Annual compounding of interest means that interest is computed at the end of every year. This means interest earned during a year is idle. This rate of compounding when increased, results in higher interest earned as the idle period for the interest is reduced.
For instance, Rs. 1,000 invested at the rate of 20% per annum for 2 years compounded annually, results in A=1000×(1+10020)2=1,440
If the above investment is compounded semi-annually (every 6 months), then the rate becomes 10% per semi-annum and the time period is for 4 semi-annums. A=1000×(1+10010)4=1,464.1
In cases where interest is provided as a percentage per annum and the compounding period is not per annum, r and n need to be converted.
In a question with interest of 8% per annum compounded quarterly for 3 years, n=3×4=12 quarters r=48%=2% per quarter
These values can then be substituted into the Compound Interest formula.
Example 9
What is the amount due on Rs. 10,000 loaned for 2 months at interest rate of 24% per annum compounded monthly?
Solution
p=Rs.10,000 n=2 months r=1224%=2% per month
A=p(1+100r)n=10000×1.022= Rs. 10,404
Answer: Rs. 10,404
3.2 Present Value of CI
Present Value of CI is nothing but the value of the Principal when the amount, rate of interest and time period are provided.
p=(1+100r)nA
This is deduced from the Compound Interest formula and need not be separately memorised.
Example 10
The value of an artwork grew at the rate of 25% per annum for 10 consecutive years, ending with a value of Rs. 6,250 at the end of the year 2020. What was the value of the artwork at the start of the year 2018?
Solution
Value of the artwork at the start of 2018 will be the same at the end of 2017. From end of 2017 to end of 2020, the number of years is 3.
Let the value at the start of 2018 be p.
p×(1+10025)3=6250
⇒ p×(45)3=6250
⇒ p=6250×12564=Rs.3200
Answer: Rs. 3,200
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