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In this section we will discuss about Simple and Compound Interest. 

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. … Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period.

When a person borrows some amount of money from another person or bank, then the borrower pays some extra money during repayment is called interest.

Let’s first deal with some terminology related to interest:

1. Interest: The extra money during the repayment

2. Principal: The money deposited for a certain time, is also known as capital.

3. Time(t/n): The duration for which money is borrowed or lent.

4. Rate of interest(R/r): The rate charged on principal, it is given in percentage term.

5. Simple Interest (SI):  For every time period, our principal is constant, on which we calculate interest.

6. Compound Interest: Every time we calculate the successive increase in the previous amount.

7.  Amount (A): Principal + Interest

SIMPLE INTEREST

 Simple interest = (Principal × Rate × Time) / 100

 SI = (P × r × t) / 100 = A – P 

E.g. Find the simple interest on rs. 100 at 12% per annum for 5 years.

Solution.

SI = (P × r × t) / 100 = (1000 × 12 × 5) / 100 = 600

Let’s discuss some more formulas of simple interest.

1. P = (100 × A) / (100 + RT) 

2. SI = (A × r × t) / (100 + r × t)

3. Two different cases can be compared by (A– N1) / (A– N2) = (P× r× t1) / (P× r× t2 )

4. Amount (A) ⇒ A = P + (P × r × t) / 100   

E.g. A sum of money doubles in 10 years. In how many years it will be treble at the same rate of interest?

Solution

A = 2P,

 SI = P (given)

SI = (P × r × 10) / 100

As SI = P so,

P = (P × r × 10) / 100 

so, r = 10%

For 3 years,

Amout = Interest + Principal

3P = Interest – P,

Interest would be 2P

2P = (P × 10 × t) / 100

t = 20 years 

E.g.  A sum was put at simple interest at a certain rate for 4 years. Had it been put at a 2% higher rate, it would have fetched more 56rs. More, find the sum.

Solution

We know that SI = (P × r × t) / 100

A simple interest in the first case,

⇒ SI = (P × r × 4) / 100 = (P × r) / 25

Simple interest in second case, 

SI = (P × (r + 2) × 4) / 100 = (P(r + 2)) / 25

The difference would be,

⇒ P(r + 2) / 25 – Pr / 25 = 56, 2P / 25 = 56

⇒ P = 700 rs(Ans.) 

COMPOUND INTEREST

Interest accrued on principal as well as interest due of principal is called compound interest.

Since principal increases after every year, the amount of interest in compound interest is always more than simple interest.

Amount (A) = P(1 + r / 100)t

Compound interest = A – P = P{(1 + r / 100)– 1}

Where signs have usual meanings, 

E.g. What will be the amount when rs. 10000 is deposited in a bank at 10% per annum compounded annually for 3 years.

Solution Amount will be A = P(1 + r / 100)t = 1000(1 + 10 / 100)3

=10000×(1.1)3 = rs.13310 (Ans.) 

E.g. The difference between the compound interest and simple interest on a certain sum at 15% per annum for 3 years is 283.50 Rs. find the sum.

Solution simple interest for 3 years would be = (P × r × t) / 100

⇒ (P × 15 × 3) / 100

⇒ (P × 45) / 100

Compound interest for 3 years would be = P{(1 + r / 100)– 1}

⇒ P{(1 + 15 / 100)– 1}

⇒ P{.520875}

According to question   P{.520875}-(P×45)/100=283.50

P{.520875-.450}=283.50

P=4000rs(Ans.) 

 1. Difference between CI and SI for two years = P(R / 100)2

2. Difference between CI and SI for 3 years = P(R / 100)× ((R / 100) +3)

Let’s try to find out with the use of this formula for faster calculation

According to formula

Difference = P(R / 100)× (R / 100 + 3)

283.50  = P(15 / 100)× [(15 / 100) + 3]

⇒  P = 100 / 315 × 283.50 × 10000 / 225

⇒ P = Rs.4000 (Ans.)

we saw we can solve questions this way much faster than conventional methods. 

SOME IMPORTANT CONCEPTS

1.  When interest is not compounded annual the amount is given by

⇒ A = P(1 + r / (n × 100)

where n = number of conversions in a year

E.g. Calculate the compound interest on rs. 2000 for 3 years at 10%, when compounded half-yearly.

Solution here n = (12 / 6) = 2

According to formula,                             

A= P(1+r/(n×100)nt

by putting the values,

⇒ A = 2000(1 + 10 / (2 × 100)3×2

⇒ A = 2000(21 / 20)6

⇒ A = rs.2680

⇒ CI = A – P

⇒ CI = 2680 – 2000

⇒ CI = 680 rs. (Ans.) 

2. When interest is not the same for every year.

 (i) A = P(1 + R / 100)(1 + R / 100)(1 + R / 100)… Up to T terms   (When the rate is the same for every year)

(ii) A = P(1 + R/ 100)T1 × (1 + R/ 100)T2 × (1 + R/ 100)T3 …..up to so on   (When the rate is not same for every year) 

E.g.  Find the amount of Rs. 40000 for 5 years compounded annually, the rate of interest is 10% for the first 3 years and 20% for the next 2 years.

Solution

According to formula  A = P(1 + R/ 100)T1 × (1 + R/ 100)T2

by putting the values

⇒ A = 4000(1 + 10 / 100)3 × (1 + 20 / 100)2

⇒ A = 4000(11 / 10)× (6 / 5)2  

⇒ A = 7667 rs. (Ans.) 

3. When the rate of interest is compounded half-yearly rate becomes R / 2% and time becomes 2T

Similarly, when compounded quarterly rate becomes R/ 4% and time becomes 4T. (Similar for simple interest calculations) 

E.g. In what time Rs. 2400 becomes Rs.2640 at 20% compounded half-yearly.

Solution.

Amount =  P(1 + (r / 2) / 100)2t

⇒ 2640 = 2400(1 + (20 / 2) / 100)2t

⇒ 2640/ 2400 = (1.1)2t

⇒ t = 1 year  (Ans.) 

Example: Find the simple interest on Rs. 68,000 at 16(2/3)% per annum for a period of 9 months?

A) Rs. 8500                  B) Rs. 3200                 C) Rs. 2100                 D) Rs. 4300

Answer: Here, P = Rs. 68000, R = 50/3% per annum and T = 9/12 years = 3/4 years. Note that the time has been converted into years as the rate is per annum. The units of rate R and the time T have to be consistent. Now using the formula for the simple interest, we have:

S.I. = [{P×R×T}/100]; therefore we may write: S.I. = Rs. [68000×(50/3)×(3/4)×(1/100)] = Rs. 8500.

Example: The simple interest on a certain sum of money for 2(1/2) years at 12% per annum is Rs. 40 less than the simple interest on the same sum for 3(1/2) years at 10% per annum. Find the sum.

A) Rs. 600                    B) Rs. 666                C) Rs. 780                D) Rs. 800

Answer: Let the sum be Rs. a. Then we can write: [{x×10×7}/{100×2}] – [{x×12×5}/{100×2}] = 40.. This can be written as: 7x/20 – 3x/10 = 4o. Therefore we have x = Rs. 800

Example: Find the compound amount which would be obtained from the interest of Rs.2000 at 6% compounded quarterly for 5 years.

TIP – To calculate interest, the day on which amount is deposited, is not counted but the day on which amount is withdrawn is counted.