RRB NTPC & Group D Maths: Master Ratio and Proportion – Concepts, Shortcuts & Solved Questions

Introduction to Ratio and Proportion for RRB Exams

Ratio and Proportion is one of the most fundamental topics in quantitative aptitude for competitive examinations conducted by the Railway Recruitment Board (RRB), including RRB NTPC, RRB Group D, and RRB Technician Grade I & III. A solid grasp of ratios is essential not only for direct questions but also because this concept forms the bedrock for several other topics like Percentages, Profit & Loss, Ages, Partnerships, Mixtures & Alligations, and Time & Work.

Understanding how quantities relate to each other quantitatively enables candidates to solve complex mathematical problems quickly during time-pressured exams. In this comprehensive guide, we will cover core concepts, essential formulas, time-saving shortcut tricks, step-by-step solved examples, and practice questions tailored specifically for upcoming RRB exams.

Topic Weightage and Importance

In almost every RRB exam, mathematical proficiency is tested rigorously. The weightage of Ratio and Proportion across various RRB exams is significant:

• RRB NTPC (CBT-1 & CBT-2): 2 to 4 direct questions; indirect application in 5 to 7 Data Interpretation (DI) questions.
• RRB Group D: 2 to 3 direct questions.
• RRB Technician (Grade I & III): 2 to 3 questions covering basics and applied ratios.

Because ratio techniques simplify lengthy arithmetic calculations, mastering this chapter boosts speed and accuracy, directly translating to higher scores.

Key Concepts and Formulas

1. What is a Ratio?

A ratio is a comparative relationship between two quantities of the same unit, showing how many times one quantity contains another. The ratio of quantity a to quantity b is expressed as a : b or as a fraction \(\frac{a}{b}\).

• Antecedent: The first term, a.
• Consequent: The second term, b (where \(b \neq 0\)).
• Note: Multiplying or dividing both terms of a ratio by the same non-zero number does not change the value of the ratio. That is, \(a : b = ka : kb\).

2. Types of Ratios

• Duplicate Ratio: The duplicate ratio of \(a : b\) is \(a^2 : b^2\).
• Sub-duplicate Ratio: The sub-duplicate ratio of \(a : b\) is \(\sqrt{a} : \sqrt{b}\).
• Triplicate Ratio: The triplicate ratio of \(a : b\) is \(a^3 : b^3\).
• Sub-triplicate Ratio: The sub-triplicate ratio of \(a : b\) is \(\sqrt[3]{a} : \sqrt[3]{b}\).
• Inverse or Reciprocal Ratio: The inverse ratio of \(a : b\) is \(b : a\) or \(\frac{1}{a} : \frac{1}{b}\).
• Compound Ratio: For ratios \(a : b\) and \(c : d\), the compound ratio is \((a \times c) : (b \times d)\).

3. What is a Proportion?

An equality of two ratios is called a proportion. If \(a : b = c : d\), then \(a, b, c,\) and \(d\) are said to be in proportion, written as \(a : b :: c : d\).

• Extremes: First and fourth terms (\(a\) and \(d\)).
• Means: Second and third terms (\(b\) and \(c\)).
• Cross-Product Rule: Product of Extremes = Product of Means, i.e., \(a \times d = b \times c\).

4. Mean, Third, and Fourth Proportional

• Fourth Proportional: If \(a : b :: c : d\), then \(d = \frac{b \times c}{a}\) is the 4th proportional to \(a, b,\) and \(c\).
• Third Proportional: If \(a : b :: b : c\), then \(c = \frac{b^2}{a}\) is the 3rd proportional to \(a\) and \(b\).
• Mean Proportional: The mean proportional between two numbers \(a\) and \(b\) is \(\sqrt{a \times b}\).

5. Properties of Proportion

Solved Examples (Step-by-Step)

Example 1: Combining Ratios

Question: If \(A : B = 3 : 4\) and \(B : C = 8 : 9\), find the ratio \(A : B : C\).

Solution:

To combine the two ratios, make the common term \(B\) equal in both ratios.

• In \(A : B = 3 : 4\), the value for \(B\) is 4.
• In \(B : C = 8 : 9\), the value for \(B\) is 8.
• LCM of 4 and 8 is 8.
• Multiply the first ratio by 2: \(A : B = (3 \times 2) : (4 \times 2) = 6 : 8\).
• Second ratio remains: \(B : C = 8 : 9\).

Therefore, combining both yields \(A : B : C = 6 : 8 : 9\).

Example 2: Division of an Amount

Question: An amount of ₹7,200 is divided among A, B, and C in the ratio \(2 : 3 : 7\). Find the share of C.

Solution:

• Total ratio parts = \(2 + 3 + 7 = 12\) parts.
• 12 parts = ₹7,200
• 1 part = \(\frac{7200}{12} = ₹600\)
• Share of C = \(7 \text{ parts} = 7 \times 600 = ₹4,200\).

Therefore, the share of C is ₹4,200.

Example 3: Mean and Third Proportional

Question: Find the mean proportional between 9 and 25, and the third proportional to 12 and 18.

Solution:

• Mean Proportional between 9 and 25 = \(\sqrt{9 \times 25} = \sqrt{225} = 15\).
• Third Proportional to 12 and 18: Let it be \(x\). Then \(12 : 18 :: 18 : x \implies x = \frac{18 \times 18}{12} = 27\).

Hence, Mean Proportional is 15 and Third Proportional is 27.

Example 4: Ratio Change on Addition

Question: Two numbers are in the ratio \(3 : 5\). If 9 is subtracted from each, the new ratio becomes \(12 : 23\). Find the smaller number.

Solution:

Let the original numbers be \(3x\) and \(5x\).

According to the problem:

\(\frac{3x - 9}{5x - 9} = \frac{12}{23}\)

Cross-multiplying gives:

\(23(3x - 9) = 12(5x - 9)\)

\(69x - 207 = 60x - 108\)

\(69x - 60x = 207 - 108\)

\(9x = 99 \implies x = 11\)

The smaller number is \(3x = 3 \times 11 = 33\).

Common Mistakes to Avoid

• Comparing Quantities with Different Units: Always convert all terms to the same unit before forming a ratio (e.g., comparing 50 paise to ₹2 require converting ₹2 to 200 paise, giving \(50 : 200 = 1 : 4\)).
• Confusing Sub-duplicate and Duplicate Ratio: Remember duplicate involves squaring (\(a^2 : b^2\)) while sub-duplicate involves square root (\(\sqrt{a} : \sqrt{b}\)).
• Misidentifying Third Proportional: For two numbers \(a\) and \(b\), third proportional is \(\frac{b^2}{a}\), NOT \(\frac{a^2}{b}\). Order matters!
• Adding Quantities Directly to Ratio Terms: Adding a constant to both terms of a ratio changes the ratio value. Do not assume \(\frac{a+k}{b+k} = \frac{a}{b}\).

Practice Questions with Solutions

Q1. If \(A : B = 2 : 3\), \(B : C = 4 : 5\), and \(C : D = 6 : 7\), find \(A : D\).

Q2. What number must be added to each of the numbers 6, 15, 20, and 43 so that the resulting numbers are in proportion?

Q3. The ratio of income of A and B is \(5 : 4\) and the ratio of their expenditure is \(3 : 2\). If each saves ₹1,600 at the end of the month, find the income of A.

Q4. Find the fourth proportional to 4, 9, and 12.

Q5. A bag contains 50p, 25p, and 10p coins in the ratio \(5 : 9 : 4\), amounting to ₹206. Find the total number of coins in the bag.

Solutions

Solution 1:

\(\frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D} = \frac{2}{3} \times \frac{4}{5} \times \frac{6}{7} = \frac{48}{105} = \frac{16}{35}\)

Therefore, \(A : D = 16 : 35\).

Solution 2:

Let the required number be \(x\).

\(\frac{6+x}{15+x} = \frac{20+x}{43+x}\)

\((6+x)(43+x) = (15+x)(20+x)\)

\(258 + 49x + x^2 = 300 + 35x + x^2\)

\(49x - 35x = 300 - 258 \implies 14x = 42 \implies x = 3\).

So, 3 must be added.

Solution 3:

Let incomes of A and B be \(5x\) and \(4x\).

Saving = Income - Expenditure \(\implies\) Expenditure = Income - Saving.

\(\frac{5x - 1600}{4x - 1600} = \frac{3}{2}\)

\(2(5x - 1600) = 3(4x - 1600)\)

\(10x - 3200 = 12x - 4800 \implies 2x = 1600 \implies x = 800\).

Income of A = \(5x = 5 \times 800 = ₹4,000\).

Solution 4:

Let 4th proportional be \(x\). Then \(4 : 9 :: 12 : x \implies 4x = 9 \times 12 \implies x = 27\).

Solution 5:

Ratio of number of coins = \(5 : 9 : 4\). Let numbers be \(5x, 9x, 4x\).

Value in rupees: 50p = ₹0.5, 25p = ₹0.25, 10p = ₹0.10.

Total Value = \(5x(0.5) + 9x(0.25) + 4x(0.10) = 2.5x + 2.25x + 0.4x = 5.15x\).

Given total value = ₹206.

\(5.15x = 206 \implies x = \frac{206}{5.15} = 40\).

Total coins = \(5x + 9x + 4x = 18x = 18 \times 40 = 720\) coins.

Frequently Asked Questions (FAQs)

Q1. Is Ratio and Proportion important for RRB Group D and NTPC?

Yes, Ratio and Proportion is a core quantitative aptitude topic that carries direct marks and helps solve questions in arithmetic and Data Interpretation across RRB NTPC, Group D, and Technician exams.

Q2. How do I quickly combine ratios like A:B and B:C?

Find the Least Common Multiple (LCM) of the common term 'B' in both ratios, multiply each ratio accordingly to equalise 'B', and then write the combined ratio \(A : B : C\).

Q3. What is Componendo and Dividendo rule?

If \(\frac{a}{b} = \frac{c}{d}\), then according to Componendo and Dividendo rule, \(\frac{a+b}{a-b} = \frac{c+d}{c-d}\). It is widely used to simplify algebraic and trigonometric expressions rapidly.

Conclusion and Final Tips

Ratio and Proportion is a high-yield topic that provides maximum return on investment during preparation. Focus on mastering shortcut methods like the LCM method for combining ratios and cross-multiplication for ratio modification problems. Practice regularly using previous years' RRB NTPC and Group D question papers to build speed and confidence. Keep practicing, maintain consistency, and success will surely be yours!