ICSE Class 9 Maths Chapter 03 Compound Interest Using Formula

Compound Interest

Using Formula

3.1 Introduction

In the previous chapter, we have learnt to calculate the amount and the compound interest on a given sum (principal) at a given rate and for a given period.

In the above process, we found compound interest as a repeated simple interest computation with a growing principal.

The computation of compound interest and amount, as found above, becomes quite tedious as the number of conversion periods (no. of years, no. of half-years, etc.) increase.

3.2 Using Formula

In order to make the above said calculation easy and fast, we use certain formulae.

First formula:

1. When the interest is compounded yearly, the formula for finding the amount is:

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

where A = amount; P = principal; r = rate of interest compounded yearly; and n = number of years.

Calculate the amount on ₹ 7,500 in 2 years and at 6% compounded annually.

Solution:

Given: P = ₹ 7,500; n = 2 years and r = 6%

\[A = ₹ 7,500\left(1 + \frac{6}{100}\right)^2\]

\[= ₹ 7,500 \times \left(\frac{106}{100}\right)^2 = ₹ 8,427\]

Required amount = ₹ 8,427

And, C.I. = ₹ 8,427 - ₹ 7,500 = ₹ 927

Calculate the compound interest on ₹ 18,000 in 2 years at 15% per annum.

Solution:

\[A = P\left(1 + \frac{r}{100}\right)^n \Rightarrow A = ₹ 18,000\left(1 + \frac{15}{100}\right)^2\]

\[= ₹ 23,805\]

Compound Interest = A - P

\[= ₹ 23,805 - ₹ 18,000 = ₹ 5,805\]

Direct method:

\[\text{Compound Interest} = ₹ 18,000\left[\left(1 + \frac{15}{100}\right)^2 - 1\right]\]

\[= ₹ 18,000 (1.3225 - 1)\]

\[= ₹ 18,000 \times 0.3225\]

\[= ₹ 5,805\]

2. When the rates for successive years are different then:

\[A = P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)\left(1 + \frac{r_3}{100}\right) \ldots \text{ and so on}\]

where r₁%, r₂%, r₃% ... and so on are the rates for successive years.

Calculate the amount and the compound interest on ₹ 12,000 in 3 years when the rates of interest for successive years are 8%, 10% and 15% respectively.

Solution:

\[\text{Required amount, } A = P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)\left(1 + \frac{r_3}{100}\right)\]

\[A = ₹ 12,000\left(1 + \frac{8}{100}\right)\left(1 + \frac{10}{100}\right)\left(1 + \frac{15}{100}\right)\]

\[= ₹ 16,394.40\]

And, C.I. = ₹ 16,394.40 - ₹ 12,000 = ₹ 4,394.40

3.3 Inverse Problems:

1. To find the principal:

What sum of money will amount to ₹ 3,630/- in 2 years at 10% per annum compound interest?

\[₹ 3,630 = P\left(1 + \frac{10}{100}\right)^2\]

\[₹ 3,630 = P \times \frac{11}{10} \times \frac{11}{10}\]

\[\text{The required sum of money, } P = ₹ 3,630 \times \frac{10}{11} \times \frac{10}{11}\]

\[= ₹ 3,000\]

On what sum of money will compound interest for 2 years at 5 percent per year amount to ₹ 164?

Solution:

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

\[₹ 164 = P\left[\left(1 + \frac{5}{100}\right)^2 - 1\right]\]

\[₹ 164 = P\left[\frac{21}{20} \times \frac{21}{20} - 1\right]\]

On further simplification, we get: P = ₹ 1,600

2. To find the rate percent:

At what rate percent per annum C.I. will ₹ 2,000 amount to ₹ 2,315.25 in 3 years?

Solution:

\[₹ 2,315.25 = ₹ 2,000\left(1 + \frac{r}{100}\right)^3\]

\[\frac{2315.25}{2,000} = \left(1 + \frac{r}{100}\right)^3\]

\[\left(\frac{21}{20}\right)^3 = \left(1 + \frac{r}{100}\right)^3\]

\[\frac{21}{20} = 1 + \frac{r}{100}\]

On further simplification, we get: r = 5%

A person invests ₹ 10,000 for two years at a certain rate of interest compounded annually. At the end of one year this sum amounts to ₹ 11,200. Calculate:

(i) the rate of interest per annum.

(ii) the amount at the end of the second year.

Solution:

(i) \[A = P\left(1 + \frac{r}{100}\right)^n \Rightarrow 11,200 = 10,000\left(1 + \frac{r}{100}\right)^1\]

\[\frac{11,200}{10,000} = 1 + \frac{r}{100}\]

\[r = \frac{112}{100} - 1 = \frac{12}{100}\]

\[r\% = \frac{12}{100} \times 100\% = 12\%\]

Rate of interest p.a. = 12%

(ii) \[A = P\left(1 + \frac{r}{100}\right)^n \Rightarrow A = ₹ 11,200\left(1 + \frac{12}{100}\right)\]

\[= ₹ 11,200 \times \frac{112}{100} = ₹ 12,544\]

Teacher's Note

Compound interest is how your savings grow exponentially over time in a bank account. Understanding the formula helps you plan for your financial future and make better investment decisions.

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