Basic RelationshipsCalling C the principal, M the amount, J the interest, and i the interest rate (from the English word interest), we have the following relationships according to the definitions above: [2\]\[ J = Ci \]Or:\[ i = \frac{J}{C} \]We also have the following equations:\[ M = C + J \]\[ M = C + Ci \]\[ M = C · (1 + i) \]\[ \frac{M}{C} = 1 + i \]When we deal with more than one time period, we must introduce n into the formula, where n represents n time periods:\[ J = Cin \]or:\[ i = \frac{J}{Cn} \]The amount formula follows immediately:\[ M = C + J \]\[ M = C + Cin \]\[ M = C · (1 + in) \]\[ \frac{M}{C} = 1 + in \]ExamplesA principal of R$5,000.00 was invested at simple interest for 3 years at a rate of 12% per year. Find the interest and the amount.\[ J = 5.000 · (0,12) · 3 = 1.800 \]\[ M = 5.000 + 1.800 = 6.800 \]A television is sold for cash at R$1,500.00 or on credit with a down payment of R$300.00 plus one installment of R$1,308.00 after three months. What is the monthly simple interest rate of the financing?\[ \text{Financed principal: } C = 1.500 - 300 = 1.200 \]\[ \text{Amount: } M = 1.308 \]\[ \text{Interest: } J = 1.308 - 1200 = 108 \]Thus:\[ 108 = 1.200 · 3 · i \]\[ i = \frac{108}{1200 · 3} = 0,03 = 3% \text{ per month} \]Equivalent RatesIn the simple interest formula, we know that the time period must be expressed in the same unit as the interest rate. The reverse is also true, that is, we can express the interest rate in the same unit as the time period. To do this, we must know how to convert rates from one period to another so that they are equivalent. \[2\]We say that two rates are equivalent when, applied to the same principal over the same period, they produce the same amount of interest.In simple interest, it is enough to multiply or divide by the equivalent time.ExamplesIn simple interest, what annual rate is equivalent to 1% per month?\[ C i_{annual} 1 = C · (0,01) · 12\]\[ i_{annual} = (0,01) · 12 = 12% \text{ per year}\]Notice that if we had adopted another period, for example, two years, we would arrive at the same result. The corresponding equation would be:\[ C i_{annual} 2 = C · (0,01) · 24 \]\[ i_{annual} = \frac{(0,01) · 24}{2} = 12% \text{ per year} \]What annual simple interest rate did an investment fund earn, knowing that the invested principal was R$ 5,000.00 and that the redemption value was R$ 5,525.00 after seven months?\[ J = M - C = 5.525 - 5.000 = 525 \]\[ i_{monthly} = \frac{J}{C n} = \frac{525}{5.000 · 7} = 0,015 = 1,5% \text{ per month} \]\[ i_{annual} = 12 · i_{monthly} = 12 · 1,5% = 18% \text{ per year}\]Exact Interest and Ordinary InterestIt is very common for financial transactions to last only one or a few days. In this case, we must use the equivalent daily rate. Two conventions may be adopted:* Considering the civil year, which has 365 days (or 366), and each month with its actual number of days.* Considering the commercial year, with 360 days, and each commercial month with 30 days.The first is called exact interest, while the second is called ordinary interest. In general, ordinary interest is adopted.Hot money transactions, which are very short-term loans, normally use compound interest with rates expressed on a monthly basis. These transactions are often carried out for a single business day, with possible renewals when necessary, in order to meet companies' temporary cash flow needs.ExamplesA principal of R$ 5,000.00 was invested for 42 days at a rate of 30% per year under the simple interest system. Find the exact interest and the ordinary interest.\[ J_{exact} = \frac{5.000 · 0,30 · 42}{365} = 172,60\]\[ J_{ordinary} = \frac{5.000 · 0,30 · 42}{360} = 175,00\]A principal of R$ 4,000.00 was invested at simple interest for 72 days; another principal of R$ 5,000.00 was also invested at simple interest, at the same rate, for 45 days. Determine the annual interest rate (using the ordinary interest convention), knowing that the difference between the interest from the first investment and the second is R$ 31.50.\[ J_{1º} = 4.000 · i · 72 = 288.000i\]\[ J_{2º} = 5.000 · i · 45 = 225.000i \]Therefore:\[ 288.000i - 225.000i = 31,50 \]\[ 63.000i = 31,50\]\[ i = \frac{31,50}{63.000} = 0,0005 = 0,05% \text{ per day} \]ExercisesA loan of $28,000 for a term of 121 days was contracted at an interest rate of 16% per 365 days, with the condition that the interest would be paid together with the repayment of the loan. Calculate the final payment using simple interest.Redo Exercise 1 considering the same simple interest rate of 16% with a 360-day period.To attract customers, the retailer offers financing for purchases of up to 60 days using simple interest at a rate of 43% per 365 days. If a customer purchased $2,650 in various goods and wants to pay 40 days after the purchase date, calculate the amount to be received by the retailer.The bank financed $20,000 for a period of 12 months at an interest rate of 3.5% per month. Calculate the interest and the final payment amount using simple interest.From an investment of $85,000 made at an interest rate of 30% per 360 days, interest will be received every 60 consecutive days under the simple interest system. Calculate the periodic interest amount.An investment of $24,000 for a period of 182 days was made at an interest rate of 23.5% per 365 days. Calculate the interest and the redemption value of this investment under the simple interest system.Given an interest rate of 29% per 360 days, calculate the proportional interest rate for 60 days.Given a daily interest rate of 0.045%, calculate the proportional interest rate for 360 days.The manufacturer finances payment for the sale up to 60 days after the purchase date with interest calculated at a rate of 78% per 365 days under simple interest. If the merchandise is worth $3,250 and the buyer will pay 35 days after the purchase date, calculate the amount to be received by the manufacturer.$15,000 was loaned for a period of 12 months at an interest rate of 4% per month. Calculate the interest and the final payment amount using simple interest.After 210 days, $97,614 was redeemed from an investment of $85,000. Calculate the interest rate for a 30-day period under the simple interest system.Roberto loaned $25,000 with the promise of receiving, before three months, the principal plus the corresponding interest at a rate of 55% per 365 days. Assuming simple interest, calculate the term of this loan if Roberto received $27,561.64.The investor receives $733.68 in interest every 31 days. Calculate the investment amount considering an interest rate of 24% per 360 days under the simple interest system.You invested $100,000 at an interest rate of 30% per 360 days with the condition of receiving interest every 90 days. Calculate the interest amount under the simple interest system.$100,000 was invested for 300 days under the simple interest system at an interest rate of 2.85% per 30 days. Calculate the redemption value under the simple interest system.$20,000 was invested for a period of 192 consecutive days. Calculate the redemption value considering a constant interest rate of 1.04% per 31 days under the simple interest system.$46,496 was redeemed from an investment of $40,000. If the investment period was 290 days under the simple interest system, calculate the interest rate for a 30-day period.$121,650.06 was redeemed from an investment made at an interest rate of 7.23% per 181 days. Calculate the invested principal if the investment period was 542 days under the simple interest system.Calculate the term of the loan with the following characteristics: amount borrowed $32,000, amount repaid $38,552, interest rate of 2.45% per 28 days under the simple interest system.Solutions\[youtube\](https://www.youtube.com/embed/qT5zt_kYLjo)$29,485.15.$29,505.78.$2,774.88.$8,400 and $28,400.$4,250.00.$2,812.27 and $26,812.27.4.833% per 60 days.16.20% per 360 days.$3,493.08.J = $7,200 and M = $22,200.i = 2.12% per 30 days.n = 68 days.With an interest rate of i = 2.0667% per 31 days, C = $35,500.65.J = $7,500.00.M = $128,500.M = $21,288.26.i = 1.68% per 30 days.C = $100,000.n = 234 days.References[1] LAPPONI, Juan Carlos. Financial Mathematics. 1998.[2] HAZZAN, Samuel; POMPEO, José Nicolau. Financial Mathematics. Saraiva, 2007.
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