Mastering Percentage Shortcuts for Competitive Exams

Competitive exams like the CAT often feature questions involving percentages. While direct calculation is possible, employing shortcut techniques can significantly save time and improve accuracy. This module focuses on understanding and applying these shortcuts for efficient problem-solving.

Understanding the Core Concept: Percentages

A percentage is a number or ratio expressed as a fraction of 100. The word 'percent' means 'per hundred'. For example, 25% means 25 out of 100, or 25/100, which simplifies to 1/4. Understanding this fundamental relationship is key to all percentage shortcuts.

What does the symbol '%' represent?

It represents 'per hundred' or a fraction out of 100.

Key Percentage Equivalents

Memorizing common percentage-to-fraction conversions is the first step towards rapid calculations. These equivalents allow you to bypass lengthy multiplication and division.

Percentage	Fraction	Decimal
100%	1/1	1.00
50%	1/2	0.50
25%	1/4	0.25
20%	1/5	0.20
10%	1/10	0.10
5%	1/20	0.05
1%	1/100	0.01

Shortcut 1: Calculating Percentages of Numbers

To find X% of Y, you can use the formula (X/100) * Y. However, shortcuts leverage the fractional equivalents or distributive properties.

To find X% of Y, convert X% to its fractional equivalent and multiply by Y.

For example, to find 20% of 150, convert 20% to 1/5. Then, (1/5) * 150 = 30.

This method is particularly effective for percentages that have simple fractional forms. For instance, 12.5% of 240 is (1/8) * 240 = 30. When the percentage is not a common fraction, you can break it down. For example, 35% of 200 can be calculated as (30% of 200) + (5% of 200). Since 10% of 200 is 20, 30% is 60. 5% is half of 10%, so it's 10. Therefore, 35% of 200 is 60 + 10 = 70.

How can you quickly find 15% of 80?

Break it down: 10% of 80 is 8. 5% of 80 is half of 10%, which is 4. So, 15% of 80 is 8 + 4 = 12.

Shortcut 2: Percentage Increase and Decrease

When a number is increased or decreased by a certain percentage, you can directly calculate the new value.

To increase a number by X%, multiply it by (1 + X/100). To decrease by X%, multiply by (1 - X/100).

If a salary of $1000 is increased by 10%, the new salary is$ 1000 * (1 + 10/100) = $1000 * 1.10 =$ 1100.

This shortcut is based on the idea that the original number represents 100%. An increase of X% means the new value is 100% + X%. Similarly, a decrease of X% means the new value is 100% - X%. For example, to find a number that is 25% more than 80, you calculate 80 * (1 + 25/100) = 80 * 1.25 = 100. To find a number that is 20% less than 50, you calculate 50 * (1 - 20/100) = 50 * 0.80 = 40.

Visualizing percentage change: Imagine a bar representing the original value (100%). An increase adds a segment representing the percentage increase, making the total bar (100% + X%). A decrease removes a segment, leaving (100% - X%). This visual helps understand why we multiply by (1 + X/100) or (1 - X/100).

📚

Text-based content

Library pages focus on text content

What factor do you multiply a number by to decrease it by 30%?

0.70 (or 1 - 0.30)

Shortcut 3: Successive Percentage Changes

When a quantity undergoes multiple percentage changes, the order matters, and a direct summation of percentages is incorrect. The shortcut involves applying each change sequentially.

Apply successive percentage changes by multiplying the original value by the factors corresponding to each change.

If a price is increased by 10% and then decreased by 20%, start with the original price, multiply by 1.10, and then multiply the result by 0.80.

Consider an item priced at $100. If it's increased by 10%, the new price is$ 100 * 1.10 = $110. If this new price is then decreased by 20%, the final price is$ 110 * (1 - 20/100) = $110 * 0.80 =$ 88. The overall change is not a simple +10% - 20% = -10%. Instead, the net effect is $100 * 1.10 * 0.80 =$ 88, which is a 12% decrease. The general formula for successive percentage changes is Original Value * (1 ± X/100) * (1 ± Y/100) * ... = Final Value.

A number is increased by 20% and then by 30%. What is the combined percentage increase factor?

1.20 * 1.30 = 1.56

Shortcut 4: When A is X% More/Less Than B

This type of question often involves comparing two quantities. The key is to correctly identify the base for the percentage calculation.

If A is X% more than B, then B is less than A by a different percentage. The base for the second percentage is A.

If A is 25% more than B, then A = 1.25 * B. To find how much less B is than A, we calculate ((A - B) / A) * 100%. Substituting A = 1.25B, we get ((1.25B - B) / 1.25B) * 100% = (0.25B / 1.25B) * 100% = (1/5) * 100% = 20%. So, B is 20% less than A.

This is a common pitfall. Always pay attention to what the percentage is 'of'. If 'A is X% more than B', it means A = B + (X/100)*B = B(1 + X/100). If you need to find 'B is how much less than A?', the calculation is (A-B)/A * 100%. Conversely, if 'A is X% less than B', then A = B - (X/100)*B = B(1 - X/100). To find 'B is how much more than A?', the calculation is (B-A)/A * 100%.

If Ram's salary is 20% less than Shyam's, by what percentage is Shyam's salary more than Ram's?

Let Shyam's salary be 100. Ram's salary is 100 * (1 - 20/100) = 80. Shyam's salary is more than Ram's by ((100 - 80) / 80) * 100% = (20/80) * 100% = 25%.

Applying Shortcuts in Mock Tests

The true test of these shortcuts is their application under timed conditions. Practice is crucial to internalize these methods and apply them instinctively during mock tests. Focus on recognizing patterns and choosing the most efficient shortcut for each problem.

Time management is as critical as accuracy. These shortcuts are your tools to conquer the clock in competitive exams.

Learning Resources

CAT Percentage Concepts and Shortcuts(blog)

This blog post provides a comprehensive overview of percentage concepts and various shortcut techniques relevant for the CAT exam.

Percentage Tricks for Competitive Exams(documentation)

IndiaBIX offers a collection of percentage tricks and formulas, explained with examples, ideal for quick revision.

Understanding Percentages - Khan Academy(video)

A foundational video explaining what percentages are and how they are used, suitable for reinforcing basic understanding.

Percentage Shortcuts for CAT Quant(blog)

This article focuses on specific shortcuts and tricks for percentage problems commonly encountered in the CAT quantitative aptitude section.

Practice Problems on Percentages(blog)

Toppr provides practice questions with detailed solutions for percentage problems, helping learners apply learned concepts.

Quantitative Aptitude for CAT: Percentages(video)

A detailed video tutorial covering percentage concepts and problem-solving strategies for the CAT exam.

Percentage Formulas and Shortcuts(documentation)

Examveda lists essential percentage formulas and shortcuts, useful for quick reference and memorization.

CAT Quantitative Aptitude - Percentages(blog)

Byjus offers a comprehensive guide to percentages for CAT, including theory, examples, and practice questions.

Mastering Percentages for Competitive Exams(blog)

A discussion forum thread where students share tips, tricks, and resources for mastering percentages in competitive exams.

Percentage Calculation Tricks(blog)

This blog post delves into various calculation tricks for percentages, aiming to speed up problem-solving for exams.