Profit, Loss and Interest Notes for Competitive Exams

Profit and loss, and simple and compound interest, are the money-related chapters of quantitative aptitude. They test how well you apply percentages to buying, selling and borrowing, and they appear in every competitive examination such as CSAT, SSC, banking and railways. The formulas are few, and the questions follow predictable patterns. This note explains the terms and formulas of profit, loss and discount, then simple interest and compound interest, with shortcuts and solved examples, in simple language so that you can master the chapter quickly.

Basic Terms in Profit and Loss

The key formulas are: Profit percent = (Profit/CP) × 100 and Loss percent = (Loss/CP) × 100. Both are always calculated on the cost price. To find the selling price: SP = CP × (100 + profit percent)/100 for a profit, and SP = CP × (100 − loss percent)/100 for a loss. For example, if CP is 400 rupees and the profit is 25 percent, then SP = 400 × 125/100 = 500 rupees. Discount is always calculated on the marked price: SP = MP × (100 − discount percent)/100. If MP is 800 rupees and the discount is 15 percent, SP = 680 rupees. If the CP was 600 rupees, the profit is 80 rupees and the profit percent is 80/600 × 100 = 13 1/3 percent.

Important Shortcuts

Marked price above cost. A shopkeeper wants a 20 percent profit after giving a 20 percent discount. Let CP be 100. SP must be 120. Since SP = 0.8 × MP, MP = 150. So the shopkeeper should mark the goods 50 percent above cost. Same selling price, equal profit and loss. If two articles are sold at the same price, one at x percent profit and the other at x percent loss, the overall result is always a loss of x²/100 percent. For x = 20, the loss is 400/100 = 4 percent. Dishonest shopkeeper. A shopkeeper who sells at cost price but uses a weight of 900 grams in place of 1 kilogram gains (100/900) × 100 = 11.11 percent. Cost of some articles equals the sale of others. If the cost of 15 articles equals the selling price of 12 articles, then the profit is (15 − 12)/12 × 100 = 25 percent. Successive discounts. Discounts of 20 percent and 10 percent together equal 20 + 10 − 2 = 28 percent.

Solved Examples on Profit and Loss

A trader buys an article for 2,000 rupees, spends 200 rupees on repairs and sells it for 2,640 rupees. The total cost is 2,200 rupees, the profit is 440 rupees, and the profit percent is 440/2,200 × 100 = 20 percent. Another example: a man sells two articles for 1,200 rupees each; on one he gains 20 percent and on the other he loses 20 percent. The overall loss is 4 percent of the total cost, which can be found by the shortcut. The total SP is 2,400 rupees; the CPs are 1,000 and 1,500 rupees (since 1,200/1.2 = 1,000 and 1,200/0.8 = 1,500), a total of 2,500 rupees, so the loss is 100 rupees, which is 4 percent. The result matches the formula.

Simple Interest (SI)

Interest is the money paid for using borrowed money. The sum borrowed is the principal (P), the yearly percentage is the rate (R), and the duration is the time (T) in years. In simple interest, interest is calculated only on the original principal. SI = (P × R × T)/100 and Amount = P + SI. If P = 5,000, R = 8 percent and T = 3 years, then SI = 5,000 × 8 × 3/100 = 1,200 rupees and the amount is 6,200 rupees. For time given in months or days, convert it to years: 6 months is 0.5 year, and 73 days is 73/365 = 0.2 year.

Useful results: a sum doubles in 100/R years at simple interest. At 8 percent, it doubles in 12.5 years. If a sum becomes 3 times in 8 years, the interest is twice the principal, so R = 200/8 = 25 percent per year. When a person lends the same sum to two people at different rates, subtract to find the difference in interest. For instalments or money lent in parts, find each part separately and add.

Compound Interest (CI)

In compound interest, the interest of each period is added to the principal, and the next interest is calculated on the new amount. Amount = P × (1 + R/100) raised to the power T, and CI = Amount − P. For P = 10,000 rupees, R = 10 percent and T = 2 years, the amount is 10,000 × 1.1 × 1.1 = 12,100 rupees, and the CI is 2,100 rupees. The SI on the same sum is 2,000 rupees. The difference for two years is P × (R/100)², which equals 10,000 × 0.01 = 100 rupees. For three years, the difference between CI and SI is P × (R/100)² × (3 + R/100).

Interest compounded half-yearly or quarterly. Divide the rate by the number of periods in a year and multiply the time by the same number. Take P = 8,000 rupees at 20 percent per year for 1 year compounded half-yearly. The rate per half-year is 10 percent and the number of periods is 2. The amount is 8,000 × 1.1 × 1.1 = 9,680 rupees. Notice that this is more than the yearly compounding (9,600 rupees), since interest is added more often.

SI versus CI: Quick Comparison

For the first year, SI and CI are equal. After that, CI is higher. The difference grows over time. Growth under CI is exponential, while under SI it is linear. When the question asks for "the difference between CI and SI for 2 years", use the shortcut P(R/100)². The same compound formula is used for population growth, depreciation (with a negative rate) and growth of investments. Remember the approximate "rule of 72": at a compound rate of R percent, money doubles in about 72/R years, so at 8 percent it takes about 9 years.

Common Mistakes

Students often calculate profit percent on the selling price, which is wrong. They ignore overhead expenses like transport and repair in the cost. They apply discount on the cost price, not on the marked price. In interest questions, they forget to convert months into years or to divide the rate when compounding is half-yearly. Another error is to find the CI amount and forget to subtract the principal. Always write the data (CP, SP, MP or P, R, T) at the start, and check the answer using a rough estimate.

More Practice With Answers

Try these on your own. First, a shopkeeper buys 20 pens for 200 rupees and sells them at 12 rupees each. The CP is 10 rupees per pen and the SP is 12 rupees, so the profit is 20 percent. Second, a television with a marked price of 25,000 rupees is sold at two successive discounts of 10 percent and 20 percent. The selling price is 25,000 × 0.9 × 0.8 = 18,000 rupees. Third, at what rate of simple interest will 6,000 rupees become 7,200 rupees in 4 years? The interest is 1,200 rupees, so R = 1,200 × 100/(6,000 × 4) = 5 percent. Fourth, find the compound interest on 5,000 rupees at 20 percent per year for 2 years. The amount is 5,000 × 1.2 × 1.2 = 7,200 rupees, so the CI is 2,200 rupees. In each case, notice how writing the data first and choosing the right formula makes the solution short. Keep practising until you can do these in your head.

Applying the Ideas in Real Life

These chapters are useful for daily money decisions. When a shop offers "buy two, get one free", it is the same as a discount of 33.33 percent. When a bank offers a loan at a flat rate, the real cost is higher than the same rate on a reducing balance. A fixed deposit that pays compound interest grows faster than one that pays simple interest. Knowing these ideas helps you avoid being cheated and make better financial choices, which is also part of the financial literacy that every citizen needs.

Quick Revision

• Profit or loss percent is on the cost price; discount is on the marked price.
• Equal sale price with x percent profit and x percent loss gives a loss of x²/100 percent.
• SI = PRT/100; CI amount = P(1 + R/100)^T; CI − SI for 2 years = P(R/100)².
• For half-yearly compounding, halve the rate and double the time.

Practise a few questions of each type every day, keep the formulas written on a single page, and revise the shortcuts. With this approach, profit, loss and interest become one of the most scoring parts of the paper.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Number System Notes for Competitive Exams
• Average, Mixture and Alligation Notes
• Mensuration Notes for Competitive Exams
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

How is profit percent calculated?

Profit percent is the profit divided by the cost price, multiplied by 100.

On which price is discount calculated?

Discount is always calculated on the marked price of an article.

What happens if two articles are sold at the same price, one at x percent profit and one at x percent loss?

There is always an overall loss of x²/100 percent.

What is the formula for simple interest?

Simple interest equals principal multiplied by rate multiplied by time, divided by 100.

What is the formula for compound interest?

The amount equals P multiplied by (1 + R/100) raised to the power T, and the compound interest is the amount minus the principal.

What is the difference between CI and SI for two years?

The difference is P × (R/100)², where P is the principal and R is the rate per cent per year.

How do I calculate compound interest half-yearly?

Divide the annual rate by 2 and multiply the number of years by 2, then use the compound interest formula.

In how many years will a sum double at simple interest?

A sum doubles in 100/R years at a simple interest rate of R percent per year.

What is the marked price?

The marked price is the listed price on an article from which a discount may be given.

What are successive discounts?

Successive discounts are discounts given one after another. Two discounts of a and b percent are equal to a + b − ab/100 percent.

What is the rule of 72?

It is an approximate rule: at a compound interest rate of R percent, money doubles in about 72/R years.

How can I avoid mistakes in profit and loss problems?

Write CP, SP and MP clearly, calculate profit or loss on the cost price, include all expenses in cost and verify the answer with a quick estimate.