Percentage: Formulas, Tricks, Shortcuts and Questions

Percentage expresses a quantity as a fraction of 100. This chapter covers percentage formulas, conversions, percentage increase and decrease, successive changes, reverse percentage, comparison methods and common percentage questions. The examples use direct calculations and standard shortcuts to solve percentage problems accurately.

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What is Percentage?

Percentage is a way to express a number or quantity out of 100. The symbol used for percentage is %.

The word percentage means per hundred. Thus, x% = x/100. For example, 35% means 35/100 = 0.35. To find p% of a quantity N, multiply N by p/100. Therefore, 20% of 250 = (20/100) × 250 = 50.

Percentage Formula & Tricks

Important Formulas

Basic percentage
Percentage = (Part / Whole) × 100

Use this formula when the part and the whole quantity are known.

Finding the part
Part = (Percentage / 100) × Whole

Use this to calculate a given percentage of a quantity.

Percentage increase
Percentage increase = [(New value − Original value) / Original value] × 100

The original value is always used as the denominator.

Percentage decrease
Percentage decrease = [(Original value − New value) / Original value] × 100

Use the positive difference between the original and new values.

Successive percentage changes
Net change = a + b + (ab / 100)

Use positive signs for increases and negative signs for decreases. The result is the net percentage change.

Reverse percentage
Original value = Final value × 100 / (100 ± p)

Use +p when the final value results from a p% increase and −p when it results from a p% decrease.

Quick Tricks

Exchange the number and percentage

The value of x% of y is equal to y% of x. This can convert an awkward percentage into an easier one.

Example: 16% of 25 = 25% of 16 = 4.
Use 100 as the base

For percentage comparison, assume the original quantity is 100 whenever only relative changes are given.

Example: After a 20% increase, 100 becomes 120. A 20% decrease from 120 gives 96, so the net change is a 4% decrease.
Convert common percentages into fractions

Fraction conversions often reduce calculation time.

Example: 25% = 1/4, 50% = 1/2, 75% = 3/4, 12.5% = 1/8 and 20% = 1/5.
Apply successive changes in order

Two percentage changes are not usually added directly because the second change applies to the changed value.

Example: A 10% increase followed by a 20% increase gives 10 + 20 + (10 × 20)/100 = 32%, not 30%.

Percentage Concepts

Percentage, fraction and decimal conversion

To convert a fraction or decimal into a percentage, express it as a value multiplied by 100.

Fraction to percentage: (Fraction) × 100. Decimal to percentage: Decimal × 100. Percentage to fraction: Percentage/100, followed by simplification. Percentage to decimal: Percentage ÷ 100.

Example: 3/8 = (3/8) × 100 = 37.5%; 0.64 = 64%; 45% = 45/100 = 9/20.

Percentage increase and decrease

Percentage change is calculated relative to the original value, not the new value.

For an increase, use [(New − Original)/Original] × 100. For a decrease, use [(Original − New)/Original] × 100. If an original value A increases by p%, the new value is A(1 + p/100). If it decreases by p%, the new value is A(1 − p/100).

Example: A price rises from ₹400 to ₹460. Increase = ₹60, so percentage increase = (60/400) × 100 = 15%.

Successive percentage changes

Successive percentage changes must be applied one after another because each change uses the latest value as its base.

For changes of a% and b%, net change = a + b + ab/100, with decreases represented by negative values. Equal increases and decreases of p% produce a net decrease of p²/100%.

Example: A value increases by 25% and then decreases by 20%: net change = 25 − 20 − (25 × 20)/100 = 0%. The final value equals the original value.

Reverse percentage calculation

Reverse percentage finds the original value when the final value and percentage change are known.

If a value becomes F after a p% increase, original value = F × 100/(100 + p). If it becomes F after a p% decrease, original value = F × 100/(100 − p).

Example: After a 20% increase, a number is 360. Original number = 360 × 100/120 = 300.

Comparison of two quantities

The percentage by which one quantity is more or less than another is calculated using the reference quantity as the denominator.

A is p% more than B when (A − B)/B × 100 = p. A is p% less than B when (B − A)/B × 100 = p. If A is p% more than B, then B is [p/(100 + p)] × 100% less than A.

Example: 500 is 25% more than 400 because 100/400 × 100 = 25%. However, 400 is 20% less than 500 because 100/500 × 100 = 20%.

Percentage Video Lessons

Watch short topic-wise lessons for quick revision.

8 Lessons
Lesson 1 of 8 Quick Revision

Percentage Basics and Conversions

Understand what percentage means and learn how to convert percentages between fractions, decimals, and related forms using clear foundational methods.

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Practice Percentage Questions

Practise published questions related to this topic.

1After allowing a discount of 60% on the marked price of an article, it is sold for ₹49,834. Find its marked price (in ₹). 647→ 2Which of the following numbers is divisible by 11?→ 3A dealer gives a 20% discount on the marked price of a book. If the selling price of the book is Rs. 840, what is its marked price?→ 4A grocery store gives a Buy 5, Get 2 Free offer. The price of each item is ₹300. A customer needs a total of 21 items and procures the items under this scheme. What is the effective cost per item? (Round off your answer to two decimal places.)→ 5Find the least value of (4K2+2M) for which the five digits number 7KM93 is divisible by 9.→ 6Nisha buys a gold ring for her mother. The ring is marked at ₹88,000 and is offered with two successive discounts of 25% and Z%, respectively. After receiving both discounts, she pays ₹33,000. Find the value of Z.→ 7Simplify: √(5 + √25 + 5² - 3² - 10) - √4→ 8The HCF and LCM of two positive integers are 36 and 7,560, respectively. If one of the numbers is 540, what is the other number?→ 9Find the greatest number that will divide 186, 321 and 402 so as to leave the same remainder in each case.→ 10How many numbers among 1210, 1331, 1442, 1553 are divisible by 11?→

Percentage Quick Quiz

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Quick Revision Notes

Percentage Revision Points

Use these rules for quick revision of percentage calculations.

  • Percentage = (Part/Whole) × 100.
  • p% of N = (p/100) × N.
  • Percentage increase and decrease are calculated using the original value as the denominator.
  • For successive changes a% and b%, net change = a + b + ab/100; use negative values for decreases.
  • An increase of p% followed by a decrease of p% gives a net decrease of p²/100%.
  • After a p% increase, multiply by (100 + p)/100; after a p% decrease, multiply by (100 − p)/100.
  • For reverse percentage, divide the final value by 1 + p/100 for an increase or by 1 − p/100 for a decrease.
  • x% of y equals y% of x.

Percentage FAQs

What is 20% of 350?

20% of 350 = (20/100) × 350 = 70.

How do you convert 0.375 into a percentage?

Multiply the decimal by 100: 0.375 × 100 = 37.5%.

A number increases from 240 to 300. What is the percentage increase?

Increase = 60. Percentage increase = (60/240) × 100 = 25%.

A price of ₹800 is reduced by 15%. What is the new price?

Reduction = 15% of ₹800 = ₹120. New price = ₹800 − ₹120 = ₹680.

What is the net effect of a 10% increase followed by a 10% decrease?

Net change = 10 − 10 − (10 × 10)/100 = −1%. Therefore, there is a 1% decrease.

A number becomes 450 after a 25% increase. What was the original number?

Original number = 450 × 100/(100 + 25) = 360.

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