RRB Group D Maths Ratio and Proportion by Manoj Sir

RRB Group D Maths Ratio and Proportion explains key concepts like combining ratios using multiplication and the Kabza method, equality–ratio relationships, types of proportionals, and ratio transformations. 

Here also covers important application problems, including changing ratios by subtraction and solving inverse speed–time questions, making it a complete and exam-focused guide for RRB Group D preparation. 

Also Read: RRB Group D Maths Syllabus

Ratio and Proportion 

Ratio and Proportion are foundational mathematical concepts vital for competitive exams like RRB Group D. Ratio and Proportion are basic maths concepts used to compare quantities: a ratio shows the comparison between two numbers, while a proportion shows that two ratios are equal.

Fundamental Rule of Combining Ratios

The most fundamental principle in ratios is that the value of a single variable cannot be different across different related ratios. If a variable, for instance 'b', appears in both ratios a:b and b:c, its value must be made consistent before combining these into a:b:c. Two primary methods achieve this consistency.

Methods for Combining Ratios

This section explains two easy and exam-oriented methods, Balancing by Multiplication and the Kabza Method, to combine ratios accurately by making the common term equal.

Method 1: Balancing by Multiplication

This method makes the shared variable's value consistent by multiplying the ratios.

Example 1:

Given a:b = 2:3 and b:c = 5:7.

• Here, b is 3 in the first ratio and 5 in the second.

• To make them equal, multiply the first ratio by 5 and the second ratio by 3:

• (a:b) * 5 → (2:3) * 5 = 10:15

• (b:c) * 3 → (5:7) * 3 = 15:21

• Now, b is consistently 15. The combined ratio a:b:c is 10:15:21.

Example 2:

Given a:b = 3:4 and b:c = 2:5.

• Here, b is 4 in the first ratio and 2 in the second.

• To make them equal, simply multiply the second ratio by 2:

• a:b = 3:4

• (b:c) * 2 → (2:5) * 2 = 4:10

• The combined ratio a:b:c is 3:4:10.

Method 2: The "Kabza" (Capture) Method

This shortcut efficiently combines ratios, especially when dealing with more than two. The method works by writing ratios in rows and filling empty adjacent spots with the nearest number from the same row (Memory Tip: The "Kabza Method" or "Capture Method" implies 'capturing' the empty space with the nearest value).

Example 1 (using Kabza Method):

Given a:b = 2:3 and b:c = 5:7.

1. Write the ratios in rows:
   a : b : c 2 : 3 : __ __ : 5 : 7

2. Fill the empty space in the first row with the number to its left (3).

3. Fill the empty space in the second row with the number to its right (5).
   a : b : c 2 : 3 : [3] [5] : 5 : 7

4. Multiply the numbers in each column vertically:

• a = 2 * 5 = 10

• b = 3 * 5 = 15

• c = 3 * 7 = 21

5. The resulting ratio a:b:c is 10:15:21.

Example 2 (using Kabza Method):

Given a:b = 3:4 and b:c = 2:5.

1. Arrange and fill the spaces:
   a : b : c 3 : 4 : [4] [2] : 2 : 5

2. Multiply vertically:

• a = 3 * 2 = 6

• b = 4 * 2 = 8

• c = 4 * 5 = 20

3. The resulting ratio is 6:8:20. A ratio must always be expressed in its simplest form. Divide all parts by their greatest common divisor (2).

4. The final ratio a:b:c is 3:4:10.

Relationship Between Equality and Ratio

A key principle is that equality and ratio are inversely related. If multiples of variables are equal (e.g., xA = yB = zC), the ratio of the variables (A:B:C) is the reciprocal of their coefficients.

Example 1:

Given 4a = 6b = 5c, find a:b:c.

1. The ratio a:b:c will be 1/4 : 1/6 : 1/5.

2. To convert this fractional ratio into integers, find the LCM (Least Common Multiple) of the denominators (4, 6, 5), which is 60.

3. Multiply each term by the LCM:

• a = (1/4) * 60 = 15

• b = (1/6) * 60 = 10

• c = (1/5) * 60 = 12

4. Therefore, a:b:c = 15:10:12.

Example 2:

Given 2A = 3B = 4C, find A:B:C.

1. The ratio is 1/2 : 1/3 : 1/4.

2. The LCM of (2, 3, 4) is 12.

3. Multiply each term by 12:

• A = (1/2) * 12 = 6

• B = (1/3) * 12 = 4

• C = (1/4) * 12 = 3

4. Therefore, A:B:C = 6:4:3.

Types of Proportionals and Their Formulas

When dealing with two or three numbers, you may need to find a missing proportion. The following are standard formulas:

Worked Examples:

1. Third Proportional of 9 and 24:

• Here, a = 9, b = 24.

• Formula: b²/a = (24 * 24) / 9 = 576 / 9 = 64.

2. Fourth Proportional of 12, 14, and 24:

• Here, a = 12, b = 14, c = 24.

• Formula: (b * c) / a = (14 * 24) / 12 = 14 * 2 = 28.

3. Fourth Proportional of 0.2, 0.12, and 0.3:

• Here, a = 0.2, b = 0.12, c = 0.3.

• Formula: (b * c) / a = (0.12 * 0.3) / 0.2 = 0.036 / 0.2 = 0.18.

4. Mean Proportional of 3.6 and 0.9:

• Here, a = 3.6, b = 0.9.

• Formula: √(a * b) = √(3.6 * 0.9) = √(36/10 * 9/10) = √(324 / 100).

• The square root is 18 / 10 = 1.8.

Solving Proportions and Advanced Ratio Types

This section covers properties of proportion and specific ratio transformations.

Product of Extremes and Means

For four numbers in proportion (a:b :: c:d), the product of the outer terms (extremes) equals the product of the inner terms (means).

• a * d = b * c

Example: Find the missing term in 6 : x :: 5 : 35.

• 6 * 35 = x * 5

• 210 = 5x

• x = 210 / 5 = 42

Specific Ratio Transformations

Application Problems

Application Problems focus on practical ratio concepts, such as finding a number to subtract from both terms of a ratio to get a new ratio, and understanding the inverse relationship between speed and time when distance remains constant.

Finding a Number to Subtract from a Ratio

When asked to find a number that, when subtracted from both parts of a ratio, results in a new ratio, the most efficient method is to test the given options rather than using algebraic equations.

Example 1: What smallest integer must be subtracted from both terms of the ratio 6:7 to get 3:4?

• Testing option 3: (6 - 3) : (7 - 3) = 3 : 4.

• This matches the target ratio, so 3 is the correct answer.

Example 2: What number should be subtracted from both terms of the ratio 5:7 to make it 3:5?

• Testing option 2: (5 - 2) : (7 - 2) = 3 : 5.

• This matches the target ratio, so 2 is the correct answer.

Inverse Relationship of Speed and Time

When distance is constant, speed and time are inversely proportional. This means if the ratio of speeds is s1:s2, the ratio of the time taken to cover the same distance will be 1/s1 : 1/s2.

Example: Three cars (P, Q, R) travel the same distance. Their speed ratio is 10:12:15. Find the ratio of the time taken.

1. Speed Ratio = 10 : 12 : 15

2. Time Ratio = 1/10 : 1/12 : 1/15

3. To convert to an integer ratio, find the LCM of (10, 12, 15), which is 60.

4. Multiply each term by 60:

• (1/10) * 60 = 6

• (1/12) * 60 = 5

• (1/15) * 60 = 4

5. The ratio of the time taken is 6:5:4.

RRB Group D Maths Ratio and Proportion Important Questions

In ratio and proportion, which statement about a variable is correct?
A. The same variable can have different values
B. A variable can change freely
C. The same variable cannot have different values
D. Variables are independent

 Correct Answer: C

If A : B = 2 : 3 and B : C = 5 : 7, then A : B : C is:
A. 10 : 15 : 21
B. 15 : 10 : 21
C. 10 : 21 : 15
D. 5 : 15 : 7

Correct Answer: A

What is the main purpose of the “KABZA Method” in ratios?
A. To add ratios
B. To subtract ratios
C. To make the common variable equal
D. To divide all terms

 Correct Answer: C

If A : B = 3 : 4 and B : C = 2 : 5, then A : B : C equals:
A. 3 : 4 : 10
B. 6 : 8 : 20
C. 9 : 12 : 30
D. 12 : 16 : 40

 Correct Answer: A

What does “ratio” mean according to the lecture?
A. Largest possible form
B. Average form
C. Smallest possible form
D. Equal form

Correct Answer: C

If 4A = 6B = 5C, then A : B : C is:
A. 10 : 12 : 15
B. 15 : 10 : 12
C. 12 : 15 : 10
D. 5 : 6 : 4

 Correct Answer: B

If A = 2, B = 3, and C = 4, then the value of A / B / C / A is:
A. 6 : 8 : 12
B. 8 : 9 : 24
C. 4 : 6 : 12
D. 12 : 18 : 36

 Correct Answer: B

If A : B = 7 : 9 and B : C = 5 : 7, what is A : C?
A. 35 : 63
B. 45 : 63
C. 7 : 9
D. 5 : 7

Correct Answer: A

If 2A = 3B = 4C, then A : B : C equals:
A. 4 : 3 : 2
B. 6 : 4 : 3
C. 3 : 4 : 6
D. 2 : 3 : 4

Correct Answer: B

What is the third proportional of 9 and 24?
A. 36
B. 48
C. 64
D. 72

Correct Answer: C

What is the third proportional of 4 and 28?
A. 98
B. 144
C. 196
D. 224

Correct Answer: C

What is the fourth proportional of 12, 14, and 24?
A. 26
B. 28
C. 30
D. 32

Correct Answer: B

What is the mean proportional of 3.6 and 0.9?
A. 1.2
B. 1.8
C. 2.4
D. 3.6

Correct Answer: B

What is the square root ratio of 16 : 25?
A. 2 : 5
B. 4 : 5
C. 5 : 4
D. 8 : 25

Correct Answer: B

What is the cube root ratio of 343 : 729?
A. 5 : 7
B. 7 : 9
C. 9 : 7
D. 14 : 27

Correct Answer: B