# 3 ways to calculate interest

Most people know the concept of interest, but not all know how to calculate it. Interest is the value we add to a loan or deposit that must be paid for using someone else's money. There are three basic ways to calculate it: simple interest, which is the simplest calculation (generally used for short-term loans); compound interest, which is a bit more complicated and more valuable; and finally the continuous compound interest, which grows at the fastest rate and is the formula used in most banks for mortgage loans. Generally, the information needed to perform any of these calculations is the same, but the mathematical procedures vary somewhat for each one.

## Steps

### Method 1 of 3: Calculate Simple Interest

#### Step 1. Determine the principal

The principal is the amount of money that you will use to calculate the interest. This amount can be what you deposit in a savings account or a bond of some kind. In that case, you will get the interest you calculate. Also, if you borrow money, such as for a mortgage payment, the principal will be the amount you asked for, and you must calculate the interest you have.

• In any case, whether you charge the interest or pay it, the principal amount is usually symbolized by the variable P.
• For example, if you made a loan of \$ 2,000 to a friend, the principal borrowed would be \$ 2,000.

#### Step 2. Determine the interest rate

Before you can calculate the amount of principal that you will appreciate, you need to know the rate at which that principal will grow. This is the interest rate, which is usually announced or agreed between the parties before the execution of the loan.

• For example, suppose you lent money to a friend knowing that, after 6 months, he will return the 2000 dollars plus 1.5%. This means that the one-time interest rate is 1.5%. However, before you can use it, you must convert it to decimals. To do this, divide the percentage by 100:

### 1, 5 % ÷ 100 = 0, 015

#### Step 3. Measure the term of the loan

The term is another name for the duration of the loan. In some cases, you will have to agree on the duration of the loan at the time it is made. For example, most mortgages have a defined term. In the case of many private loans, the borrower and the lender can agree on any term they want.

• It is important that the duration of the term coincides with the interest rate or at least is measured with the same unit. For example, if the interest rate is one year, the term must also be measured in years. For example, if the rate is fixed at 3% per year, but the loan is only for 6 months, then you will have to calculate an annual interest rate of 3% for a term of 0.5 years.
• As another example, if a rate of 1% per month is agreed and you borrow money for six months, the term you must calculate will be 6.

#### Step 4. Calculate the interest

To do this, multiply the principal by the interest rate and the term of the loan. You can express this formula algebraically as follows:

• I = P ∗ r ∗ t { displaystyle I = P * r * t}

• Utilizando el ejemplo anterior del préstamo realizado a un amigo, el principal (P{displaystyle P}
• ) es de 2000 dólares, mientras que la tasa (r{displaystyle r}

) es de 0, 015 durante seis meses. Debido a que el acuerdo realizado en este ejemplo fue durante un plazo único de seis meses, en este caso, la variable t{displaystyle t}

es de 1. A continuación, calcula el interés de la siguiente manera:

• I=Prt=(2000)(0, 015)(1)=30{displaystyle I=Prt=(2000)(0, 015)(1)=30}
• . Por lo tanto, el interés a pagar es de 30 dólares.

• Si quieres calcular la cantidad del pago total (A), con el interés y el retorno del principal, utiliza la siguiente fórmula: A=P(1+rt){displaystyle A=P(1+rt)}
• . Este cálculo se verá de la siguiente forma:

• A=P(1+rt){displaystyle A=P(1+rt)}
• A=2000(1+0, 015∗1){displaystyle A=2000(1+0, 015*1)}
• A=2000(1, 015){displaystyle A=2000(1, 015)}
• A=2030{displaystyle A=2030}

#### Step 5. Try another example

For practice purposes only, let's say you deposit \$ 5,000 into a savings account with an annual interest rate of 3%. After just three months, you withdraw the money along with the interest that has been generated in that time.

• A = P (1 + rt) { displaystyle A = P (1 + rt)}

• A=5000(1+0, 03∗0, 25){displaystyle A=5000(1+0, 03*0, 25)}
• A=5000(1, 0075){displaystyle A=5000(1, 0075)}
• A=5037, 5{displaystyle A=5037, 5}
• En tres meses, obtendrías un interés de 37, 50 dólares.
• Ten en cuenta que, en este ejemplo, t=0, 25, pues tres meses equivale a un cuarto (0, 25) del plazo original de un año.

### Método 2 de 3: Calcular el interés compuesto

#### Step 1. Understand the meaning of compound interest

In this case, as you earn interest, it returns to the account and you begin to earn (or pay) interest on top of the same interest. As a simple example, if you deposit \$ 100 with an annual interest rate of 5%, after one year you will get \$ 5 of interest. If you return that amount to the account, at the end of the second year you will get 5% of \$ 105 instead of the original 100. In the long run, this amount can increase considerably.

• The formula to calculate the value (A) of compound interest is as follows:

• A = P (1 + rn) nt { displaystyle A = P (1 + { frac {r} {n}}) ^ {nt}}

#### Step 2. Determine the principal amount

As with simple interest, begin the calculation with the amount of the principal. This calculation is the same regardless of whether you calculate the interest on the borrowed money. Generally, this quantity is expressed by the variable P { displaystyle P}

#### Step 3. Measure the interest rate

The interest rate must be agreed to from the beginning and presented in a decimal number in order to be calculated. Remember that you can convert the percentage to a decimal by dividing it by 100 (or moving the decimal point two places to the left). Make sure you know the duration of the interest rate, which is expressed algebraically as r { displaystyle r}

### Por ejemplo, una tarjeta de crédito puede tener una tasa de interés anual del 15 %. No obstante, el interés suele aplicarse cada mes, así que posiblemente quieras saber la tasa de interés mensual. En ese caso, divídelo entre 12 para hallar la tasa para hallar la tasa de interés mensual de 1, 25 %. Estas dos tasas, 15 % al año 1, 25 % al mes, son equivalentes

#### Step 4. Determine when the interest will be compounded

In this case, the interest will be calculated periodically and added back to the principal amount. For some loans, this can happen once a year, while for others it can happen every month or quarter. You will need to know how many times a year the interest will become compounded.

• If the compound interest is annual, then n = 1.
• If compound interest is quarterly, then n = 4.

#### Step 5. Determine the term of the loan

The term is the duration over which the interest will be calculated and is generally measured in years. If you need to calculate the interest for a certain period, you will have to convert to years.

• For example, for a one-year loan, t = 1 { displaystyle t = 1}

. No obstante, en un plazo de 18 meses, t=1, 5{displaystyle t=1, 5}

#### Step 6. Identify the variables of the situation

For this example, let's say you make a deposit of \$ 5,000 into a savings account with 5% interest per month. How much will this account be worth after three years?

• First, identify the variables you need to solve the problem. In this case:

• P = \$ 5,000 { displaystyle P = \ \$ 5,000}

• r=0, 05{displaystyle r=0, 05}
• n=12{displaystyle n=12}
• t=3{displaystyle t=3}

#### Step 7. Apply the formula and calculate the compound interest

Once you understand the situation and identify the variables, enter them into the formula to find the interest.

• In the case of the problem posed above, the equation will be solved as follows:

• A = P (1 + rn) nt { displaystyle A = P (1 + { frac {r} {n}}) ^ {nt}}

• A=5000(1+0, 0512)12∗3{displaystyle A=5000(1+{frac {0, 05}{12}})^{12*3}}
• A=5000(1+0, 00417)36{displaystyle A=5000(1+0, 00417)^{36}}
• A=5000(1, 00417)36{displaystyle A=5000(1, 00417)^{36}}
• A=5000(1, 1616){displaystyle A=5000(1, 1616)}
• A=5808{displaystyle A=5808}
• Por consiguiente, al cabo de tres años, el interés compuesto será de 808 dólares, que se sumará al depósito original de 5000 dólares.

### Método 3 de 3: Calcular el interés compuesto continuo

#### Step 1. Understand the concept of continuous compound interest

As you saw in the previous example, compound interest increases at a faster rate than simple interest by adding the interest back to the principal at certain times. It is more beneficial to calculate a quarterly compound interest rate than an annual one, and it is even better if it is calculated monthly. It will be much more beneficial if you calculate the compound interest rate continuously, that is, every moment. As soon as the interest can be calculated, it is returned to the account and added to the principal. Obviously this is just a theory.

• Through some calculations, mathematicians have developed a formula that simulates compound interest and returns to the account continuously. This formula, which is used to continuously calculate to calculate compound interest, is as follows:

• A = Pert { displaystyle A = Pe ^ {rt}}

#### Step 2. Determine the variables to calculate interest

The formula for calculating continuous compound interest is similar to the previous situations, only with a few slight differences. Then the variables for the formula are as follows:

• A { displaystyle A}

es el valor futuro (o el monto) de dinero que tendrá el préstamo después de aplicar el interés compuesto.

• P{displaystyle P}
• es el principal.

• e{displaystyle e}
• . Si bien puede parecer una variable, en realidad es una constante. La letra e{displaystyle e}

es un número especial llamado “constante de Euler”, conocido así en honor al matemático Leonard Euler, quien descubrió sus propiedades.

• La mayoría de las calculadoras gráficas avanzadas tienen un botón para ex{displaystyle e^{x}}
• . Al presionar este botón, con el número 1”, para representar e1{displaystyle e^{1}}

, aprenderás que el valor de e{displaystyle e}

es de aproximadamente 2, 718.

• r{displaystyle r}
• es la tasa de interés anual.

• t{displaystyle t}
• es el plazo del préstamo expresado en años.

#### Step 3. Determine the details of the loan

Banks generally use continuous compound interest for home loans. Suppose you want to borrow \$ 200,000 at an interest rate of 4.2% for a 30-year mortgage. Therefore, the variables that you will use to perform the calculation are the following:

• P = 200,000 { displaystyle P = 200,000}

• e{displaystyle e}
• , una vez más, no es una variable sino la constante 2, 718.

• r=0, 042{displaystyle r=0, 042}
• t=30{displaystyle t=30}

#### Step 4. Use the formula to calculate the interest

Apply the values in the formula to calculate the amount of interest you will have to pay on the 30-year loan.

• A = Pert { displaystyle A = Pe ^ {rt}}

• a=200000∗2, 718(0, 042)(30){displaystyle a=200000*2, 718^{(0, 042)(30)}}
• a=200000∗2, 7181, 26{displaystyle a=200000*2, 718^{1, 26}}
• a=200000∗3, 525{displaystyle a=200000*3, 525}
• a=705000{displaystyle a=705000}
• observa el enorme valor que tiene el interés compuesto continuo.