Maharashtra Board Class 8 Maths part 2 Chapter 14 Compound interest PDF Download

Compound Interest

Let's Recall

A person takes a loan from institutes like bank or patapedhi with fixed rate of interest. After stipulated time he repays the loan with some more money. The additional money paid is called interest.

We have learnt the formula for finding the interest, \(I = \frac{PNR}{100}\).

In this formula I = Interest, P = Principal, N = Number of years and R = Rate of interest at p.c.p.a. This is how the simple interest is calculated.

Teacher's Note

When you borrow money from a bank, you must pay back more money. The extra money is called interest. Just like when you borrow 100 rupees from your friend and give back 110 rupees, the extra 10 rupees is interest.

Exam Trick

Remember the formula: I = PNR/100. P means the money you borrowed. R means the percentage the bank charges each year. N means how many years you borrowed.

Points to Remember

Interest is the extra money you pay when you borrow.
Principal is the money you borrow.
The formula for simple interest is I = PNR/100.
p.c.p.a. means per cent per annum (per year).

Let's Learn

Compound Interest

We will learn how bank charges compound interest for a fixed deposit or a loan.

Teacher: Sajjanrao takes a loan of ₹10,000 from a bank at the rate of 10 p.c.p.a. for 1 year. How much money including the interest he will have to pay after one year?

Student: Here P = ₹10,000; R = 10; N = 1 year

\(I = \frac{PNR}{100} = \frac{10000 \times 10 \times 1}{100} = \) ₹1000

After one year Sajjanrao will have to pay 10,000 + 1000 = ₹11,000 with interest.

Student: If a borrower fails to pay the amount and interest after one year?

Teacher: Bank calculates the interest after each year and it is expected that the interest should be paid by the borrower every year. If the borrower fails to pay the interest for one year then the bank considers the principal and the interest of first year together as the loan for second year. Thus for second year, the interest is calculated on the amount formed by principal of first year together with its interest. That is for second year the principal for charging interest is the amount of the first year. The interest charged by this method is called as compound interest.

Student: If Sajjanarao increases the duration of loan repayment by one more year?

Teacher: Then for second year considering ₹11,000 as principal, interest and amount is to be found out.

Student: For this can we use the ratio \(\frac{\text{Amount}}{\text{Principal}} = \frac{110}{100}\) which is learnt in the previous standard?

Teacher's Note

Compound interest is when the bank adds interest to your money, and then charges interest on that too. It is like your money grows faster.

Exam Trick

Think of compound interest like this: First year you earn interest. Second year you earn interest on the interest too. It grows like a snowball rolling downhill.

Points to Remember

In compound interest, the interest is added to the principal each year.
Then interest is calculated on the new amount.
This makes your money grow faster than simple interest.
Banks use compound interest for deposits and loans.

Teacher: Surely! For every year the ratio \(\frac{\text{Amount}}{\text{Principal}}\) is constant. While finding compound interest every year, the amount (A) of previous year is the principal for next year. Hence it is convenient to find amount rather than the interest. Let us write the amount after first year as \(A_1\), after second year \(A_2\) and after third year as \(A_3\).

Originally the principal is P.

\(\frac{A_1}{P} = \frac{110}{100}\) \(\therefore A_1 = P \times \frac{110}{100}\)

For finding the amount \(A_2\) of second year,

\(\frac{A_2}{A_1} = \frac{110}{100}\) \(\therefore A_2 = A_1 \times \frac{110}{100} = P \times \frac{110}{100} \times \frac{110}{100}\)

Student: Then for finding the amount \(A_3\) of third year,

\(\frac{A_3}{A_2} = \frac{110}{100}\) \(\therefore A_3 = A_2 \times \frac{110}{100} = P \times \frac{110}{100} \times \frac{110}{100} \times \frac{110}{100}\)

Teacher: Very Good! This is the formula for finding the amount by compound interest. Here \(\frac{110}{100}\) is the amount of ₹1 after one year. To find amount after the given number of years multiply the principal by those many times by this ratio.

Student: If the ratio \(\frac{\text{Amount}}{\text{Principal}}\) is assumed to be M then the amount after one year is \(P \times M\), after second year is \(P \times M^2\), after third year \(PM^3\). In this way we can find the amount after any number of years.

Teacher: Correct! If the interest rate is R p.c.p.a. then,

the amount of ₹1 after 1 year = \(1 \times M = 1 \times \frac{100 + R}{100} = 1 \times \left(1 + \frac{R}{100}\right)\)

the amount of ₹P after 1 year = \(P \times \frac{100 + R}{100} = P \times \left(1 + \frac{R}{100}\right)\)

\(\therefore\) if the principal P, interest rate R p.c.p.a. and duration is N years then the amount after N years, \(A = P \times \left(\frac{100 + R}{100}\right)^N = P \times \left(1 + \frac{R}{100}\right)^N\)

Solved Example

Ex. (1) Find the compound interest if ₹4000 are invested for 3 years at the rate of \(12\frac{1}{2}\) p.c.p.a.

Solution: Here, P = ₹4000; R = \(12\frac{1}{2}\)%; N = 3 years.

\(A = P\left(1 + \frac{R}{100}\right)^N = P\left(1 + \frac{12.5}{100}\right)^3 = 4000\left(1 + \frac{125}{1000}\right)^3\)

\(A = 4000\left(\frac{1125}{1000}\right)^3 = 4000\left(\frac{9}{8}\right)^3 = 5695.31\) Rupees

\(\therefore\) Compound Interest after three years, I = Amount - Principal = 5695.31 - 4000 = 1695.31 rupees

Teacher's Note

Compound interest helps your savings grow. For example, if you put 4000 rupees in a bank for 3 years, you will get back more than 5695 rupees because of compound interest.

Exam Trick

Remember: Amount = P × (1 + R/100)^N. The N is the power. More years means more multiplication, so more interest.

Points to Remember

Compound Interest formula: A = P × (1 + R/100)^N
A means the total amount you get back.
The exponent N means you multiply the fraction N times.
Interest = Amount - Principal.
This type of interest grows faster than simple interest.

Practice Set 14.1

1. Find the amount and the compound interest.

2. Sameerrao has taken a loan of ₹12500 at a rate of 12 p.c.p.a. for 3 years. If the interest is compounded annually then how many rupees should he pay to clear his loan?

3. To start a business Shalaka has taken a loan of ₹8000 at a rate of \(10\frac{1}{2}\) p.c.p.a. After two years how much compound interest will she have to pay?

For More Information

1. Some times the interest is calculated at an interval of six months. For the duration of N years, if rate is R and if the interest to be calculated six monthly then the rate is to be taken as \(\frac{R}{2}\) and the duration is considered as 2N stages of six months.

2. Many banks charge the compound interest monthly. At that time they take the interest rate as \(\frac{R}{12}\) monthly and the duration is taken \(12 \times N\) stages of months and interest is calculated.

3. Now a days banks calculate compound interest daily.

Activity: Visit the bank nearer to your house and get the information regarding the different schemes and rate of interests. Make a chart and display in your class.

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