How to Calculate Simple Interest

• Introduction
• Understanding Simple Interest
• Simple Interest Formula
• How to Calculate Simple Interest?
• Formula for Monthly Simple Interest
• Simple Interest vs Compound Interest
• Applications of Simple Interest
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction 

Simple interest is a basic concept in finance. It's a way to figure out how much you'll pay or earn on a sum of money over time. Unlike compound interest, where you add the interest to the principal each time, with simple interest, the principal stays the same. The interest rate is calculated against the principal amount and that amount never changes, as long as you make payments on time.

This means that you will always pay less interest with a simple interest loan than a compound interest loan if the loan term is greater than `1` unit of time like `1` year, `1` month, `1` week, etc. Likewise, you will also end up receiving less interest on your deposit amount if the bank is offering simple interest and not compound interest on your deposit.

This lesson introduces you to terms like principal (the initial amount), rate of interest (how much interest you'll pay or earn), and time period (how long the money is borrowed or invested). You'll use these terms in a simple interest formula to calculate simple interest.

Understanding Simple Interest

Simple interest is a way of calculating interest on the original amount you invested, using the same interest rate each time. When you put money in a bank, they pay you interest on it. One type of interest they might offer is called simple interest. But first, let's talk about what a loan is. A loan is money you borrow from a bank or financial institution to meet your needs, like buying a house, car, paying for education, or personal expenses. You have to pay back the loan amount plus an extra amount, which is the interest. 

Simple interest only looks at the original amount borrowed, while compound interest considers both the original amount and the interest earned previously. Compound interest tends to be higher than simple interest over time. For example, if you invest `$100` at a `10%` interest rate for `2` years, with simple interest, you'd earn `$20` each year, but with compound interest, you'd earn more due to the interest being added to the principal each year. Likewise, if you borrow `$100` at a `5%` interest rate for `2` years, with simple interest, you'd just pay `$10` each year, but with compound interest, you would end up paying more.

Simple Interest Formula

Simple interest is a way to calculate the extra money you pay or receive when borrowing or lending money. To find simple interest `(S.I.)`, you use this formula: 

`S.I. = \frac{P \times R \times T}{100}`

where, 

`P` stands for the principal (the initial amount borrowed or invested), 
`R` is the rate of interest per year (expressed as a percentage),
`T` represents the time in years 

For instance, if you borrowed `$1000` at a rate of `5%` per year for `2` years, the simple interest would be calculated as `\frac{1000 \times 5 \times 2}{100} = \$100`. 

Amount: When someone takes out a loan from a bank, they need to pay back the amount they borrowed plus the interest. This total repayment is called the Amount. The Amount is represented by the letter "`A`".

`"Amount" = "Principal" + "Simple Interest"`

`A = P + SI`

`A = P + \frac{PRT}{100}`

`A = P(1 +\frac{RT}{100})`

How to Calculate Simple Interest?

To calculate simple interest, you use a formula: `SI = (PRT)/100`. In this formula, `P` stands for the principal amount (the initial sum of money), `R` is the rate of interest, and `T` represents the time duration. You simply plug in these values into the formula to find the simple interest. 

For instance, if you have the same principal and rate but different durations, you'd substitute those values into the formula to find the interest for each scenario. Here's how you do it:

Step `1`. First, note down the principal, rate of interest, and time duration.

Step `2`. Next, use the formula `S.I. = \frac{P \times R \times T}{100}` to find the simple interest.

Step `3`. Then, plug in the values from step `1` into the formula.

Step `4`. Finally, simplify the expression to get the simple interest.

Now, let's take an example to understand the process.

Example: Lisa deposited `$5000` into a savings account that earns a simple interest rate of `4%` annually. How much interest will she earn after `3` years?

Solution:

Principal amount, \( P = $5000 \)

Rate of interest, \( R = 4\% \) per year

Time, \( T = 3 \) years

As per the formula of simple interest  `SI = \frac{P \times R \times T}{100}`, we can calculate:

`SI = \frac{5000 \times 4 \times 3}{100}`

`SI = \frac{60000}{100}`

\( SI = $600 \)

So, Lisa will earn `$600` in interest after `3` years.

Formula for Monthly Simple Interest

In our previous discussion, we explored how to calculate simple interest `(S.I.)` on a yearly basis. Now, let's look at formulas for calculating `S.I.` for shorter time periods like months when the rate of interest is given annually. Generally, the formula for calculating `S.I.` is:

`\text{S.I.} = \frac{P \times R \times T}{100}`

Here, \( T \) represents the number of years. To adapt this formula for months, we tweak it as follows:

`\text{S.I.} = \frac{P \times R \times x}{12 \times 100`

Here, \( x \) denotes the number of months. For instance, if \( x = 3 \) (for `3` months), the formula becomes:

`\text{S.I.} = \frac{P \times R \times 3}{12 \times 100} = \frac{P \times R}{4 \times 100 }`

Similarly, for half-yearly (`6` months), the formula simplifies to:

`\text{S.I.} = \frac{P \times R \times 6}{12 \times 100} = \frac{P \times R}{2 \times 100 }`

This adjustment allows us to calculate Simple Interest for various time periods with ease.

Example: David borrowed `$8000` from a local financial institution. The interest rate on the loan amount is `6%` annually. How much interest will David pay on his loan amount after `15` months?

Solution:

Principal amount, \( P = $8000 \)

Rate of interest, \( R = 6\% \) per year

Time, \( x = 15 \) months

Using the formula for monthly simple interest `\text{S.I.} = \frac{P \times R \times x}{12 \times 100 }`, we can calculate:

`SI = \frac{8000 \times 6 \times 15}{12 \times 100}`

`SI = \frac{720000}{1200}`

\( SI = $600 \)

So, David will pay `$600` in interest after `15` months.

Simple Interest vs Compound Interest

To understand the variance between simple interest and compound interest, let's examine the table provided below:

Applications of Simple Interest

`1`. Loans: Simple interest is commonly used in loans where the interest is calculated only on the initial amount borrowed, without considering any additional interest that accumulates over time.

`2`. Savings Accounts: Some savings accounts offer simple interest on the deposited amount, where the interest earned is based solely on the principal balance.

`3`. Certificates of Deposit (CDs): CDs often utilize simple interest, where the interest is calculated on the principal investment amount without compounding.

`4`. Bonds: Certain bonds provide simple interest payments to investors, where the interest is paid periodically based on the initial investment amount.

`5`. Fixed-Rate Mortgages: In fixed-rate mortgages, the interest is typically calculated using simple interest, ensuring consistent monthly payments throughout the loan term.

`6`. Car Loans: Many car loans employ simple interest, where the interest charged is calculated based on the original loan amount without compounding.

Solved Examples

Example `1`: Sam borrows `$1000` at an annual simple interest rate of `5%` for `3` years. Calculate the total amount Sam would pay back.

Solution: 

Principal amount, \( P = $1000 \)

Rate of interest, \( R = 5\% \) per annum

Time, \( T = 3 \) years

`\text{As per the formula of simple interest:}`

`\text{Simple Interest (SI)} = \frac{P \times R \times T}{100}`

`\text{SI} = \frac{1000 \times 5 \times 3}{100} = \$150`

Total amount to be paid back:

`\text{Total Amount} = P + \text{SI}`

`= \$1000 + \$150 = \$1150`

Example `2`: If the simple interest accrued on a certain amount at `8%` per annum for `2` years is `$320`, what is the principal amount?

Solution: 

Rate of interest, \( R = 8\% \) per annum

Time, \( T = 2 \) years

Simple Interest, \( SI = $320 \)

`\text{Using the formula for simple interest:}`

`SI = \frac{P \times R \times T}{100}`

`P = \frac{SI \times 100}{R \times T}`

`= \frac{320 \times 100}{8 \times 2}`

`= $2000`

So, the principal amount is `$2000`.

Example `3`: A sum of `$4500` amounts to `$5400` in `3` years at a certain rate of simple interest. Find the rate of interest, rounded to `2` decimal places.

Solution: 

Principal amount, \( P = $4500 \)

Total amount, \( A = $5400 \)

Time, \( T = 3 \) years

As per the formula of simple interest:

`SI = A - P`

`= $5400 - $4500`

`= $900`

`SI = \frac{P \times R \times T}{100}`

We rearrange this formula to find the rate of interest:

`R = \frac{SI \times 100}{P \times T}`

`= \frac{900 \times 100}{4500 \times 3}`

`= 6.67%`

So, the rate of interest is `6.67%`.

Example `4`: A sum of money amounts to `$1650` in `3` years and to `$1800` in `4` years at simple interest. Find the sum and the rate of interest.

Solution: 

Let `P` be the principal sum and `R` be the rate of interest.

From the given data:

`P + \frac{P \times R \times 3}{100} = \$1650 \quad \text{(1)}`

`P + \frac{P \times R \times 4}{100} = \$1800 \quad \text{(2)}`

Subtracting equation `(1)` from equation `(2)`:

`\frac{P \times R \times 4}{100} - \frac{P \times R \times 3}{100} = \$1800 - \$1650`

`\frac{P \times R}{100} = \$150`

`PR = \$15000`

Now, substituting `PR = \$15000` into equation `(1)`:

`P + \frac{\$15000 \times 3}{100} = \$1650`

`P + \$450 = \$1650`

`P = \$1200`

So, the principal sum is `$1200`.

To find the rate of interest, substitute the value of `P` back into equation `(1)`:

`\$1200 + \frac{\$1200 \times R \times 3}{100} = \$1650`

`\$36 \times R = \$450`

`36R = 450`

`R = \frac{450}{36} = 12.5\%`

So, the rate of interest is `12.5%`.

Example `5`: The simple interest on a sum of money is `$312`. If the rate of interest is `6%` per annum and the time period is `5` years, find the principal.

Solution: 

Rate of interest, \( R = 6\% \) per annum

Time, \( T = 5 \) years

Simple Interest, \( SI = $312 \)

As per the formula of simple interest:

`SI = \frac{P \times R \times T}{100}`

We can rearrange this formula to find the principal amount:

`P = \frac{SI \times 100}{R \times T}`

`= \frac{312 \times 100}{6 \times 5}`

`= \$1040`

So, the principal amount is `$1040`.

Practice Problems

Q`1`. Find the simple interest on a principal amount of `$2000` at an annual interest rate of `4%` for `3` years.

1. `$240`
2. `$320`
3. `$360`
4. `$480`

Answer: a

Q`2`. If the simple interest on a certain amount at `6%` per annum for `4` years is `$480`, what is the principal amount?

1. `$600`
2. `$800`
3. `$1000`
4. `$2000`

Answer: d

Q`3`. The simple interest on a sum of money is `$375`. If the principal amount is `$5000` and the time period is `5` years, what is the rate of interest?

1. `2.4%`
2. `1.5%`
3. `4%`
4. `7%`

Answer: b

Q`4`. Find the total amount after `4` years on a principal of `$3000` at a simple interest rate of `6%` per annum.

1. `$3600`
2. `$3360`
3. `$3180`
4. `$3720`

Answer: d

Q`5`. A sum of `$4000` amounts to `$4600` at a simple interest rate of `7.5%` per annum. Find the time period.

1. `1` year
2. `2` years
3. `3` years
4. `4` years

Answer: b

Frequently Asked Questions

Q`1`. What is simple interest?

Answer: Simple interest is a type of interest calculated only on the initial principal amount over a period of time. It does not take into account any interest that accumulates on previously earned interest.

Q`2`. How is simple interest calculated?

Answer: Simple interest (\(SI\)) can be calculated using the formula: 

`SI = \frac{P \times R \times T}{100}`

Where:

\(P\) is the principal amount

\(R\) is the rate of interest per annum

\(T\) is the time period in years

Q`3`. What is the difference between simple interest and compound interest?

Answer: Simple interest is calculated only on the initial principal amount, while compound interest takes into account both the principal amount and the accumulated interest over time, resulting in interest earning interest.

Q`4`. Can simple interest be negative?

Answer: No, simple interest cannot be negative. It is always a non-negative value, representing the additional amount paid or earned on the principal amount.

Q`5`. What happens if the time period in simple interest calculation is not in years?

Answer: The time period (\(T\)) in simple interest calculation must be in years. If the time period is given in months or days, it should be converted to years before using it in the formula. For example, if the time period is given in months, it should be divided by `12` to convert it to years.