# Compound Interest

The compound interest questions hold good weightage in the CAT quantitative aptitude section. Learn the topic in detail along with its application with various solved examples.

### Definition Of Compound Interest

It is called â€śInterest on Interestâ€ť. At the end of a year or another fixed period, the interest that is due is not paid to the creditor, but it is instead added to the sum lent and the amount obtained becomes the principal for the next year or period. This process is repeated until the amount for the last year is found. The amount which accrues after obtaining Compound Interest is calculated as follows:

#### â‡’ Amount = P(1+R/100)N

Where,

Pâžś Principal,

Nâžś Number of years,

Râžś Rate of interest

Compound Interest= Amount – Principal

Example: What will be the amount for a sum of Rs 1000 at 10% for 3 years compounded annually?

Solution:

Amount = P(1+R/100)N

= 1000(1+10/100)3

=1000(110/100)3 = 1331

Note:- 1) In case interest is paid half-yearly, N will become the number of half years and R will be the rate percent per half year

Number of years x 2= Number of half years (Rate percent per year)/2= Rate percent per 6 months extrapolating from this,

If the interest is paid quarterly (once in 3 months), Number of years x 4= Number of quarters

(Rate percent per year)/4 = Rate percent per 3 months or rate percent per quarter, when rates are different for different years (say R1 and R2 for two consecutive years), we can calculate the total amount as P(1+R_1/100)x(1+R_2/100).

### Application of Compound Interest

• #### Population Appreciation/Depreciation Over Time:

if the original population of a town is P, and the annual increase is R%, the population of the town in â€śNâ€ť years will be

P(1+R/100)N If the original population of a town is P, and the annual decrease is R%, the population of the town in â€śNâ€ť years will be P(1-R/100)N

### Examples of Compound Interest

Example 1) If the annual increase in the population of a village is 4% and the population at present is 17576, then what was the population 3 years ago?

Solution:

From P(1+R/100)N , we get

Present population =Population 3 years ago x(104/100)3

Population 3 years ago x 263/253 = 17576

population 3 years ago =17576 x 253/263 = 15625

Example 2) Population of a village is 10000 in year 2000 and is 14400 in 2002. Find the Compounded Annual Growth Rate?

Solution:

CAGR implies the annual increase in growth.

CAGR = [(14400/10000)(1/2)-1]x 100 = 20 %

This means an annual increase of 20 %. To verify 10000 x 1.2 = 12000& 12000 x 1.2 = 14400

A factor of multiplication 1.2 is used to increase a number by 20% and to get the final value, one needsÂ to multiply by 120/100 = 1.2 CAGR is similar to rate percent, R in compound Interest calculations.

Final amount = P(1+R/100)n where P = initial amount, R = rate percent, n = number of years.

So, CAGR or R = [((final amount)/(initial amount))(1/n)â€“ 1] x100.

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