# Formula for Finding Compound Interest

## Formula for Finding Compound Interest

Supriya invested Rs 75000 in a bank at the rate of 10% per annum compounded annually. What is the amount received by her after 2 years?

We know the method of calculating compound interest using the concept of simple interest. However, this method is very lengthy as we have to calculate the interest and then amount for each year one by one.

We can also find the amount directly using a formula. Before solving the given problem, let us know about the formula.

When the interest is compounded annually, the amount after n years is given by

where P is the principal and R is the rate of interest per annum.

Now, using this formula, let us calculate the amount received by Supriya after 2 years.

In the given problem,

Principal, P = Rs 75000

Rate of interest, R = 10% per annum

Number of years, n = 2 years

By the formula, we obtain

A = 750 × 121

A = Rs 90750

Thus, Supriya received Rs 90750 after 2 years.

Using this formula, we can find time, rate of interest or principal, if the rest of the values are known to us.

For example, the compound interest on a certain sum invested for two years at the rate of 8% p.a. compounded annually is Rs 3328. Find the sum.

Let x be the sum.

Amount = Principal + Interest

Amount = Rs (x + 3328)

Now using the formula, we obtain

Þ (x + 3328) = x

Þ (x + 3328) =

Þ (x + 3328) =

Þ

Þ 625x + 2080000 = 729x

Þ 729x − 625x = 2080000

Þ 104 x = 2080000

Þ

Þ x = 20000

Thus, the sum is Rs 20000.

There are a lot of cases when we are not talking about interest but a value is increased or decreased (i.e., appreciated or depreciated) by a percentage after a fixed time interval as in the following example.

The value of a vacuum cleaner worth Rs 15000 is depreciated by 5% every year. What will be the value of the vacuum cleaner after two years?

In such cases, we can use the same formula that we have used for finding the compound interest, i.e.

where P is the initial value, A is the value after time period n, and R is the appreciation rate.

When R is the rate of depreciation, we will use the formula

In this example, P = Rs 15000

Rate of depreciation, R = 5% per year

Therefore, the value of vacuum cleaner at the end of two years is,

Thus, the cost of the vacuum cleaner after two years will be Rs 13537.50.

Now, let us solve some more examples.

Example 1:

Anurag borrowed Rs 50000 from his friend at the rate of 4% p.a. compounded annually. After some time, he repaid Rs 56243.20 to his friend. For how many years did Anurag borrow the money?

Solution:

Principal, P = Rs 50000

Rate, R = 4% p.a.

Amount, A = Rs 56243.2

Let n be the number of years for which Anurag borrowed the money.

Using the formula, , we obtain

Þ Rs 56243.2 = Rs 50000

Thus, Anurag borrowed the money for two years.

Example 2:

Find the rate at which Rs 700 becomes Rs 847 in two years, when the interest is compounded annually.

Solution:

Let R be the rate of interest.

Principal, P = Rs 700

Amount, A = Rs 847

Number of years, n = 2

Now, using the formula,, we obtain

Thus, the rate of interest is 10%.

Example 3:

Priya bought a diamond necklace worth Rs 300000. The value of the necklace is appreciated by 6% every year. What will be the value of the necklace after 3 years?

Solution:

The value of the necklace after three years can be calculated by using the compound interest formula.

P = Rs 300000

Rate of appreciation, R = 6% p.a.

Therefore, the value of the necklace after 3 years

Thus, the value of the necklace after 3 years is Rs 357304.80.

We know the method of calculating compound interest using the concept of simple interest. However, this method is very lengthy as we have to calculate the interest and then amount for each year one by one.

We can also find the amount directly using a formula. Before solving the given problem, let us know about the formula.

When the interest is compounded annually, the amount after n years is given by

where P is the principal and R is the rate of interest per annum.

Now, using this formula, let us calculate the amount received by Supriya after 2 years.

In the given problem,

Principal, P = Rs 75000

Rate of interest, R = 10% per annum

Number of years, n = 2 years

By the formula, we obtain

A = 750 × 121

A = Rs 90750

Thus, Supriya received Rs 90750 after 2 years.

Using this formula, we can find time, rate of interest or principal, if the rest of the values are known to us.

For example, the compound interest on a certain sum invested for two years at the rate of 8% p.a. compounded annually is Rs 3328. Find the sum.

Let x be the sum.

Amount = Principal + Interest

Amount = Rs (x + 3328)

Now using the formula, we obtain

Þ (x + 3328) = x

Þ (x + 3328) =

Þ (x + 3328) =

Þ

Þ 625x + 2080000 = 729x

Þ 729x − 625x = 2080000

Þ 104 x = 2080000

Þ

Þ x = 20000

Thus, the sum is Rs 20000.

There are a lot of cases when we are not talking about interest but a value is increased or decreased (i.e., appreciated or depreciated) by a percentage after a fixed time interval as in the following example.

The value of a vacuum cleaner worth Rs 15000 is depreciated by 5% every year. What will be the value of the vacuum cleaner after two years?

In such cases, we can use the same formula that we have used for finding the compound interest, i.e.

where P is the initial value, A is the value after time period n, and R is the appreciation rate.

When R is the rate of depreciation, we will use the formula

In this example, P = Rs 15000

Rate of depreciation, R = 5% per year

Therefore, the value of vacuum cleaner at the end of two years is,

Thus, the cost of the vacuum cleaner after two years will be Rs 13537.50.

Now, let us solve some more examples.

Example 1:

Anurag borrowed Rs 50000 from his friend at the rate of 4% p.a. compounded annually. After some time, he repaid Rs 56243.20 to his friend. For how many years did Anurag borrow the money?

Solution:

Principal, P = Rs 50000

Rate, R = 4% p.a.

Amount, A = Rs 56243.2

Let n be the number of years for which Anurag borrowed the money.

Using the formula, , we obtain

Þ Rs 56243.2 = Rs 50000

Thus, Anurag borrowed the money for two years.

Example 2:

Find the rate at which Rs 700 becomes Rs 847 in two years, when the interest is compounded annually.

Solution:

Let R be the rate of interest.

Principal, P = Rs 700

Amount, A = Rs 847

Number of years, n = 2

Now, using the formula,, we obtain

Thus, the rate of interest is 10%.

Example 3:

Priya bought a diamond necklace worth Rs 300000. The value of the necklace is appreciated by 6% every year. What will be the value of the necklace after 3 years?

Solution:

The value of the necklace after three years can be calculated by using the compound interest formula.

P = Rs 300000

Rate of appreciation, R = 6% p.a.

Therefore, the value of the necklace after 3 years

Thus, the value of the necklace after 3 years is Rs 357304.80.

**rajathadri**- LSF Member
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