Simple and Compound Interest
Interest is the price paid for the use of money, and the two ways of calculating it, simple and compound, appear in every ADRE Grade IV paper. In simple interest the interest is calculated only on the original principal for the whole period, while in compound interest the interest of each period is added to the principal and itself earns interest in the next period. The method followed in the Master Book: Maths Masti (D. Hurina) is to learn the two formulas and then to solve questions by substituting carefully, because the arithmetic is not difficult but the terms are easy to mix up. The chapter connects directly with percentage and with profit and loss, and the idea of compound interest explains why a small rate over a long time produces a large growth.
Basic Terms
- Principal: the sum of money borrowed or invested.
- Rate: the interest for one year on every hundred rupees, written as a percentage per annum.
- Time: the period for which the money is used, taken in years.
- Simple interest: interest calculated only on the principal.
- Amount: the principal together with the total interest.
- Compound interest: interest calculated on the principal and on the interest already earned.
Simple Interest and Its Formula
Simple interest equals principal multiplied by rate multiplied by time, divided by one hundred. The amount is the principal plus the simple interest, which is principal multiplied by one hundred plus rate multiplied by time, divided by one hundred. The formula assumes that the rate is per annum and the time is in years; if the time is given in months it must first be converted into years by dividing by twelve. Simple interest is easy to recognise because the interest for each year is the same.
Finding Principal, Rate and Time
The same formula can be rearranged for each unknown. Principal equals simple interest multiplied by one hundred divided by rate multiplied by time. Rate equals simple interest multiplied by one hundred divided by principal multiplied by time. Time equals simple interest multiplied by one hundred divided by principal multiplied by rate. In the examination these rearrangements appear as questions that give the interest and two of the three quantities, and the safe habit is to write the formula first and then substitute.
Compound Interest and Its Formula
Compound interest is calculated on the amount of the previous period, so the interest itself earns interest. The amount after a number of years is the principal multiplied by the quantity one plus rate over one hundred, raised to the power equal to the number of years. The compound interest is this amount minus the principal. When the time is small the calculation can be done year by year, and this step-by-step method is often safer than the formula in an examination.
Half-Yearly and Quarterly Compounding
When interest is compounded half-yearly, the rate is halved and the time is doubled. When it is compounded quarterly, the rate is divided by four and the time is multiplied by four. These two adjustments must always be made together, and forgetting one of them is a standard mistake. The same formula is then applied with the adjusted rate and the adjusted number of periods.
Difference Between Simple and Compound Interest
For two years, the difference between compound interest and simple interest is the principal multiplied by the square of rate over one hundred. For three years, the difference is the principal multiplied by the square of rate over one hundred, multiplied by three hundred plus rate, divided by one hundred. These ready results turn what looks like a long calculation into a single line, and they are among the most useful formulas of the chapter.
Compound Interest by Yearly Steps
When the number of years is small it is often quicker to build the table year by year. The interest for the first year is found by the simple interest formula, it is added to the principal to give the amount, and the second year's interest is then found on this new amount. Repeating the step gives the final amount. This method also makes it easy to answer questions that ask for the interest of a particular year rather than the whole period.
Effective Rate and Growth
Because compound interest is paid on interest, the effective growth over a year is a little more than the stated rate when the compounding is done more than once a year. This is the same successive-percentage idea studied earlier, and it explains why the same nominal rate gives a slightly larger amount under half-yearly compounding than under yearly compounding. Recognising this connection makes the chapter feel like a natural extension of percentage rather than a new subject.
Relation with Percentage
Both interest formulas are simply applications of percentage. In simple interest the same rate is applied every year to the original principal, so the interest for each year is equal. In compound interest the rate is applied each year to the amount standing at the beginning of that year, so compound interest is an example of successive percentage increase. Keeping this idea in mind makes the difference between the two, and the reason the effective rate under more frequent compounding is slightly higher, easy to understand.
How This Chapter Appears in the Examination
Interest questions in ADRE Grade IV, APSC and Assam Police papers usually ask the candidate to find the simple interest or the amount, to find the principal, rate or time, to find the compound interest for two or three years, or to find the difference between simple and compound interest. The numbers are usually chosen so that the formula simplifies neatly, and a candidate who writes the formula before substituting rarely goes wrong. The chapter is a dependable scoring area once the two main formulas are secure.
Common Mistakes to Avoid
- Using the time in months without converting it into years.
- Applying the compound interest formula when simple interest is asked, or the reverse.
- Forgetting to halve the rate and double the time for half-yearly compounding.
- Reporting the compound interest as the amount, without subtracting the principal.
- Adding the two years' simple interests when compound interest is required.
How to Prepare This Topic
- Learn the simple interest formula and the meaning of amount.
- Practise finding the principal, rate and time from the same formula.
- Learn the compound interest formula and the year-by-year method.
- Memorise the two-year and three-year differences between simple and compound interest.
- Finish with mixed questions on half-yearly and quarterly compounding.
Frequently Asked Questions
What is the main difference between simple and compound interest?
In simple interest the interest is calculated only on the original principal for the whole period. In compound interest the interest of each period is added to the principal and earns interest in the following period.
Why is half-yearly compound interest slightly more than yearly?
Because the interest is added to the principal twice in the year, so the second half also earns interest on the first half's interest. The same nominal rate therefore produces a slightly larger amount.
How do I find the compound interest for two years?
You may use the formula, or find the first year's simple interest, add it to the principal, and then find the second year's interest on that amount. The sum of the two interests is the compound interest.
What time unit must I use in the interest formulas?
The rate is per annum, so the time must be expressed in years. Convert months into years by dividing by twelve, and express any part of a year as a fraction.
What is the difference between simple and compound interest for two years?
It is the principal multiplied by the square of the rate divided by one hundred. This ready formula is much faster than calculating both interests separately.