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# Formulas for Compound Interest Problems

## Formulas for Compound Interest in Aptitude

Compound interest is the interest on interest to the principal sum of a amount. It is calculated by  result of reinvesting interest, rather than paying it out. By this interest in the next period is  earned on the principal sum plus previously accumulated interest.

### Compound Interest Formulas

• Compound interest is the interest calculated on the original principal and on the accumulated past interest of a deposit or loan. Compound interest is calculated based on the principal, interest rate (APR or annual percentage rate), and the time involved.

Formula of Compund Interest (CI) =  P $(1+ \mathbf{\frac{r}{100}})^{n}$ – P

Formula of  Amount     =    CI +P

=   P $(1+ \mathbf{\frac{r}{100}})^{n}$ – P + P

Here,  P = Principal

r = rate of interest

t = the number of years the amount is deposited or borrowed for.

n= the number of times that interest is compounded per unit ‘t’.

### Formulas for Compound Interest

• Formulas for Compound Interest (When Interest is Compound Annually)
• Amount = P$\mathbf{(1+ \frac{r}{100})^{n}}$
• Compound Interest =Total amount – Principal
• Rate of interest (R) = [$\mathbf{(\frac{A}{P})^{\frac{1}{t}}}$− 1]%

• Formulas of Compound Interest (When Interest is Compound Half Yearly)
• Amount = P $(1+ \mathbf{\frac{{\frac{r}{2}}}{100})^{2n}}$
• Compound Interest = Total amount – Principal

• Compound Interest Formulas (When Interest is Compound Quarterly)
• Amount = P $(1+ \mathbf{\frac{{\frac{r}{4}}}{100})^{4n}}$
• Compound Interest = Total amount – Principal

• Formulas of Compound Interest (When Interest is Compound Monthly)
• Amount = P $(1+ \mathbf{\frac{{\frac{r}{12}}}{100})^{12n}}$

• Compound Interest Formulas (When Interest is Compounded Annually but time is in fraction, say 2($\mathbf{\frac{3}{2}}$)years )
• Amount = P $(1+ \mathbf{\frac{r}{100}})^{2}$$(1+ \mathbf{\frac{\frac{3}{2}r}{100}})$

• Formulas of Compound Interest (When rates are different for different years)
• Amount = P (1+$\frac{r_{1}}{100}$) (1+$\frac{r_{2}}{100}$) (1+$\frac{r_{3}}{100}$)

• Formulas for Compound Interest (Present worth of Rs. x due n years)
• Present worth =    $\frac{x}{(1+\frac{r}{100})}$

### Problems Based on Formulas Of  Compound interest with solutions

Question 1 . Find the amount on Rs 16000 for 3 year at 5% per annum compounded annually n.

[A] Rs. 18562

[B] Rs. 18550

[C] Rs. 18952

[D] Rs. 18552

Solution :     According to the formula of Compound Interest

Amount = P $(1+ \mathbf{\frac{r}{100}})^{n}$

Amount = 16000 $(1+ \mathbf{\frac{5}{100}})^{3}$

Amount = 16000 $(1+ \mathbf{\frac{1}{20}})^{3}$

Amount = 16000 $(\mathbf{\frac{21}{20}})^{3}$

Amount =18552

Question 2 . Find the  compound interest on Rs. 10,000 at 20% per annum for 5 years 4 months, compounded annually is :

[A] Rs. 16600

[B] Rs. 16544

[C] Rs. 15644

[D] Rs. 16500

Solution :    According to the Formula of Compound Interest ,

CI = P $(1+ \mathbf{\frac{r}{100}})^{n}$ – P

CI = 10,000 $(1+ \mathbf{\frac{20}{100}})^{5\frac{1}{3}}$ – 10,000

CI = 10,000 $(1+ \mathbf{\frac{1}{5}})^{5\frac{1}{3}}$ – 10,000

CI = 10,000 $(1+ \mathbf{\frac{1}{5}})^{5}(1+ \mathbf{\frac{1}{3*5}})$ – 10,000

CI = 10,000 $(1+ \mathbf{\frac{1}{5}})^{5}(1+ \mathbf{\frac{1}{3*5}})$ – 10,000

CI = 10,000 $(1+ \mathbf{\frac{1}{5}})^{5}(1+ \mathbf{\frac{1}{15}})$ – 10,000

CI = 10,000 $(\mathbf{\frac{6}{5}})^{5}(\mathbf{\frac{16}{15}})$ – 10,000

On Solving ,

CI = 26544 – 10,000

CI = 16544