Introduction to Interest | Mathematics for Digital SAT PDF Download

What is an Interest?

• Interest is an extra sum of money paid by a borrower to a lender or investor as a fee for borrowing money. This additional amount is paid in addition to the original amount borrowed. For instance, if a borrower borrows $20,000, they may agree to pay an extra $200 as interest. 
• The interest rate is the percentage of the original amount that is paid or received as interest over a specific period of time.
  For example, if the borrower agrees to pay back the borrowed amount in one year, and the interest rate is 10%, then the borrower will have to pay $2,000 in interest. Interest can be simple, which means it is calculated once on the principal amount, or compounded, which means it is calculated on the principal amount as well as the interest accrued. Compound interest can be calculated daily, monthly, quarterly, half-yearly or annually. 
• It is believed that Albert Einstein once said that "compound interest is the most powerful force in the universe." This means that over time, an investment can grow to an unlimited amount, or debt can accumulate to an unmanageable level due to the effect of compound interest.

How Does Interest Work?

• Interest works in different ways depending on whether you are borrowing or lending money. When you borrow money, you must repay the amount borrowed along with additional interest to compensate the lender for the risk of lending money. On the other hand, when you lend money, you expect to earn interest on the amount you lend.
• The amount of interest you pay or earn depends on several factors such as the interest rate, the amount of the loan or deposit, and the length of time for repayment. Higher interest rates or longer repayment periods result in the borrower paying more interest.

For example, if you borrow $100 with an interest rate of 10% per year, you would pay $10 in interest per year if simple interest is used to calculate the interest amount. However, most lenders use compound interest which causes the interest amount to grow more rapidly over time. Albert Einstein famously referred to compound interest as the "most powerful force in the universe" because it has the potential to exponentially increase investment returns or accumulate debt.

How to Calculate Interest?

• Simple Interest
• Compound Interest

What is Simple Interest?
Simple interest is a straightforward method of calculating the interest on a loan or investment by considering only the original amount borrowed or invested and the interest rate for the entire duration of the loan or investment. This type of interest is not compounded, meaning that interest is only calculated on the initial principal amount and not on any previously accrued interest. Simple interest is generally applied to short-term loans or investments that have a duration of one year or less. 

The formula for calculating simple interest involves the principal amount (P), the interest rate (R), and the duration of the loan or investment (T). The final amount to be paid is the sum of the principal amount and the simple interest, which is calculated using the formula SI = P x R x T / 100.

For Example: An invested sum fetched a total interest of $5000 at the rate of 10% in one year. What was the original principal amount?

Let principal amount be P, SI be simple interest, R be the rate of interest, and T the time period.
Accordingly,
SI = PRT/100
5000 = P × 10 × 1
P = 5000/10
P = 500
Hence, the original principal amount is $ 500.

Compound Interest

• In most cases, interest rates are compound interest, which means that the interest is not just calculated based on the original principal, but on the amount of money invested or owed at the time of calculation. This compounding of interest is beneficial for investors as it can help them generate more return on their original principal amount.
• Compound interest refers to the interest charged on the interest already accrued. For instance, if someone borrows $2000 at an annual interest rate of 10%, the interest charged for the first year would be 10% of $2000, which is $200. At the beginning of the second year, the principal amount would be $2200, and the interest charged in the second year would be 10% of $2200, which is $220. Therefore, compound interest is essentially interest charged on interest.

Compound Interest Formula
The formula for calculating the amount received when interest is compounded annually:

Amount = Principal (1 + Rate/100)
The total compounded interest over the term is calculated as
Compound Interest = Amount - Principal

Solved Example 1: In how many years will an amount of $2000 will be doubled, if the interest rate is 10% per annum?

Let the principal amount be P, R be the rate of interest per annum, SI be simple interest, and T be the time period.
Accordingly,
SI = PRT/100
2000 = 2000 × 10 × T ( because SI = P)
T = 20000/ 2000
T = 10 years
Hence, the amount of Rs.1000 will be doubled in 10 years.

Solved Example 2: What would be the amount Rs.100 in a year if the interest rate is 10% half-yearly ( every 6 months).

Let A be the amount received,  P be the principal amount, R be the rate of interest per annum, SI be simple interest and T be time period.
Accordingly,
A = P (1 + R)T
= 100 × (1 + 10/100)²
= 100 × (21/20)²
= 100 × ( 441/400)²
= 441/4
= 110.25