Percentage Notes for Competitive Exams

Percentage is the most important chapter in quantitative aptitude. It is the base of profit and loss, discount, simple and compound interest, data interpretation, mixtures, population problems and many other topics, and almost every competitive examination in India, from SSC, banking and railways to state exams, asks questions on it. If you can handle percentages quickly, you will find the rest of arithmetic much easier. This note gives you a complete, practical guide: the meaning, the shortcuts, the multiplier method, 13 types of questions with solved examples and the common traps, in simple language.

Meaning and Basic Formulas

"Per cent" means "for every hundred". So 35 percent means 35 out of every 100, or 35/100. Percentage is only a way to compare quantities on the common base of 100. The basic formulas are: Percentage = (Part/Whole) × 100, Part = Percentage × Whole/100, and Whole = Part × 100/Percentage. To convert a fraction into a percentage, multiply by 100; to convert a percentage into a fraction, divide by 100 and simplify. A decimal is converted to percentage by moving the decimal point two places to the right.

Fraction-Percentage Table (Learn This First)

Multiples follow easily: 3/8 is 3 × 12.5 = 37.5 percent, 5/6 is 5 × 16.67 = 83.33 percent, and 2/7 is 28.57 percent. Knowing this table lets you convert most calculations into simple fractions and avoid long multiplication.

The Multiplier Method

A very useful method is to convert every percentage change to a multiplier. An increase of r percent multiplies by (1 + r/100), and a decrease of r percent multiplies by (1 − r/100). So 15 percent increase is × 1.15, and 20 percent decrease is × 0.8. For successive changes, simply multiply the multipliers. For example, a number 800 increased by 15 percent becomes 800 × 1.15 = 920.

Type 1: Basic Percentage Questions

15 percent of 360 = 360 × 15/100 = 54. What percent of 250 is 40? It is 40/250 × 100 = 16 percent. If 3/5 of a number is 120, the number is 120 × 5/3 = 200. A useful trick: x percent of y = y percent of x, so 16 percent of 25 equals 25 percent of 16, which is 4.

Type 2: "More Than" and "Less Than" Comparison

If A is 25 percent more than B, then B is less than A by 25/125 × 100 = 20 percent. If A is 20 percent less than B, then B is more than A by 20/80 × 100 = 25 percent. The same quantity is the base in one statement and the other in the next, so the percentages differ. Remember the pairs: 25 and 20, 20 and 16.67, 50 and 33.33, 10 and 9.09.

Type 3: Percentage of Percentage

20 percent of 30 percent of 500: first 30 percent of 500 = 150, and 20 percent of 150 = 30. Or multiply the multipliers: 500 × 0.3 × 0.2 = 30.

Type 4: Successive Percentage Change

For two successive changes of a percent and b percent, the net change is a + b + ab/100. Two successive discounts of 20 percent and 10 percent equal 20 + 10 − 2 = 28 percent. On an article of marked price 500 rupees, the final price is 500 × 0.8 × 0.9 = 360 rupees, which is 28 percent lower. A rise of 20 percent followed by a fall of 20 percent gives a net decrease of 4 percent.

Type 5: Population and Growth

If a town has 50,000 people, the population decreases by 4 percent in the first year and increases by 10 percent in the second year. The population after two years is 50,000 × 0.96 × 1.10 = 52,800. For a constant growth rate of r percent per year for n years, the formula is P × (1 + r/100) raised to n, which is the compound pattern.

Type 6: Expenditure and Savings

A man spends 40 percent of his salary on rent and 25 percent of the remaining on food, and saves 13,500 rupees. After rent, 60 percent of salary remains; food takes 25 percent of that, which is 15 percent of the salary; so the savings are 45 percent of the salary. Therefore the salary is 13,500 × 100/45 = 30,000 rupees. Always work step by step with the salary as 100 units.

Type 7: Marks and Results

A candidate scores 35 percent and fails by 20 marks. Another scores 45 percent and gets 40 marks more than the pass marks. The difference between the two scores is 10 percent of the total, which equals 20 + 40 = 60 marks. So the total marks are 600. The pass marks are 35 percent of 600 plus 20, that is, 210 + 20 = 230. Check: 45 percent of 600 is 270, and 270 − 230 = 40, which matches.

Type 8: Votes and Elections

In an election with 10,000 votes, 20 percent are invalid. The valid votes are 8,000. If the winner got 60 percent of the valid votes, the winner got 4,800, the loser got 3,200, and the margin is 1,600 votes. Note that the percentage is on valid votes, not on the total votes.

Type 9: Mixture and Solution

If 10 litres of a solution contains 20 percent alcohol and 10 litres of water is added, the alcohol remains 2 litres, but the total becomes 20 litres, so the strength falls to 10 percent. The key idea is that the amount of the pure substance does not change when only water is added.

Type 10: Area and Geometry

If the side of a square increases by 10 percent, the area increases by 10 + 10 + (10 × 10)/100 = 21 percent. If the length of a rectangle increases by 20 percent and the breadth decreases by 10 percent, the area changes by 20 − 10 − (20 × 10)/100 = 8 percent, that is, an increase of 8 percent. For a circle, a 10 percent increase in the radius gives a 21 percent increase in the area.

Type 11: Price and Consumption

If the price of sugar increases by 25 percent, a family can keep the expenditure the same by reducing consumption by 25/125 × 100 = 20 percent. If the price falls by 20 percent, consumption can rise by 20/80 × 100 = 25 percent without changing expenditure.

Type 12: Profit, Loss and Discount in Percentage

Profit percent is always on the cost price, and discount percent is on the marked price. If a shopkeeper marks an item 30 percent above cost and allows a 10 percent discount, the profit is 1.3 × 0.9 = 1.17, which is 17 percent. We shall study this in more detail in the profit and loss chapter.

Type 13: Percentage in Data Tables

When data are given in tables, find the percentage change using (new − old)/old × 100, and remember the difference between percent and percentage point. Rising from 10 percent to 12 percent is a 2 percentage-point increase, which is a 20 percent rise.

Common Traps

Students often use the wrong base when the question reverses the comparison; add successive percentages without the product term; forget to use valid votes; or calculate on the original value after a change. Always ask, "Percent of what?" before you compute.

Mental Maths Practice Routine

Speed in percentage comes from small daily habits. Every morning, do ten quick conversions such as 37.5 percent to 3/8 and 62.5 percent to 5/8. Take any price you see in a shop and calculate a 10 percent discount, a 15 percent tip or 18 percent tax mentally. For 18 percent of a number, find 10 percent, add 5 percent (half of 10 percent) and then add 3 percent. For 15 percent, find 10 percent and add half of it. For 5 percent, halve 10 percent. These breaking-down tricks work for any number and remove the fear of long multiplication. Also practise comparing percentages without a calculator: ask which is bigger, 7/12 or 5/9, and use cross-multiplication. When you review your mistakes, mark whether they came from the wrong base, wrong conversion or a calculation slip, and solve the same type again after two days. Over a few weeks, this routine builds the mental agility that competitive exams demand.

Quick Revision

• Learn the fraction-percentage table; use multipliers for increase and decrease.
• Successive change = a + b + ab/100.
• A is r percent more than B: B is r/(100 + r) × 100 percent less than A.
• Always identify the base before calculating.

Practise five questions of each type daily, and within two weeks you will notice that percentage questions take less than 30 seconds. This confidence will support every other chapter of arithmetic.

Related Notes to Read Next

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

• Ratio and Proportion Notes: Formulas, Tricks and Solved Examples
• Time and Work Notes for Competitive Exams
• Profit, Loss and Interest Notes for Competitive Exams
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

What is percentage?

Percentage is a way of expressing a number as a fraction of 100. The symbol % means "out of every hundred".

What is the formula for percentage?

Percentage equals the part divided by the whole, multiplied by 100.

How do I find the percentage increase?

Percentage increase is the increase divided by the original value, multiplied by 100.

If A is 25 percent more than B, by what percent is B less than A?

B is less than A by 20 percent, calculated as 25/125 × 100.

What is the formula for successive percentage change?

For successive changes of a and b percent, the net change is a + b + (ab/100), with negative values for decreases.

What is the net discount for successive discounts of 20 percent and 10 percent?

The net discount is 28 percent, since 20 + 10 − (20 × 10)/100 equals 28.

How does a 10 percent increase in the side of a square affect its area?

The area increases by 21 percent, because 1.1 × 1.1 equals 1.21.

What is the multiplier method?

It converts a percentage change into a multiplying factor, such as 1.15 for a 15 percent increase and 0.8 for a 20 percent decrease, to make calculations quick.

How do I calculate the reduction in consumption when the price rises?

If the price rises by r percent, consumption must fall by r/(100 + r) × 100 percent to keep expenditure unchanged.

What is the difference between percent and percentage point?

A percentage point is the arithmetic difference between two percentages, while a percent change is the difference relative to the original value.

Which fractions should I memorise for percentage?

Memorise fractions from 1/2 to 1/16 along with 1/20, 1/25 and 1/50 with their percentage values for quick conversion.

How can I improve speed in percentage problems?

Learn the fraction table, use multipliers, practise the standard question types daily and always identify the base before you calculate.