divide rupees 21866 into two parts such that the amount of 1 year in 3 years is the same as the amount of the second in 5 years. The rate of CI being 5% per annum..
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Interest: It is the additional money besides the original money paid by the borrower to the money lender in lieu of the money used.
Principal: The money borrowed (or the money lent) is called principal.
Amount: The sum of the principal and the interest is called amount.
Thus, amount = principal +interest.
Rate: It is the interest paid on Rs 100 for a specified period.
Time: It is the time for which the money is borrowed.
Simple Interest: It is the interest calculated on the original money (principal) for any given time and rate.
Formula: Simple Interest = (Principal x Rate x time)/100
Compound interest
Compound interest (abbreviated C.I.) can be easily calculated by the following formula:
A = P where A is the final amount, P is the principal, r is the rate of interest compounded yearly and n is the number of years.
C.I. = A -P =
Remark.
Interest may be converted into principal annually, semiannually, quarterly, monthly etc. The number of times interest is converted into principal in a year is called the frequency of conversion, and the period of time between two conversions is called the conversion period or interest period. Thus "rate of 5%" means a rate of 5% compounded annually; 12% compounded semi-annually means that each interest period of 6 months earns an interest of 6%. Thus the rate of interest per interest period is
r = (annual rate of interest) / (frequency of conversion)
and the number of interest periods is
n = (given number of years) x (frequency of conversion).
In solving problems on compound interest, remember the following:
1. A = P and C.I. =
where A is the final amount, P is the principal, r is the rate of interest compounded yearly (or every interest period) and n is the number of years (or terms of the interest period).
2. When the interest rates for the successive fixed periods are r1 %, r2 %, r3 %, ..., then the final amount A is given by
A =
3. S.I. (simple interest) and C.I. are equal for the first year (or the first term of the interest period) on the same sum and at the same rate.
4. C.I. of 2nd year (or the second term of the interest period) is more than the C.I. of Ist year (or the first term of the interest period), and C.I. of 2nd year -C.I. of Ist year = S.I. on the interest of the first year.
5. Equivalent, nominal and effective rates of interest
Two annual rates of interest with different conversion periods are called equivalent if they yield the same compound amount at the end of the year. For example, consider an amount of Rs 10,000 invested at 4% interest compounded quarterly. So, the amount at the end of one year = 10000(1·01)4 = 10406. This is equivalent to interest of 4·06% compounded annually because 10000(1·0406) = 10406.
When interest is compounded more than once in a year, the given annual rate is called nominal rate or nominal annual rate. The rate of interest actually earned is called effective rate. In the above example, nominal rate is 4% while effective rate is 4·06%.
If nominal rate is r% compounded p times in year, then effective rate of interest is
6. Present value or present worth of a sum of Rs P due n years hence at r% compound interest is
P.V. =
In particular, present value of sum of a Rs P due one year hence (i.e. n = 1) at r% (compound) interest is
P.V.=
7. Equal instalments (with compound interest)
Loan amount = , where
P = each equal instalment
R = rate of interest per annum (or per interest period)
T = time, say 3 years (or 3 interest terms).
Note. If T = n years (or interest terms), then there will be n brackets.
8. Formulae for population
If the present population of a town is P and its annual increase is r%, the population after n years will be P , and n years ago, the population was .
If, however, there is annual decrease of r% per annum, the population after n years will be , and n years ago, the population was .
Depreciation
All fixed assets such as machinery, building, furniture etc. gradually diminish in value as they get old
Principal: The money borrowed (or the money lent) is called principal.
Amount: The sum of the principal and the interest is called amount.
Thus, amount = principal +interest.
Rate: It is the interest paid on Rs 100 for a specified period.
Time: It is the time for which the money is borrowed.
Simple Interest: It is the interest calculated on the original money (principal) for any given time and rate.
Formula: Simple Interest = (Principal x Rate x time)/100
Compound interest
Compound interest (abbreviated C.I.) can be easily calculated by the following formula:
A = P where A is the final amount, P is the principal, r is the rate of interest compounded yearly and n is the number of years.
C.I. = A -P =
Remark.
Interest may be converted into principal annually, semiannually, quarterly, monthly etc. The number of times interest is converted into principal in a year is called the frequency of conversion, and the period of time between two conversions is called the conversion period or interest period. Thus "rate of 5%" means a rate of 5% compounded annually; 12% compounded semi-annually means that each interest period of 6 months earns an interest of 6%. Thus the rate of interest per interest period is
r = (annual rate of interest) / (frequency of conversion)
and the number of interest periods is
n = (given number of years) x (frequency of conversion).
In solving problems on compound interest, remember the following:
1. A = P and C.I. =
where A is the final amount, P is the principal, r is the rate of interest compounded yearly (or every interest period) and n is the number of years (or terms of the interest period).
2. When the interest rates for the successive fixed periods are r1 %, r2 %, r3 %, ..., then the final amount A is given by
A =
3. S.I. (simple interest) and C.I. are equal for the first year (or the first term of the interest period) on the same sum and at the same rate.
4. C.I. of 2nd year (or the second term of the interest period) is more than the C.I. of Ist year (or the first term of the interest period), and C.I. of 2nd year -C.I. of Ist year = S.I. on the interest of the first year.
5. Equivalent, nominal and effective rates of interest
Two annual rates of interest with different conversion periods are called equivalent if they yield the same compound amount at the end of the year. For example, consider an amount of Rs 10,000 invested at 4% interest compounded quarterly. So, the amount at the end of one year = 10000(1·01)4 = 10406. This is equivalent to interest of 4·06% compounded annually because 10000(1·0406) = 10406.
When interest is compounded more than once in a year, the given annual rate is called nominal rate or nominal annual rate. The rate of interest actually earned is called effective rate. In the above example, nominal rate is 4% while effective rate is 4·06%.
If nominal rate is r% compounded p times in year, then effective rate of interest is
6. Present value or present worth of a sum of Rs P due n years hence at r% compound interest is
P.V. =
In particular, present value of sum of a Rs P due one year hence (i.e. n = 1) at r% (compound) interest is
P.V.=
7. Equal instalments (with compound interest)
Loan amount = , where
P = each equal instalment
R = rate of interest per annum (or per interest period)
T = time, say 3 years (or 3 interest terms).
Note. If T = n years (or interest terms), then there will be n brackets.
8. Formulae for population
If the present population of a town is P and its annual increase is r%, the population after n years will be P , and n years ago, the population was .
If, however, there is annual decrease of r% per annum, the population after n years will be , and n years ago, the population was .
Depreciation
All fixed assets such as machinery, building, furniture etc. gradually diminish in value as they get old
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