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Arithmetic I

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Interest & Growth
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CAT 2025 Lesson : Interest & Growth - Basics & Simple Interest

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1. Introduction

When a certain amount is lent, the lender needs to be compensated for parting with the money lent for the time period, possible inflation or even the risk of the borrower defaulting. Therefore, the borrower is required to pay an interest as a compensation. This interest is almost always applied as a percentage on the principal or amount and directly varies with the time period. The following are the key components in the interest calculation.

Principal: The amount lent or invested

Interest Rate: Rate or percentage of the principal at which the borrower needs to pay as interest for a given time period. Interest rate is typically written as, say, 12%12 \% per annum, which means interest is to be paid at 12%12 \% of the principal or amount (as the case maybe) every year.

Time: This is the duration of the loan or investment. In the case of compound interest, time is broken down into time periods for compounding. This is detailed in later sections.

Type of Interest: There are typically two forms of interest – Simple Interest and Compound Interest. Simple Interest is wherein the interest rate is applied on the principal component only. Compound Interest applies the interest rate on the principal component as well as the interest accrued till the previous time period (when interest was compounded).

Compounding Period: This is applicable only in the case of Compound Interest. Interest can be compounded for any defined time period. Banks in India typically compound their loans on a monthly basis. For example, an interest rate of 12%12 \% per annum compounded semi-annually for a 22 year period, effectively means an interest of 6%6 \% per semi annum for 44 time periods of 66 months each. This is detailed in later sections.

Amount: This is the sum of principal and interest that the borrower needs to pay to the lender at the end.

Example 1

Ram borrowed Rs. 1,0001,000 from Shyam on 1st1^{\text{st}} January 20182018. This sum is to be repaid with interest on 31st31^{\text{st}} December 20202020. Interest to be paid is 12%12 \% per annum of the amount initially borrowed. How much did Ram pay Shyam on 31st31^{\text{st}} December 20202020?

(1) Rs. 1,1201,120            (2) Rs. 1,2401,240            (3) Rs. 1,3601,360            (4) Rs. 1,4801,480           

Solution

Interest due per year =1000×12100= 1000 \times \dfrac{12}{100} = Rs. 120

Note that the amount was borrowed at the start of 20182018 till the end of 20202020.

∴ The amount has been borrowed for 3 years.

Total interest =120×3== 120 \times 3 = Rs. 360

Total Amount repaid = Principal + Interest
=1,000+360== 1,000 + 360 = Rs. 1,360

Answer: (3) 13601360

The above example pertains to Simple Interest as interest is a percentage of the initial principal borrowed and not the accumulated interest. We can directly calculate the interest as follows.

Total Interest =1000×12%×3years=1000×12100×3= 1000 \times 12 \% \times 3 \text{years} = 1000 \times \dfrac{12}{100} \times 3 = Rs. 360

2. Simple Interest

In Simple Interest, the interest is calculated only on the principal lent or invested.

Simple Interest is not paid on interest accumulated during the duration of the loan/investment.

Simple Interest (SI) =pnr100= \dfrac{pnr}{100}

where p\bm{p} is the principal;
n\bm{n} is the number of time periods of the loan/investment; and
r\bm{r} is the rate of interest for a given time period, expressed as r\bm{r}%.

Time period is typically a year. Therefore, rate of interest is generally provided as r\bm{r}% per annum for n number of years. Note that if the time period is provided in months, the same needs to be converted to the time period of the interest rate.

Example 2

Rohan invests Rs. 24,00024,000 for 2020 months in a deposit which yields simple interest of 8%8 \% per annum. What is the amount that Rohan receives at maturity?

Solution

Interest is 8%8 \% per year and time period is 2020 months. We need to convert one of the units to the other. In this case, we shall convert the time period from months to years.

p\bm{p} = Rs. 24,000
n\bm{n} = 20 months = 2012\dfrac{20}{12} years = 53\dfrac{5}{3} years
r=8%\bm{r} = 8 \% p.a.
Simple Interest =pnr100=24000×53×8100= \dfrac{pnr}{100} = \dfrac{24000 \times \dfrac{5}{3} \times 8}{100} = Rs. 3200

Rohan receives the principal and interest on maturity.

Amount = Principal + Interest = 24,000+3,200=24,000 + 3,200 = Rs. 27,20027,200

Answer: Rs. 27,20027,200

Example 3

Ram places Rs. 18,00018,000 in a deposit yielding simple interest. At what interest rate per annum will the deposit grow to Rs. 19,89019,890 in 1818 months?

Solution

p=18,000\bm{p} = 18,000
n=18\bm{n} = 18 months =32= \dfrac{3}{2} years

Let r%\bm{r \%} be the interest rate per annum.

Simple Interest (SI) =19,890−18,000=1,890= 19,890 - 18,000 = 1,890

SI =pnr100= \dfrac{pnr}{100}

⇒ 1890=18000×32×r100 1890 = \dfrac{18000 \times \dfrac{3}{2} \times r}{100}

⇒ r=1890270=7 r = \dfrac{1890}{270} = 7

Rate of interest =7%= 7 \% per annum

Answer: 7%7 \%

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