Basic relations of simple interestCalling C the capital, M the amount (from the Portuguese montante), J the interest (from juro) and i the rate (from the English interest), we have, for a single period, the following relations [2]:\[ J = C \cdot i \]or:\[ i = \frac{J}{C} \]And, for the amount:\[ M = C + J \]\[ M = C + C \cdot i \]\[ M = C \cdot (1 + i) \]\[ \frac{M}{C} = 1 + i \]When the transaction lasts more than one period, we introduce n, the number of periods. Under simple interest, the interest of each period is always calculated on the initial capital, so it is the same in every period and we just multiply by n:\[ J = C \cdot i \cdot n \]or:\[ i = \frac{J}{C \cdot n} \]The formula for the amount follows:\[ M = C + J \]\[ M = C + C \cdot i \cdot n \]\[ M = C \cdot (1 + i \cdot n) \]\[ \frac{M}{C} = 1 + i \cdot n \]Note: i and n must be in the same unit of time (a monthly rate with the term in months, an annual rate with the term in years).ExamplesA capital of R$ 5,000.00 (Brazilian reais) was invested at simple interest for 3 years, at a rate of 12% p.a. Find the interest and the amount.\[ J = 5{,}000 \cdot 0.12 \cdot 3 = 1{,}800 \]\[ M = 5{,}000 + 1{,}800 = 6{,}800 \]A television is sold for R$ 1,500.00 in cash or in instalments, with a down payment of R$ 300.00 plus one payment of R$ 1,308.00 after three months. What is the monthly simple interest rate of the financing?The down payment is made at once, so only the rest is financed:\[ \text{Financed capital: } C = 1{,}500 - 300 = 1{,}200 \]\[ \text{Amount: } M = 1{,}308 \]\[ \text{Interest: } J = 1{,}308 - 1{,}200 = 108 \]Thus:\[ 108 = 1{,}200 \cdot i \cdot 3 \]\[ i = \frac{108}{1{,}200 \cdot 3} = 0.03 = 3\% \text{ p.m.} \]A capital of R$ 4,000 invested at simple interest of 2% p.m. earns how much interest in 5 months, and what is the amount?Answer: J = R$ 400 and M = R$ 4,400 — J = 4,000 · 0.02 · 5 = 400 and M = 4,000 + 400 = 4,400. R$ 80 is the interest of a single month; R$ 416.32 would be compound interest, in which interest also earns interest.Equivalent ratesIn the simple interest formula, the term must be in the same unit as the rate. We can also do the opposite: express the rate in the unit of the term. To do this, we need to convert rates from one period to another so that they are equivalent [2].Two rates are equivalent when, applied to the same capital for the same term, they produce the same interest.Under simple interest, the conversion is proportional: to go from a monthly to an annual rate, multiply by 12; from annual to monthly, divide by 12; from monthly to daily (commercial month), divide by 30. That is why, under simple interest, equivalent rates and proportional rates are the same thing.ExamplesUnder simple interest, what annual rate is equivalent to 1% p.m.?Equating the interest of one year (12 months) at both rates:\[ C \cdot i_{annual} \cdot 1 = C \cdot 0.01 \cdot 12 \]\[ i_{annual} = 0.01 \cdot 12 = 0.12 = 12\% \text{ p.a.} \]Had we used another term, for example two years, we would reach the same result:\[ C \cdot i_{annual} \cdot 2 = C \cdot 0.01 \cdot 24 \]\[ i_{annual} = \frac{0.01 \cdot 24}{2} = 12\% \text{ p.a.} \]What annual simple interest rate did an investment fund yield, given that the capital invested was R$ 5,000.00 and the redemption value was R$ 5,525.00 after seven months?\[ J = M - C = 5{,}525 - 5{,}000 = 525 \]\[ i_{monthly} = \frac{J}{C \cdot n} = \frac{525}{5{,}000 \cdot 7} = 0.015 = 1.5\% \text{ p.m.} \]\[ i_{annual} = 12 \cdot i_{monthly} = 12 \cdot 1.5\% = 18\% \text{ p.a.} \]Under simple interest, what quarterly rate is equivalent to 30% p.a.?Answer: 7.5% per quarter — a year has 4 quarters and, under simple interest, the conversion is proportional: 30% / 4 = 7.5%. The 2.5% is the monthly rate (30% / 12), and 6.8% would be the compound interest equivalent.Exact interest and commercial interestMany financial transactions last only a few days. In these cases we use the equivalent daily rate, and there are two conventions:Calendar year: 365 days (or 366 in a leap year), with each month having its actual number of days.Commercial year: 360 days, with every month having 30 days.Interest calculated with the first convention is called exact interest; with the second, commercial interest (also known as ordinary interest). In textbooks and exercises, unless stated otherwise, commercial interest is used. Note that, for the same annual rate, commercial interest is slightly higher, because it divides the rate by 360 and not by 365.An example of a very short-term transaction is hot money: loans made by banks to companies, often for a single business day and renewed if necessary, to cover temporary cash needs. In these, the rate is usually quoted per month, and the cost of each day is derived from that monthly rate.ExamplesA capital of R$ 5,000.00 was invested for 42 days at 30% p.a. under simple interest. Find the exact interest and the commercial interest.\[ J_{exact} = \frac{5{,}000 \cdot 0.30 \cdot 42}{365} = 172.60 \]\[ J_{commercial} = \frac{5{,}000 \cdot 0.30 \cdot 42}{360} = 175.00 \]A capital of R$ 4,000.00 was invested at simple interest for 72 days; another capital, of R$ 5,000.00, was invested at the same rate for 45 days. Find the annual rate (commercial interest convention), given that the difference between the interest of the 1st investment and that of the 2nd is R$ 31.50.Calling i the daily rate:\[ J_{1} = 4{,}000 \cdot i \cdot 72 = 288{,}000\,i \]\[ J_{2} = 5{,}000 \cdot i \cdot 45 = 225{,}000\,i \]Therefore:\[ 288{,}000\,i - 225{,}000\,i = 31.50 \]\[ 63{,}000\,i = 31.50 \]\[ i = \frac{31.50}{63{,}000} = 0.0005 = 0.05\% \text{ per day} \]Since the question asks for the annual rate in the commercial year:\[ i_{annual} = 0.05\% \cdot 360 = 18\% \text{ p.a.} \]A note of R$ 10,000 earns simple interest of 36% p.a. for 20 days. What is the difference between commercial interest and exact interest?Answer: R$ 2.74 — commercial interest: 10,000 · 0.36 · 20 / 360 = 200.00; exact interest: 10,000 · 0.36 · 20 / 365 = 197.26. The difference is 200.00 - 197.26 = 2.74. The annual rate is the same, but commercial interest splits the year into 360 days, so each day earns slightly more.ExercisesA loan of $ 28,000 for a term of 121 days was closed at an interest rate of 16% per 365 days, with the interest to be paid together with the repayment of the loan. Calculate the final payment with simple interest.Redo Exercise 1 considering the same simple interest rate of 16%, but per 360 days.To attract customers, a shopkeeper finances purchases for up to 60 days at simple interest of 43% per 365 days. If a customer bought $ 2,650 in goods and wants to pay 40 days after the purchase, calculate the amount the shopkeeper will receive.The bank financed $ 20,000 for 12 months at an interest rate of 3.5% per month. Calculate the interest and the final payment considering simple interest.From an investment of $ 85,000, made at an interest rate of 30% per 360 days, interest will be received every 60 calendar days under simple interest. Calculate the periodic interest.An investment of $ 24,000 for 182 days was closed at an interest rate of 23.5% per 365 days. Calculate the interest and the redemption value of this investment under simple interest.Given an interest rate of 29% per 360 days, calculate the proportional rate per 60 days.Given a daily interest rate of 0.045%, calculate the proportional rate per 360 days.A manufacturer finances payment for up to 60 days from the purchase date, with simple interest at 78% per 365 days. If the goods cost $ 3,250 and the buyer will pay 35 days after the purchase, calculate the amount the manufacturer will receive.$ 15,000 was lent for 12 months at an interest rate of 4% per month. Calculate the interest and the final payment considering simple interest.After 210 days, $ 97,614 was redeemed from an investment of $ 85,000. Calculate the interest rate per 30 days under simple interest.Roberto lent $ 25,000 with the promise of receiving, within three months, the amount lent plus interest at 55% per 365 days. Considering simple interest, calculate the term of this loan if Roberto received $ 27,561.64.An investor receives interest of $ 733.68 every 31 days. Calculate the value of the investment considering an interest rate of 24% per 360 days under simple interest.You invested $ 100,000 at an interest rate of 30% per 360 days, receiving interest every 90 days. Calculate the interest under simple interest.$ 100,000 was invested for 300 days at an interest rate of 2.85% per 30 days. Calculate the redemption value under simple interest.$ 20,000 was invested for 192 consecutive days. Calculate the redemption value considering a constant interest rate of 1.04% per 31 days under simple interest.From an investment of $ 40,000, $ 46,496 was redeemed. If the term of this investment was 290 days under simple interest, calculate the interest rate per 30 days.$ 121,650.06 was redeemed from an investment made at an interest rate of 7.23% per 181 days. Calculate the capital invested if the term was 542 days under simple interest.Calculate the term of a loan with the following characteristics: amount lent $ 32,000, amount repaid $ 38,552, interest rate of 2.45% per 28 days under simple interest.Solutions$ 29,485.15$ 29,505.78$ 2,774.88J = $ 8,400 and M = $ 28,400$ 4,250.00J = $ 2,812.27 and M = $ 26,812.274.833% per 60 days16.20% per 360 days$ 3,493.08J = $ 7,200 and M = $ 22,200i = 2.12% per 30 daysn = 68 daysWith the interest rate i = 2.0667% per 31 days, C = $ 35,500.65J = $ 7,500.00M = $ 128,500M = $ 21,288.26i = 1.68% per 30 daysC = $ 100,000n = 234 daysReferences[1] LAPPONI, Juan Carlos. Matemática financeira. 1998.[2] HAZZAN, Samuel; POMPEO, José Nicolau. Matemática financeira. Saraiva, 2007.
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