Definitions

Compound Interest: If the borrower and the lender agree to fix up a certain interval of time (say, a year or a half year or a quarter of a year etc) so that the amount ( = Principal + Interest) at the end of an interval becomes the principal for the next interval, then the total interest over all the intervals, calculated in this way is called the compound interest and is abbreviated as C.I.

Clearly, compound interest at the end of certain specified period is equal to the difference between the amount at the end of the period and the original principal i.e.
                                                          C.I. = Amount — Principal

Conversion Period: The fixed interval of time at the end of which the interest is calculated and added to the principal at the beginning of the interval is called the conversion period.

In other words, the period at the end of which the interest is compounded is called the conversion period.

When the interest is calculated and added to the principal every six months, the conversion period is six months. Similarly, the conversion period is 3 months when the interest is calculated and added quarterly.

Finding CI When Interest Is Compounded Annually

when Interest is compounded yearly, the interest accrued during the first year is added to the principal and the amount so obtained becomes the principal for the second year. The amount at the end of the second year becomes the principal for the third year, and so on.

Example 1: Maria invests Rs 93750 at 9.6% per annum for 3 years and the interest is compounded annually. Calculate:
    (i) The amount standing to her credit at the end of second year.
    (ii) The interest for the third year.
Solution.    (i) We have,
                    Principal for the first year Rs 93750

                    Rate of interest = 9.6% per annum.

Therefore,          Interest for the first year = Rs (\(\frac{93750 \times 9.6 \times 1}{100}\)) = Rs 9000

                    Amount at the end of the first year = Rs 93750 +Rs 9000

                                                                                  = Rs 102750

                    Principal for the second year = Rs 102750

                    Interest for the second year = Rs (\(\frac{102750 \times 9.6 \times 1}{100}\)) = Rs 9864

                    Amount at the end of second year = Rs 102750 + Rs 9864

                                                                                = Rs 112614

            (ii) Principal for the third year = Rs 112614

                    Interest for the third year =Rs (\(\frac{112614 \times 9.6 \times 1}{100}\) ) = Rs 10810.94

Example 2: Find the compound interest on Rs 25000 for 3 years at 10% per annum, compounded annually.

Solution.    Principal for the first year = Rs 25000

            Interest for the first year  = (\(\frac{25000 \times 10 \times 1}{100}\)) = Rs 2500

            Amount at the end of the first year = (25000 + 2500) = Rs 27500

            Principal for the second year = Rs 27500

            Interest for the second year = (\(\frac{27500 \times 10 \times 1}{100}\)) = Rs 2750

            Amount at the end of the second year = (27500 + 2750) = Rs 30250.

            Principal for the third year = Rs 30250.

            Interest for the third year = (\(\frac{30250 \times 10 \times 1}{100}\)) = Rs 3025

            Amount at the end of the third year = (30250 + 3025) = Rs 33275.

    Therefore, compound interest = (33275 — 25000) = Rs 8275

Finding CI When Interest Is Compounded Half-Yearly

If the rate of Interest is R% per annum then it is clearly (\(\frac{R}{2}\))% per half-year.
The amount after the first half-year becomes the principal for the next half-year, and so on.

Example 3: Find the compound interest on Rs 5000 for 1 year at 8% per annum, compound half-yearly.

Solution.    Rate of interest             = 8% per annum = 4% per half-year.

            Time                                          = 1 year = 2 half-years.

            Original principal                    = Rs 5000.

            Interest for the first half-year = (\(\frac{5000 \times 4 \times 1}{100}\)) = Rs 200.

            Amount at the end of the first half-year = (5000 + 200) = 5200.
            Principal for the second half-year = Rs 5200.

            Interest for the second half-year = ((\(\frac{5200 \times 4 \times 1}{100}\))) = Rs 208.

            Amount at the end of the second half-year = Rs (5200 + 208) = Rs 5408.
Therefore,        compound interest = Rs (5408 — 5000) = Rs 408.

Example 4: Find the compound interest on Rs 8000 for \(1\frac{1}{2}\) years at 10% per annum, interest being payable half-yearly.

Solution.    We have,

   Rate of interest                = 10% per annum = 5% per half-year.

   Time                                  = \(1\frac{1}{2}\) years = \(\frac{3}{2}\) x 2 = 3 half- years

   Original principal             = Rs 8000

   Interest for the first half-year = Rs ( \(\frac{8000 \times 5 \times 1}{100}\)) = Rs 400

   Amount at the end of the first half-year = Rs 8000 + R 400 = Rs 8400
   Principal for the second half-year = Rs 8400

   Interest for the second half-year = Rs (\(\frac{8400 \times 5 \times 1}{100}\)) = Rs 420

   Amount at the end of the second half-year = Rs 8400 + Rs 420 = Rs 8820
   Principal for the third half-year = Rs 8820

   Interest for the third half-year = Rs (\(\frac{8820 \times 5 \times 1}{100}\)) = Rs 441

   Amount at the end of third half-year = Rs 8820 + Rs 441 = Rs 9261
Therefore,        Compound interest = Rs 9261 — Rs 8000 = Rs 1261

Finding CI When Interest Is Compounded Quarterly

If the rate of interest is R % per annum and the interest is compounded quarterly, then it is \(\frac{R}{4}\) %  per quarter.

Example 5: Find the compound interest on Rs 10000 for 1 year at 20% per annum compounded quarterly.

Solution.    We have,

                Rate of interest             = 20% per annum = \(\frac{1}{5}\)% = 5% per quarter

                Time                         = 1 year = 4 quarters.

                Principal for the first quarter = Rs 10000

                Interest for the first quarter = Rs (\(\frac{10000 \times 5 \times 1}{100}\)) = Rs 500

                Amount at the end of first quarter = Rs 10000 + Rs 500 = Rs 10500

                Principal for the second quarter = Rs 10500

                Interest for the second quarter = Rs (\(\frac{10500 \times 5 \times 1}{100}\)) = Rs 525

                Amount at the end of second quarter = Rs 10500 + Rs 525 = Rs 11025
                Principal for the third quarter = Rs 11025

                Interest for the third quarter = Rs (\(\frac{11025 \times 5 \times 1}{100}\)) = Rs 551.25

                Amount at the end of the third quarter = Rs 11025 + Rs 551.25

                                                                                     = Rs 11576.25

                Principal for the fourth quarter = Rs 11576.25

                Interest for the fourth quarter = Rs (\(\frac{11576.25 \times 5 \times 1}{100}\)) = Rs 578.8125

                Amount at the end of fourth quarter = Rs 11576.25 + Rs 578.8125

                                                                                 = Rs 12155.0625

Therefore,        Compound interest = Rs 12155.0625 — Rs 10000

                                                             = Rs 2155.0625

                                                             = Rs 2155.06