Compound Interest
Compound interest can be defined as interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Depreciation definition, decrease in value due to wear and tear, decay, the decline in price, etc.
Summary
Compound interest can be defined as interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Depreciation definition, decrease in value due to wear and tear, decay, the decline in price, etc.
Things to Remember
a. The amount of the previous year becomes the principal for the coming year.
b. The compound interest for every succeeding year is always greater than compound interest for the previous year.
c. The final amount is equal to the sum of the original principal and the interest for all the years.
d. The compound interest for the entire period is the sum of the interest for all the year that is a difference between the final amount and the original principal.
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Compound Interest
Suppose, Deepak borrows Rs. 1000 at 10% interest from Luna. The simple interest on this sum at the end of one year will be \(Rs. \frac {1000 \times 1 \times 10} {100}\) = Rs. 100. If Deepak pays this interest to Luna, Luna can get back Rs. 1100. In case, Deepak pays this interest to Luna then Luna has right to charge interest on Rs. 1100 for next year. The compound interest is infact a simple interest computed on the previous simple amount. When Deepak calculates the simple interest for 2 years then,
\(I = \frac {1000\times 2\times 10} {100}\) = Rs. 200 but when he calculates the compound interest for 2 years then compound interest for 2 years = simple interest for second year
\(=\frac{1000 \times 1 \times 10} {100} + \frac{1100\times 1 \times 10} {100}\) (Principal for 2nd year = Rs. 1000 + interest of 1st year)
=100 + 110
= Rs. 210
In this process we get Rs. 10 profit by the way of compound interest.
The following points should be remembered before calculating the compound interest.
- The compound interest for every succeeding year is always greater than the compound interest for the previous year.
- The amount of the previous year becomes the principal for the coming year.
- The final amount is equal to the sum of the original principal and the interest for all the years.
- The compound interest for the entire period is the sum of the interest for all the years that is the difference between the final amount and the original principal.
The installment is the regular interval of time in which the compound interest is calculated. The payment might be yearly, half-yearly, quarterly, monthly, daily etc. Here we use only yearly and half-yearly installments.
Derivation of yearly compound interest
| Year | Principal | Time | Rate | Interest | Amount |
| 1st | P | 1 year | R% | \(\frac {PR}{100}\) | \(P +\frac {PR}{100} = P((1 + \frac {R} {100})\) |
| 2nd | \(P (1 + \frac {R}{100})\) | 1year | R% | \(P\times(1 + \frac {R} {100})\times \frac {R}{100}\) |
\(P\times(1 + \frac {R} {100}) + P\times(1 + \frac {R} {100})\times \frac {R}{100}\) \(= P\times(1 + \frac {R} {100})\times (1 + \frac {R} {100})\) \(= P\times(1 + \frac {R} {100})^2\) |
| 3rd | \(P (1 + \frac {R}{100})^2\) | 1year | R% | \(P\times(1 + \frac {R} {100})^2\times \frac {R}{100}\) |
\(P\times(1 + \frac {R} {100})^2 + P\times(1 + \frac {R} {100})^2 \times\frac {R}{100}\) \(=P\times(1 + \frac {R} {100})^2 \times(1 + \frac {R} {100}) \) \(=P\times(1 + \frac {R} {100})^3\) |
| 4th | \(P\times(1 + \frac {R} {100})^{T-1} \) | 1year | R% | \(P\times(1 + \frac {R} {100})^{T-1} \times \frac {R} {100}\) |
\(P\times(1 + \frac {R} {100})^{T + 1-1}\) \(=P\times(1 + \frac {R} {100})^T \) |
So, the yearly compound amount for T years at R% p.a. = \(P\times \left (1 + \frac {R} {100}\right)^T \)
Compound interest for T years = Compound amount for T years - original principal
$$ = P \times \left(1 + \frac {R} {100}\right)^T - P $$
$$ =P \left \lbrace \left (1 + \frac {R} {100} \right )^T - 1 \right \rbrace$$
Half-yearly compound interest
In case of half-yearly compoundinterest, time will be double and rate will be halved.
Since yearly compound interest = \( P \left (1 + \frac {R} {100} \right ) ^ T - P \)
$$ = P \times \left(1 + \frac {R} {100}\right)^2T - P $$
$$ =P \left \lbrace \left (1 + \frac {R} {100} \right )^2T - 1 \right \rbrace$$
Lesson
Arithmetic
Subject
Compulsory Mathematics
Grade
Grade 10
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