Relation between SI and CI

The entire economic world is based on one principle that is ‘The value of money is not constant’ but keeps changing with time. This is known as the concept of time value money. In simple words the value that a rupee holds today will not be equal to the value of a rupee by tomorrow. Therefore, a rupee borrowed today cannot be repaid by a rupee tomorrow which leads to the basic concept of interest. 

In lame language the rate of interest helps to determine the difference or extra amount that is to be paid on account of money which was borrowed yesterday.

Interests are calculated on two different bases that are: 

1. Simple interest
2. Compound interest

Simple Interest

If the interest is reckoned uniformly on the original principal throughout the loan period, the interest is called simple interest, abbreviated as S.I.

The interest of one year for every 100 is called the Interest rate per annum. If we say “the rate of interest per annum is r%”, we mean that is the interest on a principal of *100 for one year.

Relation among Principal, Time, Rate Percent of Interest per Annum and Total Interest

Suppose, Principal = RP, Time = 1 years, Rate of interest per annum = % and Total interest = I

Then,   

I = (P*r*t)/100

i.e. Total interest = Principal x Rate of interest per annum x Time /100

Amount= Principal + Total Interest 

Logical Understanding of Simple Interest: The students must have got through the understanding of the mathematical formulae pertaining to Simple Interests above. Before we move onto a discussion of compound interest however, let’s revisit Simple Interest logically. We can do this through a few examples.

Suppose a man invests ₹100 @ 8% simple interest. He would be paid an interest of ₹8 per year (simply because 8% of 100 is ₹8). Further, going forward, if it is mentioned to us that the man kept 8% simple interest for five years on an investment of ₹100’, the straightforward logical reaction is: He earns ₹8 interest per year for five years. Hence, his interest is ₹40. It is a straight logical thought and you would never fail to see it in any Q4 problem. Hence, there is absolutely no reason for you to remember the mathematical formulae of simple interests. Just treat every simple interest question like a percentage calculation situation and you would be logically able to work out all the reactions that you need to have for every kind of problem situation on percentages. In order to strengthen these logical structures, let us take a look at various standard statements and situations that you would regularly face in the percentages. 

Compound interest 

In compound interest the interest from the previous years is added to the principal in order to get a new principal and in this way the interest is calculated by compounding interest to principal at the end of each year.

By the end of the first year or any other fixed time period, if the calculated interest is not paid to the moneylender then the interest is added to the principal, and this amount becomes the principal for the next year or any other fixed time period and so on. This process is repeated until the borrower pays the full amount and this is known as compound interest.

Formulae

Case 1:

Suppose principal = P, time = n years and rate = r% per annum and let A be the total amount at the end of n years, then A = P * [1 + r/100] ^ n

Case 2: When compound interest is reckoned half-yearly If the annual rate is r 0 per annum and is to be calculated for n years,

Then in this case, rate= (r/2) % half-yearly and time (2n) half-years. =

From the above, we get

A = P * [1 + (r / 2)/100] ^ (2n)

Case 3: When compound interest is reckoned quarterly In this case, rate = (r/4) % quarterly and time (4n) = quarter years.

As before,

A = P * [1 + (r / 4)/100] ^ (4n)

Difference between Simple interest and compound interest

• If the interest is reckoned uniformly on the original principal throughout the loan period, the interest is called simple interest whereas, In compound interest the interest from the previous years is added to the principal in order to get new principal and in this way the interest is calculated by compounding interest to principal at the end of each year.
• In simple interest the interest is levied upon the principal amount whereas, in compound interest the interest is calculated upon the principal amount plus the interest accumulated at the end of each year.
• In simple interest grows steadily whereas, in compound interest we observe an exponential growth.
• The formula for simple interest is (p*t*r)/100 whereas, formula for compound interest is P * [1 + r/100] ^ n
• Simple interest bears a lesser interest in comparison to compound interest.

The difference between the compound interest and the simple interest over two years is given by P * r ^ 2 / (100 ^ 2) or P * (r/100) ^ 2

Compound interest and simple interest formula

Formula for compound interest is given below:

Let principal = P, time = n years and rate = r\% per annum and let A be the total amount at the end of n years, then A = P * [1 + r/100] ^ n

Formula for simple interest is given below:

Suppose, Principal = RP, Time = 1 years, Rate of interest per annum = % and Total interest = I

Then,   

I = (P*r*t)/100

The rate of interest is generally specified in terms of annual rate of interest. In such a case, we consider the time t in years.

However, if the rate of interest is specified in terms of six-monthly rate, we take time in terms of six months.

Also, the half-yearly rate of interest is half the original annual rate of interest. That is, if the interest is 10% per annum to be charged six-monthly, we have to add interest every six months @ 5%

Conclusion

If the interest is reckoned uniformly on the original principal throughout the loan period, the interest is called simple interest whereas, At the end of the first year (or any other fixed period), if the interest accrued is not paid to the moneylender but is added to the principal, then this amount becomes the principal for the next year (or any other fixed period) and so on. This process is repeated until the amount for the whole time is repaid; this is called compound interest.