ICSE Class 10 Maths Chapter 01 Compound Interest

Unit I Commercial Mathematics

Chapter 1

Compound Interest

Points To Remember

I Formulae

1. When Interest is Reckoned Annually

(i) Let Principal = P, Rate = R% p.a. and Time = n (in years)

Then, the amount after n years is given by:

\[A = P\left(1 + \frac{R}{100}\right)^n\]

\[\therefore C.I. = (A - P) = P\left(1 + \frac{R}{100}\right)^n - P\]

\[\Rightarrow C.I. = P\left[\left(1 + \frac{R}{100}\right)^n - 1\right]\]

(ii) Let the rate of interest for two successive years be R₁% and R₂% respectively. Then, the amount after 2 years is given by:

\[A = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\]

(iii) Let the rate of interest for three successive years be R₁%, R₂% and R₃% respectively. Then, the amount after 3 years is given by:

\[A = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)\]

(iv) When the period is not a complete number of years, say it is 2 years 7 months and let the rate of interest be R% p.a., compounded annually. Then,

\[A = P\left(1 + \frac{R}{100}\right)^2\left(1 + \frac{7}{12} \cdot \frac{R}{100}\right)\]

2. When Interest is Reckoned Half-yearly (or Semi-annually)

Let principal = P, Rate = R% p.a. and Time = n years.

And, let the interest be reckoned half-yearly. Then,

Principal = P, Rate = \(\frac{R}{2}\)% per half-year and Time = (2n) half-years.

So, the amount after n years is given by:

\[A = P\left(1 + \frac{R}{2 \times 100}\right)^{2n}\]

Note. The S.I. and the C.I. for first unit of time are always equal.

3. Formulae for Population Growth and Population Decrease

It is easy to derive the following formulae by unitary method.

(i) Let there be a growth of r% p.a. in the population of a place. Then,

\[(Population after n years) = (Present Population) \times \left(1 + \frac{r}{100}\right)^n\]

Similarly,

\[(Population n years ago) \times \left(1 + \frac{r}{100}\right)^n = (Present Population)\]

(ii) If there is a growth of r₁% during first year and r₂% during second year, then:

\[(Population after 2 years) = (Present Population) \times \left(1 + \frac{r_1}{100}\right) \times \left(1 + \frac{r_2}{100}\right)\]

This formula can be extended for a period of more than 2 years.

(iii) When there is a regular decrease in population, we use minus sign instead of plus sign in the above formulae.

4. Formulae for Depreciation

It is easy to derive the following formulae by unitary method.

If the value of a machine depreciates by r% per annum, then:

\[(Value of the machine after n years) = (Its Present Value) \times \left(1 - \frac{r}{100}\right)^n\]

Similarly,

\[(Value of the machine n years ago) \times \left(1 - \frac{r}{100}\right)^n = (Its Present Value)\]

Teacher's Note

Compound interest is used everywhere from bank savings accounts to investment returns, making it essential for understanding how money grows over time in real financial planning.

Maturity Tables For R.D.

For A Deposit Of Rs 100 P.M. At Rates 6% To 11% (Compounded Quarterly)

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