8th Class Mathematics Related to Competitive Exam Compound Interest

       Compound Interest

If the interest is calculated at the end of certain fixed period and principal for the next is the amount after adding the interest of the previous period to the principal.  

Formula for Calculating Compound Interest

\[A=P{{\left( 1+\frac{r}{100} \right)}^{n}}\]

\[C.I.=P\left[ {{\left( 1+\frac{r}{100} \right)}^{n}}-1 \right]\]

Thus,   \[\mathbf{C}\mathbf{.I}\mathbf{.=A-P}\]

Where,

P = Principal Amount

r = Interest Rate

n = Number of times the amount is compounded

A = Amount after time t

C.I. = Compound interest.    

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• The interest is either calculated quarterly, half yearly or annually.
• The simple interest is calculated with the help of the relation \[\frac{P\times R\times T}{100}\].
• The compound interest is calculated with the help of the relation \[C.I.=P\left[ {{\left( 1+\frac{r}{100} \right)}^{n}}-1 \right]\]
• The amount is calculated with the help of the relation \[A=P{{\left( 1+\frac{r}{100} \right)}^{n}}\] where n denotes the number of times the principal has been compounded.         
• If the rate of interest decreases at a certain rates the amount is calculated with; the help of \[A=P{{\left( 1-\frac{r}{100} \right)}^{n}}\].  

  Arwin lent Rs. 10,000 on first January 2000 to his friend Robert at 5% simple interest which amounts to Rs. 18000 after certain period of time. Find the date on which he will get the amount.

(a) \[\text{3}{{\text{1}}^{\text{st}}}\] December 2016 

(b) \[\text{3}{{\text{1}}^{\text{st}}}\] December 2015

(c) \[\text{3}{{\text{1}}^{\text{st}}}\] December 2014                    

(d) \[\text{3}{{\text{1}}^{\text{st}}}\] December 2013

(e) None of these

Answer: (a)  

  Codi borrows Rs. 10000 for 120 days at 10% simple interest per year from his friend Lorentz for admission of his son in the school. Find the interest he has to pay to his friend after 120 days.

(a) Rs. 382.20                                    

(b) Rs. 255.65     

(c) Rs. 328.76                                     

(d) Rs. 192.72

(e) None of thee

Answer: (c)  

  Johnson has Rs. 4000 he wants to invest it in two types of bond. The first bond pays him 7% and second pays an interest 9% per annum. He uses the first bond for 12 years and other for 6 years such that interest on first type is double to that of other. Find the amount in each type of bond.

(a) Rs. 1275, & Rs. 725                   

(b) Rs. 2250, & Rs. 1750

(c) Rs. 925, & Rs. 1075                    

(d) Rs. 1325, & Rs. 675

(e) None of these

Answer: (b)  

  If m, n, p are the three sums of money such that n is the simple interest on m and p is the simple interest on n for the same time period and at the same rate of interest. The relation among m, n, p is given by:

(a) \[{{\text{n}}^{\text{2}}}=\text{mp}\]                                             

(b) \[{{\text{m}}^{\text{2}}}=\text{np}\]     

(c) \[{{p}^{\text{2}}}\text{=}\,\text{nm}\]                                          

(d) \[\text{p}=\text{nm}\]

(e) None of these

Answer: (a)  

  If the simple interest on a certain sum of money is one - ninth of the principal and rate of interest is equal to the time for which interest is found. Find the rate of interest on that sum of money.

(a) 10 %                                               

(b) \[\frac{10}{3}%\]         

(c) 5.5 %                                              

(d) 7.25 %

(e) None of these

Answer: (b)      

When the Interest is Compounded Annually but Rates are Different for Different Years

Let principal = Rs. P, time = 2 years and let the rate of interest be m% per annum (p.a.) during the first year and n% p. a. during the \[{{\text{2}}^{\text{nd}}}\]years

Then the formula of amount after 2 years. \[=P\left( 1+\frac{{{R}_{1}}}{100} \right)\left( 1+\frac{{{R}_{2}}}{100} \right)\]

This formula may similarly be extended for any number of years  

When the Interest is Compounded Annually but Time is a Fraction

Let time is 2 years and 4 month, then

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

Compounded Half Yearly

Let principal = p, Rate = r%. Time = n years.

If compounded half yearly then, rate = \[\frac{r}{2}\]% per half year, time = 2n

Then amount \[=P\times {{\left( 1+\frac{r}{2\times 100} \right)}^{2n}}\]

Compound Interest = Amount - Principal  

Compounded Quarterly

Let principal = p. Rate = r% per annum, Time = n years.

If compounded quarterly then, rate = \[\frac{r}{4}\]% per quarter, time = 4n

Then amount \[A=P\times {{\left( 1+\frac{r}{2\times 100} \right)}^{4n}}\]

Compound Interest = Amount - Principal