Simple and Compound Interest

Simple and compound interest are calculations performed in order to correct the amounts involved in the transactions financial, that is, the correction made when lending or investing a certain amount during a period of time.

The amount paid or redeemed will depend on the fee charged for the transaction and the period the money will be borrowed or invested. The higher the rate and time, the higher this value.

Difference between simple and compound interest

In simple interest, correction is applied to each period and considers only the initial value. In compound interest, correction is made on already corrected amounts.

For this reason, compound interest is also called interest on interest, that is, the amount is adjusted on an amount that has already been adjusted.

Therefore, for longer periods of investment or loan, correction by compound interest will cause the final amount to be received or paid to be greater than the amount obtained with simple interest.

Difference between simple and compound interest over time.
Difference between simple and compound interest over time.

Most financial operations use the correction by the compound interest system. Simple interest is restricted to short-term operations.

Simple interest formula

Simple interest is calculated using the following formula:

bold italic J bold equals bold italic C bold. bold italic i bold. bold italic t

Being,

J: interest
C: initial transaction value, called capital financial mathematics
i: interest rate (amount usually expressed as a percentage)
t: transaction period

We can also calculate the total amount that will be redeemed (in the case of an investment) or the amount to be repaid (in the case of a loan) at the end of a predetermined period.

This value, called the amount, is equal to the sum of the principal plus the interest, that is:

bold italic M bold equals bold italic C bold bolder italic J

We can substitute the value of J in the formula above and find the following expression for the amount:

bold italic M bold equals bold italic C bold plus bold italic C bold. bold italic i bold. bold italic t bold italic M bold equals bold italic C bold space bold left parenthesis bold 1 bold bolder italic i bold. bold italic t bold right parenthesis

The formula we found is an affine function, so the value of the amount grows linearly as a function of time.

Example

If capital of $1000.00 monthly yields $25.00, what is the annual interest rate in the simple interest system?

Solution

First, let's identify each quantity indicated in the problem.

C = BRL 1000.00
J = BRL 25.00
t = 1 month
i = ?

Now that we have identified all the quantities, we can substitute in the interest formula:

J equals C. i. t 25 equals 1000. i.1 i equal to 25 over 1000 i equal to 0 point 025 equal to 2 point 5 percent sign

However, please note that this fee is monthly as we use the 1 month period. To find the annual fee we need to multiply this value by 12, so we have:

i = 2.5.12 = 30% per year

Compound interest formula

The amount capitalized to compound interest is found by applying the following formula:

bold italic M bold equals bold italic C bold space bold left parenthesis bold 1 bold bolder italic i bold right parenthesis to bold power t

Being,

M: amount
C: capital
i: interest rate
t: time period

Unlike simple interest, in this type of capitalization, the formula for calculating the amount involves an exponential variation. Hence it is explained that the final value increases considerably for longer periods.

Example

Calculate the amount produced by R$2,000 applied at the rate of 4% per quarter, after one year, in the compound interest system.

Solution

Identifying the information given, we have:

C = 2000
i = 4% or 0.04 per quarter
t = 1 year = 4 quarters
M = ?

Replacing these values ​​in the compound interest formula, we have:

M equals 2000 space left parenthesis 1 plus 0 comma 04 right parenthesis to the power of 4 M equals 2000.1 comma 1698 M equals 2339 comma 71

Therefore, at the end of one year the amount will be equal to R$2,339.71.

Solved Exercises

question 1

Calculation of amount

What is the amount of an investment of R$500.00, at a rate of 3% per month, in a period of 1 year and 6 months, in simple and compound interest systems?

simple interest

Data:

C = 500

i = 0.03

t = 18 months (1 year + 6 months)

The amount will be the starting capital plus interest.

M = C + J

The interest being:

J = C.i.t

J = 500.0.03.18 = 270

So the amount will be:

M = C+J

M = 500+270

M = 770

Answer: The amount of this application will be R$770.00.

Compound interest

Applying the values ​​in the formula, we have:

M equals C left parenthesis 1 plus i right parenthesis to the power of t space M equals 500 parenthesis left 1 comma 03 right parenthesis to the power of 18 M equal to 500.1 comma 70 M equal to 851 comma 21

Answer: The investment amount under the compound interest regime is R$851.21.

question 2

Capital calculation

A certain capital was applied for a period of 6 months. The rate was 5% per month. After this period, the amount was R$5000.00. Determine the capital.

simple interest

Putting C in evidence in the simple interest formula:

M = C + J

M = C + C.i.t

M = C(1+i.t)

Isolating C into the equation:

C space equal to numerator space M space over denominator left parenthesis 1 plus i. t right parenthesis space end of fraction C space equal to space 4854 comma 37

Compound interest

Isolating C in the compound interest formula and replacing the values:

C equals numerator M over denominator left parenthesis 1 plus i right parenthesis to the power of t end of fraction C equals numerator 5000 over denominator left parenthesis 1 comma 03 right parenthesis to power of 6 end of fraction C equal to numerator 5000 over denominator 1 comma 19 end of fraction C equal to 4201 comma 68

Answer: The capital must be R$4201.68.

question 3

Interest rate calculation

What would the monthly interest rate be on a $100,000 investment over an eight-month period that earned an amount of $1600.00.

simple interest

Applying the formula and putting C in evidence:

M = C + J

M = C + C.i.t

M = C(1+i.t)

Replacing the values ​​and doing the numerical calculations:

m over C space minus 1 space equal to i space. t space space 1 comma 6 space minus space 1 space equal to i space. t space space 0 comma 6 space equal to i space. t space space numerator 0 comma 6 over denominator 8 end of fraction space equal to space i space space 0 comma 075 space equal to space i

in percentage

I = 7.5%

Compound interest

Let's use the formula for compound interest and divide the amount by the principal.

M over C equals left parenthesis 1 plus i right parenthesis to the power of t 1600 over 1000 equals left parenthesis 1 plus i right parenthesis a power of 8 1 comma 6 equals left parenthesis 1 plus i right parenthesis to power 8 radical index 8 of 1 comma 6 end of root equals 1 plus i

question 4

Calculation of application period (time)

A capital of R$8000 was invested at a monthly interest of 9%, obtaining an amount of R$10360.00.

How long was this capital invested?

simple interest

Using the formula

M space equals C space space plus J space space M space minus C space space equals C space. i. t space numerator M space minus space C space space over denominator C. i end of fraction space equal to space t space space numerator 10360 space minus space 8000 space space over denominator 8000.0 comma 09 end of fraction space equals space t space space 3 comma 27 space equals space t

Therefore, the time is approximately 3.27 months.

Compound interest

M equals C left parenthesis 1 plus t right parenthesis cubed M over C equals 1 comma 09 cubed 1 comma 295 equals 1 comma 09 to the power of t

In this step, we are faced with an exponential equation.

To solve it, we will use the logarithm, applying a logarithm of the same base, to both sides of the equation.

l o g 1 comma 295 equal to lo g 1 comma 09 to the power of t

Using a property of the logarithms on the right side of the equation, we have:

log space 1 comma 295 space equals space t space. space log space 1 comma 09 space t space equal to space numerator log space 1 comma 295 space over denominator log space 1 comma 09 end of fraction space space t space equal to space numerator 0 comma 1122 over denominator 0 comma 0374 end of fraction space space t space equal to space 3

question 5

UECE - 2018

A store sells a TV set, with the following payment terms: down payment of R$800.00 and a payment of R$450.00 two months later. If the price of the spot TV is R$1,200.00, then the simple monthly interest rate embedded in the payment is
A) 6.25%.
B) 7.05%.
C) 6.40%.
D) 6.90%.

When comparing the price of the TV in cash (R$1,200.00) and the amount paid in two installments, we observe that there was an increase of R$50.00, as the amount paid was equal to R$1,250.00 (800 +450) .

To find the rate charged, we can apply the simple interest formula, considering that the interest was applied on the debit balance (TV value less down payment). So we have:

C = 1200 - 800 = 400
J = 450 - 400 = 50
t = 2 months

J = C.i.t
50 = 400.i.2
i equal to numerator 50 over denominator 400.2 end of fraction i equal to 50 over 800 i equal to 0 comma 0625 equal to 6 comma 25 percent sign

Alternative: a) 6.25%

Equivalence of capital

In Financial Mathematics, it is essential to keep in mind that the amounts involved in a transaction will be shifted in time.

Given this fact, making a financial analysis implies comparing present values ​​with future values. Thus, we must have a way to make the equivalence of capital at different times.

When we calculate the amount, in the compound interest formula, we are finding the future value for t time periods, at a rate i, from a present value.

This is done by multiplying the term (1+i)no at present value, that is:

bold V with bold F subscript bold equal to bold V with bold P subscript bold left parenthesis bold 1 bold plus bold i bold right parenthesis to the power of bold t

On the contrary, if we want to find the present value knowing the future value, we will do a division, that is:

bold V with bold p subscript bold equal to bold V with bold F subscript over bold left parenthesis bold 1 bold plus bold i bold right parenthesis to the power of bold t

Example:

To buy a motorcycle at a great price, a person asked for a loan of R$ 6,000.00 from a finance company at 15% monthly interest. Two months later, he paid R$3,000.00 and paid off the debt the following month.

What was the amount of the last installment paid by the person?

Solution

If the person has paid off the amount owed on the loan, then the amount paid in the first installment plus the second installment is equal to the amount owed.

However, the installments were adjusted over the period by monthly interest. Therefore, to match these amounts, we have to know their equivalent values ​​on the same date.

We will carry out the equivalence considering the time of the loan, as shown in the diagram below:

Example of compound interest equivalence

Using the formula for two and three months:

V with p subscript equal to V with F subscript over left parenthesis 1 plus i right parenthesis to the power of t 6000 equal to 3000 over left parenthesis 1 plus 0 comma 15 parenthesis right squared plus x over left parenthesis 1 plus 0 comma 15 right parenthesis cubed 6000 space equals space numerator 3000 over denominator 1 comma 3225 end of fraction plus straight numerator x over denominator 1 comma 520875 end of fraction straight numerator x over denominator 1 comma 520875 end of fraction space equal to space 6000 space minus space numerator 3000 over denominator 1 comma 3225 end of fraction straight numerator x over denominator 1 comma 520875 end of fraction space equals space 6000 space minus space 2268 comma 43 straight numerator x over denominator 1 comma 520875 end of fraction space equal to space 3731 comma 56 bold x bold bold space equal to bold bold space 5675 bold bold comma 25

Therefore, the last payment made was R$5,675.25.

Exercise solved

question 6

A loan was made at the monthly rate of i%, using compound interest, in eight fixed installments equal to P.

The debtor has the possibility of repaying the debt in advance at any time, paying for this the current value of the installments still to be paid. After paying the 5th installment, it decides to pay off the debt upon paying the 6th installment.

The expression that corresponds to the total amount paid for the repayment of the loan is:

Question Enem 2017 Compound interest

Answer: Letter a

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