Ratio and Proportions: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction to Ratio and Proportions

Ratio and proportion are arguably the most crucial fundamental concepts in the Quantitative Aptitude section of competitive examinations like the UPSC Civil Services Aptitude Test (CSAT) and MPSC exams. Mastering these topics is not just about solving direct questions; it acts as a universal key to unlocking a multitude of other topics, including Data Interpretation (DI), Time and Work, Time Speed and Distance, Mixtures and Alligations, Percentages, and Profit & Loss. A robust understanding of ratio and proportion can drastically reduce calculation times and eliminate the need for complex algebraic equations, thereby increasing both your speed and accuracy in the high-pressure environment of the CSAT.

In simple terms, a Ratio is a mathematical comparison between two or more quantities of the same unit, indicating how many times one quantity is present in the other. It is a way of expressing a fraction. For example, if a class has 30 boys and 40 girls, the ratio of boys to girls is 30:40, which simplifies to 3:4. This means for every 3 boys, there are 4 girls.

Proportion, on the other hand, is an equation which defines that two given ratios are equivalent to each other. For instance, the ratio 2:3 is equivalent to the ratio 4:6. When we write 2/3 = 4/6, we are stating a proportion. When four quantities a, b, c, and d are in proportion, they are written as a:b :: c:d. Understanding how to manipulate these proportions is a game-changer for solving complex word problems.

Core Concepts, Formulas, and Tricks

To master Ratio and Proportions, you must be thoroughly familiar with the foundational terminology and properties.

1. Basic Terms: In the ratio a:b, ‘a’ is called the first term or antecedent, and ‘b’ is called the second term or consequent. The ratio remains unchanged if both the antecedent and consequent are multiplied or divided by the same non-zero number.

2. Important Types of Ratios:

• Duplicate Ratio: The duplicate ratio of a:b is a²:b². (e.g., Duplicate ratio of 3:4 is 9:16)
• Sub-duplicate Ratio: The sub-duplicate ratio of a:b is √a:√b. (e.g., Sub-duplicate ratio of 25:36 is 5:6)
• Triplicate Ratio: The triplicate ratio of a:b is a³:b³. (e.g., Triplicate ratio of 2:3 is 8:27)
• Sub-triplicate Ratio: The sub-triplicate ratio of a:b is ∛a:∛b. (e.g., Sub-triplicate ratio of 64:125 is 4:5)
• Inverse or Reciprocal Ratio: The inverse ratio of a:b is b:a. The inverse ratio of a:b:c is 1/a : 1/b : 1/c, which simplifies to bc : ac : ab.
• Compound Ratio: For ratios a:b and c:d, the compound ratio is ac:bd.

3. Properties of Proportion: When a, b, c, and d are in proportion (a:b :: c:d or a/b = c/d):

• Product of Extremes equals Product of Means: a × d = b × c. This is the most frequently used property.
• Fourth Proportional: ‘d’ is the fourth proportional to a, b, and c.
• Third Proportional: If a:b = b:c, then ‘c’ is the third proportional to a and b. Here, b² = ac.
• Mean Proportional: If a:b = b:c, then ‘b’ is the mean proportional between a and c. It is calculated as b = √(a × c).

4. Advanced Operations:

• Invertendo: If a/b = c/d, then b/a = d/c.
• Alternendo: If a/b = c/d, then a/c = b/d.
• Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d.
• Dividendo: If a/b = c/d, then (a-b)/b = (c-d)/d.
• Componendo and Dividendo (C&D Rule): If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). This is an incredibly powerful shortcut for complex algebra questions in CSAT.

5. The “N” Trick for Combining Ratios: If you are given A:B = x:y and B:C = p:q, and you need to find A:B:C, write them aligned:
A : B = x : y
B : C = p : q
Then A:B:C = (x×p) : (y×p) : (y×q). This vertical and diagonal multiplication pattern resembles an inverted ‘N’ and saves a lot of time.

Solved Examples with Step-by-Step Explanations

Let’s dive into some practical applications of these concepts through solved examples commonly seen in UPSC/MPSC examinations.

Example 1: Combining Ratios
If A:B = 3:4, B:C = 5:7, and C:D = 8:9, find the ratio of A:D.

Step-by-Step Solution:
Step 1: Understand that to find the ratio of the first term to the last term, you can simply multiply all the given ratios together.
Step 2: Setup the multiplication:
(A/B) × (B/C) × (C/D) = (3/4) × (5/7) × (8/9)
Step 3: Notice that the intermediate terms (B and C) cancel out on the left side, leaving A/D.
Step 4: Perform the calculation on the right side.
A/D = (3 × 5 × 8) / (4 × 7 × 9)
A/D = (120) / (252)
Step 5: Simplify the fraction by dividing numerator and denominator by their highest common factor (12).
A/D = 10 / 21.
Therefore, A:D = 10:21.

Example 2: Value-Based Ratio Problem
Two numbers are in the ratio 3:5. If 9 is subtracted from each, the new numbers are in the ratio 12:23. What is the smaller number?

Step-by-Step Solution:
Step 1: Assume the common multiplier to be ‘x’. Let the two numbers be 3x and 5x.
Step 2: Formulate the equation according to the condition given in the problem. When 9 is subtracted from both, the new ratio is 12:23.
So, (3x – 9) / (5x – 9) = 12 / 23
Step 3: Cross-multiply to solve for x.
23(3x – 9) = 12(5x – 9)
69x – 207 = 60x – 108
Step 4: Bring like terms to one side.
69x – 60x = 207 – 108
9x = 99
x = 11
Step 5: Find the smaller number. The smaller number is 3x.
Smaller number = 3 × 11 = 33.

Example 3: Distribution of Amounts
Divide ₹1162 among A, B, and C in the ratio 35:28:20.

Step-by-Step Solution:
Step 1: Find the sum of the ratio terms to determine the total number of parts.
Total parts = 35 + 28 + 20 = 83 parts.
Step 2: Find the value of one part.
Value of 1 part = Total Amount / Total Parts = 1162 / 83 = 14.
Step 3: Multiply the value of one part by the respective ratio share of each individual.
A’s share = 35 parts × 14 = ₹490
B’s share = 28 parts × 14 = ₹392
C’s share = 20 parts × 14 = ₹280
Check: 490 + 392 + 280 = 1162. The calculation is correct.

Example 4: Mean Proportional
Find the mean proportional between 0.08 and 0.18.

Step-by-Step Solution:
Step 1: Recall the formula for mean proportional between two numbers ‘a’ and ‘b’, which is √(a × b).
Step 2: Substitute the given values.
Mean Proportional = √(0.08 × 0.18)
Step 3: To simplify calculation, convert decimals to fractions.
= √(8/100 × 18/100)
Step 4: Multiply the numerators and denominators.
= √(144 / 10000)
Step 5: Find the square roots.
= 12 / 100 = 0.12.

Example 5: Coin Problems (A CSAT Favorite)
A bag contains 50p, 25p, and 10p coins in the ratio 5:9:4, amounting to ₹206. Find the number of coins of each type.

Step-by-Step Solution:
Step 1: Let the number of 50p, 25p, and 10p coins be 5x, 9x, and 4x respectively.
Step 2: Convert the value of coins into rupees (or convert the total amount into paise; here we will use rupees).
Value of 50p coins = (5x × 0.50) = 2.5x rupees
Value of 25p coins = (9x × 0.25) = 2.25x rupees
Value of 10p coins = (4x × 0.10) = 0.40x rupees
Step 3: Form the equation for the total amount.
2.5x + 2.25x + 0.40x = 206
5.15x = 206
Step 4: Solve for x.
x = 206 / 5.15 = 40
Step 5: Calculate the number of each type of coin.
Number of 50p coins = 5 × 40 = 200
Number of 25p coins = 9 × 40 = 360
Number of 10p coins = 4 × 40 = 160.

Pro-Tips to Avoid Common Mistakes

• Unit Consistency is Non-Negotiable: A ratio is a dimensionless quantity. Therefore, before finding a ratio between two quantities, you absolutely must ensure they are expressed in the exact same unit. Finding the ratio of 3 hours to 45 minutes requires converting hours to minutes (180 mins : 45 mins = 4:1) or vice versa. Ignoring this is the most common pitfall.
• The Order is Sacred: The ratio A:B is strictly not equal to B:A (unless A=B). If a question asks for the ratio of water to milk, do not accidentally provide the ratio of milk to water. Read the final line of the question twice.
• Always Simplify: Leaving a ratio as 15:25 instead of 3:5 can lead to missing the correct option in multiple-choice questions or complicating subsequent calculations.
• Don’t Confuse Parts with Actual Values: If the ratio of boys to girls is 2:3, it does not mean there are 2 boys and 3 girls. It means there are 2x boys and 3x girls. Always attach a variable (like ‘x’) when moving from ratio to actual quantities in equations.
• Utilize Options: Often, CSAT questions can be solved by simply checking which of the multiple-choice options satisfies the given ratio conditions. This “reverse engineering” approach saves immense time.

Practice Questions

Test your understanding with these practice questions designed around the UPSC/MPSC CSAT pattern.

1. If 0.75 : x :: 5 : 8, then find the value of x.
2. The salaries of A, B, and C are in the ratio 2:3:5. If increments of 15%, 10%, and 20% are allowed respectively in their salaries, what will be the new ratio of their salaries?
3. What number must be added to each term of the ratio 7:11 so that it becomes 3:4?
4. In a mixture of 60 litres, the ratio of milk and water is 2:1. If this ratio is to be 1:2, then the quantity of water to be further added is?
5. The sum of three numbers is 98. If the ratio of the first to the second is 2:3 and that of the second to the third is 5:8, then what is the value of the second number?