CAT Quant Percentage: Concepts, Shortcuts, and Solved Examples

Percentage is one of the most critical foundational topics in the Quantitative Aptitude section of the CAT exam. It acts as the backbone of arithmetic, appearing repeatedly in topics like CAT Profit and Loss, Simple and Compound Interest, Discounts, Mixtures and Alligations, Averages, and CAT Ratio and Proportion. Mastering this single unit allows you to solve both Quant questions and Data Interpretation sets much faster.

A solid grasp of percentages helps you break down complex word problems into simple mental steps. Instead of relying on slow, mechanical calculations, learning fraction equivalents and multiplier tricks will significantly boost your calculation speed and accuracy during the test.

What Is Percentage and Why Is 100 the Base?

The word "percent" comes from the Latin phrase per centum, which literally translates to "per hundred" or "out of 100." When you calculate a percentage, you convert different quantities to a shared platform with a common denominator of 100. This makes it possible to compare quantities that originally had completely different starting points.

Consider two students: Student A scores 300 marks, and Student B scores 400 marks. Looking only at these raw scores, you might assume Student B performed better. However, you cannot make a valid comparison without knowing the total marks possible.

Suppose Student A scored 300 out of 600, while Student B scored 400 out of 1000. When you bring both scores to a base of 100, Student A earned 50%, whereas Student B earned only 40%. Converting fractions to a common base of 100 reveals the true performance.

How to Convert Between Fractions and Percentages

A percentage calculation always identifies what part a specific number represents out of a whole reference quantity. The basic mathematical structure is:

Percentage = (Part / Whole) * 100

Here, the numerator is the specific quantity you are measuring, while the denominator is the total or reference base. The complete, original quantity is always treated as 100%.

To turn any fraction A/B into a percentage, multiply it by 100: (A / B) * 100. Conversely, to remove the percent sign (%) and turn a percentage back into a fraction or decimal, divide the number by 100.

The reference values below show the exact relationship between standard fractions and their percentage equivalents for quick calculations:

Smart Mental Calculation Hacks for CAT Percentages

In the CAT exam, calculating percentages using manual long division wastes valuable time. Instead of multiplying x * N and dividing by 100 on scratch paper, break difficult numbers into standard benchmark percentages: 10%, 1%, 50%, and 5%.

• 10% of a number: Shift the decimal point one place to the left. For example, 10% of 3980 is 398.

• 1% of a number: Shift the decimal point two places to the left. For example, 1% of 3980 is 39.80.

• 50% of a number: Simply divide the number by 2. For 3980, 50% is 1990.

• 5% of a number: Take half of the 10% value. Since 10% of 3980 is 398, 5% is 398 / 2 = 199.

You can combine these basic building blocks using addition or subtraction to evaluate complex percentages mentally:

• Calculating 39% of 480:
  Break 39% into (40% - 1%) or (50% - 10% - 1%).
  50% of 480 = 240
  10% of 480 = 48
  1% of 480 = 4.8
  39% = 240 - 48 - 4.8 = 187.2

• Calculating 48% of 720:
  Break 48% into (50% - 2%).
  50% of 720 = 360
  1% of 720 = 7.2, so 2% = 14.4
  48% = 360 - 14.4 = 345.6

• Calculating 99% of 350:
  Break 99% into (100% - 1%).
  100% of 350 = 350
  1% of 350 = 3.5
  99% = 350 - 3.5 = 346.5

Percentage to Fraction Conversion Chart

Recurring decimal percentages can look intimidating in word problems. Converting these decimals into simple fractions eliminates messy arithmetic and simplifies equations.

Recognising the patterns behind these fractions makes them easy to recall during the exam:

• The 1/7 family follows multiples of 14: 1/7 ≈ 14.28%, 2/7 ≈ 28.56%.

• The 1/9 family follows multiples of the 11 times table: 1/9 ≈ 11.11%, 2/9 ≈ 22.22%.

• The 1/11 family follows multiples of the 9 times table: 1/11 ≈ 9.09%, 2/11 ≈ 18.18%.

The table below summarises the most common fraction-to-percentage conversions tested in CAT:

Solving Common CAT Percentage Word Problems

Learn how to break down and solve the 5 most common percentage word problem patterns in CAT Quant using intuitive shortcuts and visual models. 

1. Subject Marks vs. Overall Percentage

When different subjects carry the same maximum marks, calculate the actual marks for each subject before summing them up. If Sunanda scores 40% in Hindi, 50% in English, 60% in Math, and 70% in Science, with each paper out of 60:

• Total maximum marks = 4 * 60 = 240.

• Marks obtained = (0.40 * 60) + (0.50 * 60) + (0.60 * 60) + (0.70 * 60) = 24 + 30 + 36 + 42 = 132.

• If the question asks for the total marks, the answer is 132. If it asks for the overall percentage, divide total marks by maximum marks: (132 / 240) * 100 = 55%.

2. Chained Number Relationships

When multiple numbers depend on one another via percentages, set the baseline reference number to 100:

• Suppose the first number is 20% more than the third number, and the second number is 50% more than the first number.

• Let the third number = 100.

• First number = 100 + 20% of 100 = 120.

• Second number = 120 + 50% of 120 = 120 + 60 = 180.

• The ratio of the first number to the second number is 120 / 180 = 2:3. Remember that the second percentage applies to 120, not the original 100.

3. Passing Marks and the Number Line Approach

If a candidate scores 240 marks and fails by 6%, while another candidate scores 320 marks and passes by 4%, visualise this along a linear scale:

[240 marks] <---- 6% ----> [Passing Boundary] <---- 4% ----> [320 marks]

|------------------------------- Total Gap = 10% ------------------------------|

• 
  The score gap between the two students is 320 - 240 = 80 marks.

• The percentage gap is 6% + 4% = 10%.

• Since 10% of the total marks = 80, the maximum marks (100%) are (80 / 10) * 100 = 800.

• The first candidate's percentage was (240 / 800) * 100 = 30%. Since they fell short by 6%, the required passing percentage is 30% + 6% = 36%.

4. Direct Ratio Comparisons

When two different percentage conditions refer to the same underlying value, you do not need to calculate the original number. For example, if 200% of a number is 90, what is 120% of that same number?

200% / 120% = 90 / x

5 / 3 = 90 / x

x = (90 * 3) / 5 = 54

Similarly, if 5/8 (62.5%) of a number equals 900, finding 7/8 (87.5%) of the number can be set up as:

(5/8) / (7/8) = 900 / x

5 / 7 = 900 / x

5x = 6300

x = 1260

5. Two-Candidate Election Problems

In an election with only two candidates, the total votes cast represent 100%. If the winner secures 65% of the total votes and wins by a margin of 300 votes:

• The loser automatically receives 100% - 65% = 35% of the votes.

• The winning margin in percentage points is 65% - 35% = 30%.

• Since 30% of total votes = 300, the total number of votes cast (100%) is (300 / 30) * 100 = 1000.

Percentage Increase, Decrease, and Net Change

Whenever a problem states that a quantity is "increased by x%", the new value becomes (100 + x)% of the original. When it is "decreased by x%", the new value becomes (100 - x)% of the original.

• An increase of 20% on X produces 1.20 * X.

• A decrease of 30% on Y produces 0.70 * Y.

If an algebraic fraction X / Y undergoes a 20% increase in the numerator and a 30% decrease in the denominator to yield 4/5, set up the equation using multipliers:

(1.20 * X) / (0.70 * Y) = 4 / 5

12X / 7Y = 4 / 5

X / Y = (4 / 5) * (7 / 12) = 7 / 15

Understanding the Comparison Base

When comparing two numbers, always verify which value belongs in the denominator. The reference value mentioned after words like "than," "compared to," or "over" forms the denominator:

Percentage Change = (Absolute Difference / Reference Base) * 100

• If A earns 300 and B earns 500, B's income is higher than A's by:
  ((500 - 300) / 300) * 100 = (200 / 300) * 100 = 66.67%

• However, A's income is less than B's by:
  ((500 - 300) / 500) * 100 = (200 / 500) * 100 = 40%

The absolute difference (200) remains identical, but the percentage changes because the reference base changes from 300 to 500.

Successive Percentage Changes and Multiplier Methods

A common trap in CAT questions is assuming that successive percentage changes can be added together directly. For instance, two successive increases of 10% do not equal a flat 20% increase.

Suppose a town has an initial population of 1000, and it increases by 10% each year for two consecutive years:

1. Year 1: 1000 + (10% of 1000) = 1000 + 100 = 1100.

2. Year 2: The second 10% increase applies to the new base of 1100, not the original 1000:
   1100 + (10% of 1100) = 1100 + 110 = 1210.

Adding 10% + 10% gives 1200, which is incorrect. The compounding effect produces 1210.

The Successive Change Formula

For two successive percentage changes of x% and y%, the net percentage change is:

Net Change = x + y + (x * y / 100)

Apply a positive sign for an increase and a negative sign for a decrease or discount.

• Case 1: +10% then -10%
  Net Change = 10 + (-10) + ((10 * -10) / 100) = 0 - 1 = -1% (a net decrease of 1%).

• Case 2: +20% then -20%
  Net Change = 20 - 20 + ((20 * -20) / 100) = -4% (a net decrease of 4%).

• Case 3: Successive Discounts of 10% and 20%
  Net Change = -10 - 20 + ((-10 * -20) / 100) = -30 + 2 = -28% (an effective discount of 28%, not 30%).

The table below outlines common successive percentage scenarios and their actual net effects on an initial value of 1000:

Handling Three or More Successive Changes: The Multiplier Method

When a problem includes three or more successive changes, combining them two at a time using the formula can become tedious. Instead, convert each percentage change into a decimal multiplier:

Final Value = Initial Value * M1 * M2 * M3 * ...

• An increase of 10% has a multiplier of 1.10.

• A decrease of 20% has a multiplier of 0.80.

• An increase of 25% has a multiplier of 1.25.

Example 1: A price of 600 is reduced three times in a row by 10% each time.

Final Price = 600 * 0.90 * 0.90 * 0.90 = 600 * 0.729 = 437.40

Example 2: An item faces successive discounts of 10%, 20%, and 10%.

Final Multiplier = 0.90 * 0.80 * 0.90 = 0.648

• The item now sells for 64.8% of its original price.

• The effective single discount is 100% - 64.8% = 35.2%.

• The percentage of the original cost recovered by the seller is 64.8%.

Practice Problems with Detailed Step-by-Step Solutions

Strengthen your conceptual understanding and calculation speed with these step-by-step solved practice problems on percentages. Working through these varied question types will help you master smart multiplier techniques and eliminate common calculation traps. 

• Problem 1

Question: In a school, 25% of the students are girls, and the actual number of girls is 420. What is the total number of boys in the school?

Solution:

• Total students = 100%.

• Since girls account for 25%, the boys account for 100% - 25% = 75%.

• Notice that the percentage of boys (75%) is exactly 3 times the percentage of girls (25%).

• Number of boys = 3 * 420 = 1260.

• Problem 2

Question: If 25% of a number is 19 less than 18% of 650, find the value of that number.

Solution:

• First, calculate 18% of 650:
  (18 / 100) * 650 = 9 * 13 = 117.

• Subtract 19 from 117:
  117 - 19 = 98.

• We know that 25% of the number = 98.

• Since the full number is 100% (which is 4 * 25%):
  Number = 4 * 98 = 392.

• Problem 3

Question: A student was asked to multiply a number by 5. By mistake, he divided the number by 5. Find the percentage error in the resulting value.

Solution:

• Choose a convenient starting number that is divisible by 5, such as 20.

• Correct operation: 20 * 5 = 100.

• Incorrect operation performed: 20 / 5 = 4.

• The error is the difference between the correct and incorrect results: 100 - 4 = 96.

• The base for comparison must always be the correct expected value:
  Percentage Error = (96 / 100) * 100 = 96%.

• Problem 4

Question: A tree was planted 2 years ago. Its height increases at a constant rate of 20% every year. If its present height is 540 cm, what was its height when it was first planted?

Solution:

• Let the initial height be H.

• A 20% increase each year corresponds to a multiplier of 1.20 = 6/5.

• After 2 successive years of growth:
  H * (6/5) * (6/5) = 540
  H * (36/25) = 540
  H = 540 * (25/36)

• Divide 540 by 36: 540 / 36 = 15.

• H = 15 * 25 = 375 cm.

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