Simple Interest Compound Interest Notes for Competitive Exams Preparation
Interest:
Interest is money paid to the lender by the borrower for using the money for a specified period of time. Various terms and their general representation are as follows:
- Principal : The original sum borrowed = P
- Time : Time for which money is borrowed = n. (n is expressed in number of periods, which is normally one year)
- Rate of Interest : Rate at which interest is calculated on the original sum = r
- Amount : Sum of Principal and Interest = A
Simple Interest:
When interest is calculated every year (or every time period) on the original principal. i.e., the sum at the beginning of first year, such interest is called Simple Interest. Here, year after year, even though the interest gets accumulated and is due to the lender, this accumulated interest is not taken into account for the purpose of calculating interest for later years.
Simple Interest = Pnr/100 where P, n, r are explained above.
Total Amount A = P+(Pnr/100)
= P(1+(nr/100))
Compound Interest :
In Compound Interest, the interest is added to the principal at the end of each period to arrive at the new principal for the next period.
In other words, the amount at the end of first year (or period) will become the principal for the second year (or period); the amount at the end of second period becomes the principal for the third period and so on.
If P denotes the principal at the beginning of period 1, then,
P at the beginning of year 1= P{1+(r/100)}= PR
P at the beginning of year 2=P{1+(r/1000}2= PR2
P at the beginning of year (n=1) = P{1+(r/100)}n= PRn
Where R={1+(r/100)}
Repayment in equal installments - Simple Interest:
If sum P borrowed, is repaid in 'n' equal instalments, Simple Interest being calculated at r% per year, we can find out the value of each instalment. Normally, this is observed in the sale/purchase of consumer durables. The installments are also paid as monthly instalments.
Instead of buying the item under instalment scheme, if full payment is made at the time of purchase itself, then the amount so paid is referred to as the 'cash down price'.
Under the instalment scheme, the money paid at the time of purchase is called 'down payment'. The difference between the total money paid under the instalment scheme (down payment + all instalments together) and the cash down price is called the 'Instalment charge (I)'.
If n is the number of instalments, then the rate (percent) charged under the instalment scheme can be calculated using the following formula:
r = (24.I.100/n(F+L))
where F= principal left during the first month = cash down price- down payment
and L = principal left during the last month = F-(n-1)xI
Repayment in equal installments- Compound Interest:
If a sum P borrowed, is repaid in 'n' equal instalments compound interest being calculated at r% per period of instalment, we can find out the value of each instalment. The value of teach instalment X (paid at the end of each period) is given by
X = P.r/100[1-{100/(100+r)}n]
Worked out Examples:
1) What is the simple interest on a sum of Rs. 8,000 at an interest of 6% per annum for a period of five years ?
Sol:-
The formula for simple interest (S.I) = PNR/100
Where P is the principal = Rs.8,000/-
R si the rate of interest = 6% p.a.
N is the time period = 5 years
Hence SI = (8000x6x5)/100
= 2400
Simple Interest = Rs.2,400/-
2) A sum of money was invested in a bank ar 8% p.a. simple interest for three years. Instead had it been invested in mutual fund at 8.5% p.a. simple interest for four years, the earnings would have been Rs.500 more. What is the sum invested ?
Sol:-
Let the sum be Rs= p
S.I occurred in the bank = px8x3/100
= 24p/100
Earnings in the form of interest from mutual fund = px8.5x4/100
= 34p/100
Given that = (34p/100)-(24p/100)= Rs.500
= p = 5000
Therefore sum invested = Rs.5,000/-
3) What is the interest earned on a sum of Rs. 20,000 at 10% per annum, for second year, interest being compounded annually ?
Sol:-
Interest for the first year = 10% of 20,000 = Rs.2,000/-
Amount at the end of first year is equal to principal at the beginning of second year.
= 20000+2000= 22000
Interest for the second year = 10% of 22000= 2200
Interest earned on Rs.20,000 for the second year = Rs.2,200/-
4) The simple interest and compound interest on a sum of money at the same at the same rate of interest for two years are Rs.4,800 and Rs.5,160 respectively. If interest is compounded annually, find the rate of interest.
Sol:-
Different in simple interest and compound interest = 5160-4800= Rs.360
The difference is the interest on the first year interest.
Interest for first year = 4800/2= (i.e., S.I/2)= Rs.2,400
360 = r% of 2400
r = 15% p.a.
Important Links:
In other words, the amount at the end of first year (or period) will become the principal for the second year (or period); the amount at the end of second period becomes the principal for the third period and so on.
If P denotes the principal at the beginning of period 1, then,
P at the beginning of year 1= P{1+(r/100)}= PR
P at the beginning of year 2=P{1+(r/1000}2= PR2
P at the beginning of year (n=1) = P{1+(r/100)}n= PRn
Where R={1+(r/100)}
Repayment in equal installments - Simple Interest:
If sum P borrowed, is repaid in 'n' equal instalments, Simple Interest being calculated at r% per year, we can find out the value of each instalment. Normally, this is observed in the sale/purchase of consumer durables. The installments are also paid as monthly instalments.
Instead of buying the item under instalment scheme, if full payment is made at the time of purchase itself, then the amount so paid is referred to as the 'cash down price'.
Under the instalment scheme, the money paid at the time of purchase is called 'down payment'. The difference between the total money paid under the instalment scheme (down payment + all instalments together) and the cash down price is called the 'Instalment charge (I)'.
If n is the number of instalments, then the rate (percent) charged under the instalment scheme can be calculated using the following formula:
r = (24.I.100/n(F+L))
where F= principal left during the first month = cash down price- down payment
and L = principal left during the last month = F-(n-1)xI
Repayment in equal installments- Compound Interest:
If a sum P borrowed, is repaid in 'n' equal instalments compound interest being calculated at r% per period of instalment, we can find out the value of each instalment. The value of teach instalment X (paid at the end of each period) is given by
X = P.r/100[1-{100/(100+r)}n]
Worked out Examples:
1) What is the simple interest on a sum of Rs. 8,000 at an interest of 6% per annum for a period of five years ?
Sol:-
The formula for simple interest (S.I) = PNR/100
Where P is the principal = Rs.8,000/-
R si the rate of interest = 6% p.a.
N is the time period = 5 years
Hence SI = (8000x6x5)/100
= 2400
Simple Interest = Rs.2,400/-
2) A sum of money was invested in a bank ar 8% p.a. simple interest for three years. Instead had it been invested in mutual fund at 8.5% p.a. simple interest for four years, the earnings would have been Rs.500 more. What is the sum invested ?
Sol:-
Let the sum be Rs= p
S.I occurred in the bank = px8x3/100
= 24p/100
Earnings in the form of interest from mutual fund = px8.5x4/100
= 34p/100
Given that = (34p/100)-(24p/100)= Rs.500
= p = 5000
Therefore sum invested = Rs.5,000/-
3) What is the interest earned on a sum of Rs. 20,000 at 10% per annum, for second year, interest being compounded annually ?
Sol:-
Interest for the first year = 10% of 20,000 = Rs.2,000/-
Amount at the end of first year is equal to principal at the beginning of second year.
= 20000+2000= 22000
Interest for the second year = 10% of 22000= 2200
Interest earned on Rs.20,000 for the second year = Rs.2,200/-
4) The simple interest and compound interest on a sum of money at the same at the same rate of interest for two years are Rs.4,800 and Rs.5,160 respectively. If interest is compounded annually, find the rate of interest.
Sol:-
Different in simple interest and compound interest = 5160-4800= Rs.360
The difference is the interest on the first year interest.
Interest for first year = 4800/2= (i.e., S.I/2)= Rs.2,400
360 = r% of 2400
r = 15% p.a.
Important Links:
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tnax sir
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