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Introduction to Business Math

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|This link is concerned with interest, interest rates and their effect on the value of money. Companies pay millions of rupees as interest each year for use |

|of money which they borrowed. We earn profit on money which we have invested in saving accounts, certificates of deposit etc… Interest is a fee (rent), |

|which is paid for the use of money. We pay interest on the loans drawn from banks. Similarly, bank pays us interest on money deposited in savings accounts, |

|etc… The money which is invested or lent is called principal or Capital. Interest is paid usually in proportion to the principal over the period for which |

|money is used. Interest rate specifies the rate at which interest accumulates. Interest rate normally expressed as a percentage of principal per period of |

|time e.g. 18% per month. 5% per year etc and sum of principal of interest is called amount. Interest rate actually tells us that how much amount is |

|increased for every Rs.100. Interest is further divided into two parts: Simple Interest & Compound Interest |

Concept of Simple Interest

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|            When interest is calculated for every period only on the principal then the total interest gained on all the period is called simple interest. |

|            The principal plus simple interest gained is called amount. |

|                        i.e. Amount = Principal + Simple Interest |

|                                     A = P + S.I |

|            Before derive the formula for finding simple interest and the amount, we consider an example first. Suppose a person borrowed Rs.100 with 10% |

|rate of simple interest for 3 years. |

|            Then simple interest for 1st year = [pic] |

|            Amount at the end of 1st year = 100 + 10 = 110 |

|            Amount at the end of 2nd year = 110 + 10 = 120 |

|            Amount at the end of 3rd year = 120 + 10 = 130 |

|             |

|            i.e. every time the interest of Rs.10 calculated on the principal of Rs.100 is added to get the amount, after 1st, 2nd and 3rd year. |

|            Total simple interest gained in 3 years = 3 × 10 = 30 |

|            It can be obtained as |

|                        S.I = Amount – Principal = 130 – 100 = Rs.30 |

| |

|Now, using above concept we can drive the formula of simple interest and the amount. |

|            Suppose, |

|                        Principal = P |

|                        Rate of Interest = r |

|                        Number of periods = n |

|                        Simple Interest of 1st period = Pr (same for all periods) = S.I |

|                        Amount at the end of 1st period = Principal + Interest |

|                                                                            = P + Pr = A1 |

|                        Amount at the end of 2nd period = A1 + S.I = (P + Pr) + Pr |

|                                                                             = P + 2Pr = A2 |

|                        Amount at the end of 3rd period = A2 + S.I = (P + 2Pr) + Pr |

|                                                                            = P + 3Pr = A3 |

|                        And so on |

|              |

Examples of Simple Interest

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|Example 1: |

|            Find simple interest on Rs. 3000 at 7% rate of interest for one year. |

|Solution: |

|Let       Principal = 3000 |

|            Rate of interest = 7% |

|            [pic] |

|            Simple interest  [pic] |

|                                       [pic] |

|                                       [pic] |

|                                       [pic] |

| |

|Example 2: |

|            Find simple interest on Rs. 10,000 at the rate of 5% for 5 years. Also find the amount for 5 years. |

|Solution: |

|Let       Principal = 10,000 Rs. |

|            Rate = 5% |

|            Time [pic] |

|            Amount of simple interest for 5 years is |

|            Interest    [pic] |

|                            [pic] |

|                            [pic] |

|            Hence the amount after 5 years    [pic] |

|                                                                   [pic] |

|                                                                   [pic] |

| |

|Example 3: |

|            Find simple interest on Rs. 156,00 for [pic] years at the rate of 5% per annum. Also find total amount. |

|Solution: |

|Let       Principal = 15,600 |

|            Rate = 5%   [pic]   [pic] |

|            Time = [pic]years   [pic]     [pic] |

|            Simple interest for 5 years     [pic] |

|                                                            [pic] |

|                                                            [pic] |

|            Amount    [pic] |

|                              [pic]   [pic] |

| |

|Example 4: |

|            Find simple interest on Rs. 8,000 for 40 days, at 10% per annum. |

|Solution: |

|Let       Principal = 8,000 Rs. |

|            Rate = 10% per annum |

|            Time = 40 days  [pic]    [pic] |

|            Simple interest   [pic] |

|                                       [pic]    [pic] |

| |

| |

More Examples of Simple Interest

[pic]

|To Find Principal |

|            When interest, time, rate are given: |

|Example 5: |

|What amount was borrowed, for which amount of interest is 200 at the rate of 5% for 4 years. |

|Solution: |

|Let       P = Principal               Rate = 5% |

|            Interest = 200                         time = n = 4 |

|            I = Interest |

|            We know that |

|                        [pic] |

|            [pic] |

|To Find Principal |

|            When amount, rate, time are given: |

|Example 6: |

|            If the amount for 2 years at 6% is 4,000, what was the principal? |

|Solution: |

|Let       P = Principal               Rate = 6% |

|            n = 2                            A = 4,000 |

|            We know that |

|            [pic] |

|            [pic] |

|[pic]       [pic] |

|            [pic] |

|            [pic] |

|To Find Rate |

|            When interest, time, principal are given: |

|Example 7: |

|            The amount of simple interest for Rs.15,000 for 2 years is 1000, find rate of interest. |

|Solution: |

|Let       Principal = 15,000 |

|            Simple interest = I = 1000 |

|            Time = 2 years |

|            Interest = P r n |

|[pic]       [pic] |

|[pic]       [pic] |

|Example 8: |

|            At what rate would a sum of money double in 20 years? |

|Solution: |

|Let       Principal = P |

|            Amount after 20 years = A = 2P |

|            Time = n = 20 years |

|            But, we know |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] per annum. |

|To Find Time |

|            When simple interest, principal and rate are given: |

|            We have |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] |

|Example 9: |

|            Suppose Rs.700 are invested at 4% per annum. How long will it take for the amount to reach 784? |

|Solution: |

|Let       P = 700 |

|            A = 784 |

|            r = 4% |

|            We know that |

|            [pic] |

|Example 10: |

|            How long will it take for Rs. 2000 to amount 2360 at 6% per annum simple interest. |

|Solution: |

|Let       P = 2000 |

|            A = 2360 |

|            r = 6% |

|            we know that |

|            [pic] |

| |

|Concept of Compound Interest |

|[pic] |

|Compound Interest: |

|            When the interest is calculated for every period on the total previous amount, then the total amount of interest gained on all the periods is |

|called compound interest. |

|Let       Principal = P |

|            Rate of interest = r |

|            No. of years = n |

|            After 1st year,             interest     [pic] |

|            After 1st year, Amount   [pic] |

|            Now principal becomes, |

|                                    Principal   [pic] |

|            After 2nd year,            interest      [pic] |

|            After 2nd year, Amount    [pic] |

|                                                      [pic] |

|                                                      [pic] |

|            and so on. |

|            Amount after n-years is |

| |

|            [pic] |

|Hence |

|            [pic] |

| |

| |

Examples of Compound Interest

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|Example 11: |

|            Find the compound amount and compound interest on the principal 20,000 borrowed at 6% compounded annually for 3 years. |

|Solution: |

|            Let       P = 20000 |

|                        r  = 6% |

|                        n = 3 |

|                        using [pic] |

|                                      [pic] |

|                                      [pic] (compound amount) |

|Compound interest      [pic] |

| |

|Example 12: |

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

|Solution: |

|            Let       Principal = 2000 |

|                        r = 6%             [pic] |

|            [pic] |

|[pic]         [pic] |

| |

|Example 13: |

|            Find compound interest on Rs.2500 invested at 6% per annually, compound semi-annually for 8 years. |

|Solution: |

|            Let       Principal = 2500 |

|                        r =6% [pic] |

|                        [pic] |

|            We know that |

|                        [pic] |

|                            [pic] |

|                            [pic] |

|[pic]         Compound interest   [pic] |

| |

More Examples of Compound Interest

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|To Find Principal |

|            We are given n, A, and r |

|            We have to find P by the formula |

|                        [pic] |

|            [pic]       [pic] |

|Example 14: |

|            Find principal, when compound interest for 4 years at 6% is 525. |

|Solution: |

|            Let       Principal = P |

|                        [pic] |

|                        r = 6% |

|            C.I = 525 |

|We know that  |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] |

|            [pic] |

|Example 15: |

|            Find principal of Rs.4635.50 due at the end of 3 years at 5% annually. |

|Solution: |

|            Let       Principal = P |

|                        A = 4635.50 |

|                        r = 5% |

|                        [pic] |

|            Using [pic] |

|            [pic]       [pic] |

|            [pic]       [pic] |

|To Find Time |

|            When P, A, r are given |

|            We have to find time by the same formula. |

|            [pic] |

|Example 16: |

|            In how many years a sum of 600 would amount to 757.48 at 6% compounded annually. |

|Solution: |

|            Let       Principal = 600 |

|                        A = 757.48 |

|                        [pic] |

|                        r = 6% |

|            Using [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|            Taking log both sides |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|To Find Rate |

|            When P, n, A are given |

|            We have to find r by the using same formula |

|                        [pic] |

|Example 17: |

|            Find the rate of interest compounded per annum for 3 years so that 4000 becomes 4635.50 |

|Solution: |

|            Let       A = 4635.50 |

|                        P = 4000 |

|                        [pic] |

|                        r = ? |

|            Using [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

|                        [pic] |

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