400} \right )$

= Rs. 10816

$\large \mapsto$ When interest is compound monthly, then

A = P $\left (1 + \frac{R}{100 \times 12} \right ) ^{12N}$ ( $\because$ K = 12 )

e.g. Find amount on Rs. 14400 at 16 % per annum for 2 months, compounded monthly.

Soln. Here, K = 12 No. of years = N = 12 $\times \frac{2}{12}$ = 2

A = 14400 $\frac{1216}{1200}\times \frac{1216}{1200}$ $\left (\because k=12 \Rightarrow \textup{use } \frac{1200+R}{1200} \right )$

= Rs. 14786.56

(ii) When interest is compounded Annually but time is in fraction,

( say 4 $\frac{2}{3}$ years )

A = P $\left (1+\frac{R}{100} \right ) ^{4} \times \left [1+\frac{2\times R}{3\times 100} \right ]$

e.g. Find C. I. on Rs. 10000 at 24 % per annum for 2 years and 5 months.

Soln. Here, P = Rs. 10000, R = 24, N = 2 yrs.

A = 10000

( K = 1 Use )

= Rs. 16913.60

I. = A – P = 16913.60 – 10000 = Rs. 6913.60

(iii) When Rates are different for different years, say R₁ %, R₂ %, R₃ % for N yrs., N yrs. and N yrs. respectively.

Then, A = P

e.g. Find C. I. on Rs. 10000 at 10 % per annum for 1^st year and 20 % for another 2 years.

Soln. Here, P = Rs. 10000, R = 10, R = 20, N = 1 yr, N = 2 $\frac{5}{12}$ yrs.

A = $10000\times \frac{124}{100} \times \frac{124}{100}\times 1+\left (\frac{5\times 24}{12\times100} \right )$ $\left (\because K=1 \Rightarrow use \frac{100+R}{100} \right )$