ICSE Class 9 Maths Chapter 02 Compound Interest Without Using Formula

Unit 2: Commercial Mathematics

Compound Interest

Without Using Formula

Introduction

Sometimes, in need, we borrow money from a bank or some other agency doing financial business. In general, the money is borrowed for a specified period and has to be returned at the end of that period. At the end of the period, we pay the money borrowed plus some extra money for utilising the money of the lender.

The money borrowed is called the principal, the extra money paid for using lender's money is called interest and the total money, paid to the lender at the end of the specified period is called the amount.

Amount = Principal + Interest i.e. A = P + I

Interest (Simple Interest)

Interest is said to be simple, if it is calculated on the original principal throughout the loan period, irrespective of the length of the period for which it is borrowed.

We know, S.I. = \(\frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}\) i.e. I = \(\frac{P \times R \times T}{100}\)

When we say, interest, it always means simple interest.

Compound Interest (C.I.)

Money is said to be lent at compound interest, when the interest, which has become due at the end of a certain fixed period (one year, half year, etc., as given), is not paid to the money lender, but is added to the sum lent. The amount thus obtained becomes the principal for the next period. This process is repeated until the amount for the last period is found.

The difference between the final amount and the original principal is the required compound interest.

Compound Interest = Final Amount - Original Principal i.e. C.I. = A - P

The difference between simple interest (S.I.) and compound interest (C.I.), is made clear by the table given on next page.

Here, in the table, we have taken sum borrowed (principal) = \(\text{₹}\) 1,000 at 10% per annum and for 3 years.

Comparison Table: Simple Interest vs Compound Interest

Compound Interest As A Repeated Simple Interest Computation With A Growing Principal

As shown in the table, given above, the principal for 1st year is \(\text{₹}\) 1,000 and interest (C.I.) on it is \(\text{₹}\) 100. The principal for 2nd year is \(\text{₹}\) 1,100 and interest (C.I.) on it is \(\text{₹}\) 110; whereas, the principal for 3rd year is \(\text{₹}\) 1,210 and the interest (C.I.) on it is \(\text{₹}\) 121.

It is observed that the compound interest is growing (increasing) every year which increases the principal for next year.

As shown in the table, given above, compound interest in 3 years = C.I. of 1st year + C.I. of 2nd year + C.I. of 3rd year = \(\text{₹}\) 100 + \(\text{₹}\) 110 + \(\text{₹}\) 121 = \(\text{₹}\) 331

Also, compound interest in 3 years. = Amount at the end of 3 years - Original sum (Principal for 1st year) = \(\text{₹}\) 1,331 - \(\text{₹}\) 1,000 = \(\text{₹}\) 331

Teacher's Note

When you open a savings account at a bank, your money earns compound interest - the interest earned each month gets added to your balance, so you earn interest on your interest, much like how a snowball grows as it rolls downhill.

Worked Examples

Example 1

\(\text{₹}\) 8,000 is lent at 5 percent compound interest per year for 2 years. Find the amount and the compound interest.

Solution

For the first year:

Principal (P) = \(\text{₹}\) 8,000; Rate (R) = 5% and, Time (T) = 1 year

Interest = \(\frac{P \times R \times T}{100}\) = \(\frac{8,000 \times 5 \times 1}{100}\) = \(\text{₹}\) 400

Amount = Principal + Interest = \(\text{₹}\) 8,000 + \(\text{₹}\) 400 = \(\text{₹}\) 8,400

According to the definition of the compound interest, the amount of the first year will work as principal for the next (second) year.

For the second year:

Principal (P) = \(\text{₹}\) 8,400; R = 5% and T = 1 year

Interest = \(\text{₹}\) \(\frac{8,400 \times 5 \times 1}{100}\) = \(\text{₹}\) 420

Amount at the end of 2nd year = \(\text{₹}\) 8,400 + \(\text{₹}\) 420 = \(\text{₹}\) 8,820

Compound Interest = Final Amount - Initial Principal = \(\text{₹}\) 8,820 - \(\text{₹}\) 8,000 = \(\text{₹}\) 820

Also, C.I. of 2 years = C.I. of 1st year + C.I. of 2nd year = \(\text{₹}\) 400 + \(\text{₹}\) 420 = \(\text{₹}\) 820

Example 2

Find the amount and the compound interest on \(\text{₹}\) 10,000 at 8 per cent per annum and in 1 year; interest being compounded half-yearly.

Solution

For 1st \(\frac{1}{2}\) year: P = \(\text{₹}\) 10,000; R = 8% and T = \(\frac{1}{2}\) year

Interest, I = \(\text{₹}\) \(\frac{10,000 \times 8 \times 1}{100 \times 2}\) = \(\text{₹}\) 400

And, A = P + I = \(\text{₹}\) 10,000 + \(\text{₹}\) 400 = \(\text{₹}\) 10,400

For 2nd \(\frac{1}{2}\) year: P = \(\text{₹}\) 10,400; R = 8% and T = \(\frac{1}{2}\) year.

I = \(\text{₹}\) \(\frac{10,400 \times 8 \times 1}{100 \times 2}\) = \(\text{₹}\) 416

And, A = P + I = \(\text{₹}\) 10,400 + \(\text{₹}\) 416 = \(\text{₹}\) 10,816

Required amount = \(\text{₹}\) 10,816

and, compound interest = A - P = \(\text{₹}\) 10,816 - \(\text{₹}\) 10,000 = \(\text{₹}\) 816

It is clear from examples, given above, that:

1. When the interest is compounded yearly, the principal changes (increases) every year. 2. When the interest is compounded half-yearly, the principal increases every six months. 3. The period (time), after which the principal changes, is called the conversion period. In example 1, given above, the conversion period is one year. And, in example 2, given above, the conversion period is half-year.

Teacher's Note

Understanding conversion periods helps explain why high-yield savings accounts that compound interest daily will give you more money than those that compound monthly - the more frequently interest is calculated and added, the faster your money grows.

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