Master Percentages for RRB NTPC & Group D: Concepts, Tricks, and Solved Questions

Introduction: Why Mastering Percentages is Crucial for RRB Exams

Welcome, future railway professionals! If you're gearing up for the highly competitive RRB NTPC, RRB Group D, or RRB Technician exams, you know that every single mark counts. The Quantitative Aptitude section is often the make-or-break part of these exams, and within it, the topic of Percentages forms the very foundation. It's not just a standalone topic; its concepts are intricately woven into other crucial areas like Profit and Loss, Simple & Compound Interest, and Data Interpretation. Typically, you can expect 2-3 direct questions from Percentages and several more indirect questions in the RRB exams.

Mastering this single topic can unlock your ability to solve a significant portion of the mathematics paper with speed and accuracy. This comprehensive guide is designed to take you from the basic concept of percentages to advanced problem-solving techniques. We will cover everything from fundamental formulas and conversion tricks to solved examples from previous years' papers and practice questions to solidify your understanding. Let's begin your journey to becoming a Percentage expert!

Understanding the Core Concept: What is a Percentage?

The word 'Percentage' is derived from the Latin phrase 'per centum,' which means 'per hundred' or 'for every hundred.' In essence, a percentage is a fraction or a ratio where the denominator is always 100. It is denoted by the symbol '%'.

For example, if you score 85 marks out of a total of 100, your score is 85 percent, written as 85%. This means you got 85 parts for every 100 parts of the total.

The fundamental formula to calculate a percentage is:

Percentage = (Value / Total Value) × 100

Understanding this simple idea is the first step. In competitive exams, however, speed is key. This is where converting percentages to fractions and vice-versa becomes an invaluable tool.

Key Concepts, Formulas, and Shortcuts for RRB Exams

To solve percentage problems quickly and efficiently, you need to be well-versed with some key formulas and shortcuts. Memorizing these will save you precious seconds during the exam.

1. Essential Conversions: The Foundation of Speed

The fastest way to handle percentages in calculations is to use their fractional equivalents. You should memorize the following table until it becomes second nature.

Table: Percentage to Fraction Conversion

How to use this? Instead of calculating 25% of 360 as (25/100) * 360, you can simply calculate (1/4) * 360, which is much faster and equals 90.

2. Percentage Increase and Decrease

This is a very common type of question in RRB exams. The formula is straightforward:

• Percentage Increase = [(New Value - Original Value) / Original Value] × 100
• Percentage Decrease = [(Original Value - New Value) / Original Value] × 100

A quicker way is to use multiplying factors. For an 'x%' increase, the new value is (100+x)% of the original value. For an 'x%' decrease, the new value is (100-x)% of the original value.

• For a 20% increase, new value = 120% of original = 1.2 × Original Value.
• For a 15% decrease, new value = 85% of original = 0.85 × Original Value.

3. Successive Percentage Change

What happens when a value is changed by a certain percentage, and then the new value is changed by another percentage? This is called successive percentage change.

If a value is first changed by x% and then by y%, the net percentage change is given by the formula:

Net % Change = x + y + (xy / 100)

Important Note: Use a positive sign for increase and a negative sign for decrease. For example, if a price is increased by 20% and then decreased by 10%:

Here, x = +20 and y = -10.

Net % Change = 20 + (-10) + (20 * -10 / 100) = 10 - (200 / 100) = 10 - 2 = 8%.

So, the net change is an 8% increase.

4. The Concept of Base (A is what % of B)

This concept confuses many aspirants. The key is to identify the 'base' value, which is the value you are comparing against. The value that comes after 'of' is usually the base.

• A is what % of B? Formula: (A / B) × 100
• A is what % more than B? Formula: [(A - B) / B] × 100
• B is what % less than A? Formula: [(A - B) / A] × 100

Notice how the denominator (the base) changes depending on the question.

5. Price and Consumption Relationship

This is a classic problem type. If the price of a commodity increases by R%, the percentage reduction in consumption required to keep the expenditure the same is:

Reduction % = [R / (100 + R)] × 100

Conversely, if the price of a commodity decreases by R%, the percentage increase in consumption required to keep the expenditure the same is:

Increase % = [R / (100 - R)] × 100

Solved Examples (Step-by-Step Solutions)

Let's apply these concepts to questions similar to those asked in RRB exams.

Example 1: Basic Percentage Calculation

Question: In an examination, 35% of the students passed, and 455 students failed. How many students appeared for the examination?

Solution:

1. Understand the information: Percentage of students who passed = 35%.
2. Calculate the percentage of failed students: The total percentage of students is always 100%. So, Percentage of students who failed = 100% - 35% = 65%.
3. Relate percentage to the actual number: We are given that the number of failed students is 455. This means 65% of the total students is equal to 455.
4. Set up the equation: Let 'T' be the total number of students. Then, 65% of T = 455.
5. Solve for T: (65 / 100) * T = 455 => T = (455 * 100) / 65.
6. Simplify the calculation: 455 can be divided by 65. 65 * 7 = 455. So, T = 7 * 100 = 700.

Answer: A total of 700 students appeared for the examination.

Example 2: Successive Percentage Change

Question: The salary of a worker was first increased by 10% and thereafter decreased by 10%. What was the net change in his salary?

Solution:

1. Identify the percentage changes: First change (x) = +10% (increase). Second change (y) = -10% (decrease).
2. Apply the successive percentage change formula: Net % Change = x + y + (xy / 100).
3. Substitute the values: Net % Change = 10 + (-10) + (10 * -10 / 100).
4. Calculate the result: Net % Change = 0 + (-100 / 100) = -1.

Answer: The net change is -1%, which means his salary decreased by 1%.

Alternative Method (Using 100 as base): Assume the initial salary is ₹100. After a 10% increase, it becomes ₹110. Now, decrease this new salary by 10%. 10% of 110 is 11. So, the final salary is 110 - 11 = ₹99. The initial salary was ₹100, and the final is ₹99. This is a decrease of ₹1 on ₹100, which is a 1% decrease.

Example 3: Election Based Question

Question: In an election between two candidates, one candidate got 55% of the total valid votes. 20% of the votes were invalid. If the total number of votes was 7500, what is the number of valid votes that the other candidate got?

Solution:

1. Calculate the total valid votes: Total votes = 7500. Invalid votes = 20%. So, valid votes = 80% of 7500.
2. Calculation: Valid Votes = (80 / 100) * 7500 = 80 * 75 = 6000.
3. Find the votes for the winning candidate: The winning candidate got 55% of the *valid votes*. Votes for winner = 55% of 6000 = (55 / 100) * 6000 = 55 * 60 = 3300.
4. Find the votes for the other candidate: The other candidate would get the remaining percentage of the valid votes. Percentage for the other candidate = 100% - 55% = 45% of valid votes.
5. Calculation: Votes for other candidate = 45% of 6000 = (45 / 100) * 6000 = 45 * 60 = 2700.

Answer: The other candidate got 2700 valid votes.

Example 4: Price and Consumption

Question: If the price of sugar is increased by 25%, by what percentage must a household reduce its consumption so as not to increase its expenditure?

Solution:

1. Identify the price increase (R): R = 25%.
2. Use the direct formula for reduction in consumption: Reduction % = [R / (100 + R)] × 100.
3. Substitute the value of R: Reduction % = [25 / (100 + 25)] × 100.
4. Calculate the result: Reduction % = (25 / 125) × 100 = (1 / 5) × 100 = 20%.

Answer: The household must reduce its consumption by 20%.

Fraction Method: A 25% increase means the price went from 1 to 1 + 1/4 = 5/4 of the original. To keep expenditure constant, consumption must become the reciprocal, i.e., 4/5 of the original. A change from 1 to 4/5 is a reduction of 1/5. And 1/5 in percentage terms is 20%.

Practice Questions with Solutions

Now it's your turn! Try solving these questions. The detailed solutions are provided at the end for you to check your work.

Practice Questions

1. If A's salary is 50% more than B's, then by what percent is B's salary less than A's?
2. A number is increased by 20% and then decreased by 20%. Find the net increase or decrease percent.
3. In a class of 60 students, 40% are girls. How many boys are there in the class?
4. A man spends 75% of his income. If his income is increased by 20% and his expenditure is increased by 10%, find the percentage increase in his savings.
5. The population of a town was 160,000 three years ago. If it has increased by 3%, 2.5%, and 5% respectively in the last three years, what is its present population?
6. If 40% of a number is equal to two-thirds of another number, what is the ratio of the first number to the second number?
7. A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, how many apples did he have?
8. What percentage of numbers from 1 to 70 have 1 or 9 in the unit's digit?
9. If x is 80% of y, what percent of x is y?
10. The value of a machine depreciates at the rate of 10% every year. It was purchased 3 years ago. If its present value is ₹8748, what was its purchase price?

Solutions to Practice Questions

1. Solution:

Let B's salary be ₹100. Then A's salary = 100 + 50% of 100 = ₹150. Now, we need to find how much B's salary is less than A's. Difference = 150 - 100 = ₹50. The comparison is with A's salary. Percentage less = (Difference / A's salary) * 100 = (50 / 150) * 100 = (1/3) * 100 = 33.33%. Answer: 33.33%

2. Solution:

Using the successive change formula: x = +20, y = -20. Net % Change = 20 - 20 + (20 * -20 / 100) = 0 - (400/100) = -4%. Answer: 4% decrease.

3. Solution:

Total students = 60. Percentage of girls = 40%. Number of girls = 40% of 60 = (40/100) * 60 = 24. Number of boys = Total students - Number of girls = 60 - 24 = 36. Answer: 36 boys.

4. Solution:

Let the initial income be ₹100. Expenditure = 75% of 100 = ₹75. Savings = 100 - 75 = ₹25. New income = 100 + 20% increase = ₹120. New expenditure = 75 + 10% increase on 75 = 75 + 7.5 = ₹82.5. New savings = New income - New expenditure = 120 - 82.5 = ₹37.5. Increase in savings = 37.5 - 25 = ₹12.5. Percentage increase in savings = (Increase / Original Savings) * 100 = (12.5 / 25) * 100 = (1/2) * 100 = 50%. Answer: 50%.

5. Solution:

This is a successive percentage increase problem. Present Population = 160000 * (1 + 3/100) * (1 + 2.5/100) * (1 + 5/100) = 160000 * (103/100) * (102.5/100) * (105/100) = 160000 * 1.03 * 1.025 * 1.05 = 177366. Answer: 177,366.

6. Solution:

Let the numbers be A and B. Given: 40% of A = (2/3) of B. So, (40/100) * A = (2/3) * B => (2/5) * A = (2/3) * B. To find the ratio A/B, we rearrange: A/B = (2/3) / (2/5) = (2/3) * (5/2) = 5/3. The ratio is 5:3. Answer: 5:3.

7. Solution:

He sells 40% of the apples. The remaining percentage is 100% - 40% = 60%. We are given that the remaining apples are 420. So, 60% of Total Apples = 420. (60/100) * Total = 420 => Total = (420 * 100) / 60 = 7 * 100 = 700. Answer: 700 apples.

8. Solution:

Numbers from 1 to 70 with unit digit 1 are: 1, 11, 21, 31, 41, 51, 61 (7 numbers). Numbers from 1 to 70 with unit digit 9 are: 9, 19, 29, 39, 49, 59, 69 (7 numbers). Total favorable numbers = 7 + 7 = 14. Total numbers = 70. Required Percentage = (Favorable Numbers / Total Numbers) * 100 = (14 / 70) * 100 = (1/5) * 100 = 20%. Answer: 20%.

9. Solution:

Given: x = 80% of y => x = (80/100) * y => x = (4/5) * y. We need to find what percent of x is y. This means we need to calculate (y/x) * 100. From the equation, y = (5/4) * x. So, (y/x) * 100 = ((5/4) * x / x) * 100 = (5/4) * 100 = 125%. Answer: 125%.

10. Solution:

Let the purchase price be P. The value depreciates by 10% for 3 years. This is a successive decrease. Present Value = P * (1 - 10/100)^3. Given Present Value = 8748. So, 8748 = P * (90/100)^3 = P * (9/10)^3 = P * (729/1000). P = (8748 * 1000) / 729. 8748 / 729 = 12. So, P = 12 * 1000 = 12000. Answer: ₹12,000.

Conclusion: Your Path to Success

Percentages are more than just a chapter in your RRB exam syllabus; they are a fundamental building block for your entire quantitative aptitude preparation. By understanding the core concepts, memorizing the fraction-to-percentage conversion table, and practicing the various types of problems, you can turn this topic into one of your strongest scoring areas.

Remember, consistency is key. Dedicate some time every day to practice percentage questions. Analyze your mistakes in the practice set and revisit the concepts you find challenging. With dedicated effort, you can confidently tackle any percentage-based question that comes your way in the RRB NTPC, Group D, or Technician exams. Best of luck with your preparation!