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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
[pic]
| 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
[pic]
|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
[pic]
|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
[pic]
|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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