# Compound Interest

Pratice Compound Interest Questions and answers.

Rule Description
1

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

2

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

3

If rate of compound interest differs from year to year, then

Amount (A) = $P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\left(1+\frac{r_3}{100}\right).....$

4

Compound interest – when interest is compounded annually but time is in fraction

If time = $t \frac{p}{q}$ years, then

$P\left(1+\frac{r}{100}\right)^t\left(1+\frac{\frac{p}{q}r}{100}\right)$

5

Compound interest – when interest is calculated half-yearly

Since r is calculated half-yearly therefore the rate per cent will become half and the time period will become twice, i.e.,

Rate per cent when interest is paid half-yearly = $\frac{r}{2}$%

Time = $2 \times \text{time given in years}$

Hence,

Amount (A) = $P\left(1+\frac{r}{200}\right)^{2n}$

6

Compound interest – when interest is calculated quarterly

Since 1 year has 4 quarters, therefore rate of interest will become $\frac{1}{3}th$ of the rate of interest per annum, and the time period will be 4 times the time given in years, Hence, for quarterly interest

Amount (A) = $P\left(1+\frac{r}{400}\right)^{4n}$

7

Difference between Compound Interest (C.I.) and Simple Interest (S.I.) When T = 2

• $S.I.=P\left(\frac{R}{100}\right)^2$
• $C.I.-S.I.=\frac{R\times S.I.}{200}$
8

Difference between Compound Interest (C.I.) and Simple Interest (S.I.) When T = 3

• $C.I.-S.I.=\frac{PR^2}{10^4}\left(\frac{300+r}{100}\right)$
• $C.I.-S.I.=\frac{S.I.}{3}\left[\left(\frac{R}{100}\right)^2+3\left(\frac{R}{100}\right)\right]$
9

NOTE: SI and CI for one year on the same sum and at same rate are equal.

10

If a certain sum at compound interest becomes x times in n1 yr and y times in n2 yr, then

$x^{\frac{1}{n_1}}=y^{\frac{1}{n_2}}$

11

If the population of a city is P and it increases with the rate of R% per annum, then

• Population after n yr = $P\left(1+\frac{r}{100}\right)^n$
• Population n yr ago = $\frac{P}{\left(1+\frac{r}{100}\right)^n}$
12

If the population of a city is P and it decreases with the rate of R% per annum, then

• Population after n yr = $P\left(1-\frac{r}{100}\right)^n$
• Population n yr ago = $\frac{P}{\left(1-\frac{r}{100}\right)^n}$
13

If the rate of growth per year is R1%, R2%, R3%, ......., Rn%,

then Population after n yr =

$P\left(1+\frac{R_1}{100}\right)\left(1+\frac{R_2}{100}\right)\left(1+\frac{R_3}{100}\right).....\left(1+\frac{R_n}{100}\right)$

NOTE: This formula can also be used, if there is increase/decrease in the price of an article.