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Basic Ratio & Proportion

CISF Constable · Quantitative Aptitude

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Core Concept

Read this first — the foundation of the topic

Basic Ratio & Proportion is the foundation of many SSC CGL problems. Think of ratio as a way to compare quantities. If you have 3 apples and 6 oranges, the ratio is 3:6, which simplifies to 1:2.

This means for every 1 apple, there are 2 oranges. A ratio compares parts to parts. A proportion states that two ratios are equal. For example, 2:3 = 4:6 is a proportion because both ratios equal 2/3 when simplified. Key Rules: (1) Ratios have no units - they are pure numbers. (2) Always simplify ratios by dividing by the HCF. (3) In a:b, 'a' is the first term and 'b' is the second term. (4) The ratio a:b can be written as the fraction a/b.

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Formula Block

Memorise — at least one formula appears in every paper

• If a:b = c:d, then ad = bc (Cross multiplication rule)

• If a:b = m:n, then a = (m×total)/(m+n) and b = (n×total)/(m+n)

• For three quantities in ratio a:b:c, if total is T, then parts are aT/(a+b+c), bT/(a+b+c), cT/(a+b+c)

• Compound ratio of a:b and c:d is ac:bd

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Exam Patterns

What examiners ask — read before attempting PYQs

→Common types include

age ratios, mixture problems, salary divisions, and proportion chains. Questions often involve finding actual values when ratios and total/difference are given

⚡Powerful Shortcut - The K Method

When dealing with ratios, use 'K' as a multiplier. If ratio is 3:5, write quantities as 3K and 5K. This makes calculation much easier as you can find K first, then multiply

✏️Worked Example 1

1

Let numbers be 4K and 7K (using K method)

2

Difference = 7K - 4K = 3K

3

Given difference = 21, so 3K = 21

4

Therefore K = 7

5

First number = 4K = 4×7 = 28

6

Second number = 7K = 7×7 = 49 Answer: 28 and 49 Worked Example 2: Rs. 850 is divided among A, B, C in ratio 2:3:5. Find each person's share.

1

Ratio parts are 2, 3, 5

2

Total ratio parts = 2+3+5 = 10

3

A's share = (2/10) × 850 = Rs. 170

4

B's share = (3/10) × 850 = Rs. 255

5

C's share = (5/10) × 850 = Rs. 425 Answer: A gets Rs. 170, B gets Rs. 255, C gets Rs. 425 Shortcut for Direct Proportion: If A varies directly as B, and you know A₁, B₁, A₂, then B₂ = (A₂ × B₁)/A₁. Cross multiply and solve instantly

⚡Trick for Inverse Proportion

If A varies inversely as B, then A₁B₁ = A₂B₂. When one increases, other decreases proportionally. Common Mistake #1: Students often forget to check if the proportion is direct or inverse. In direct proportion, both quantities change in the same direction. In inverse proportion, they change in opposite directions.

Missing this distinction leads to wrong answers in 40% of cases. Always read the question twice to identify the relationship type.

🔑 Key Points

Ratio compares quantities without units, always simplify by dividing by HCF
Use K method: if ratio is a:b, write quantities as aK and bK for easy calculation
Cross multiplication rule: if a:b = c:d, then ad = bc
For ratio a:b with total T, parts are aT/(a+b) and bT/(a+b)
Direct proportion: A₁/B₁ = A₂/B₂, quantities change in same direction
Inverse proportion: A₁B₁ = A₂B₂, quantities change in opposite directions
Compound ratio of a:b and c:d equals ac:bd
In three-way ratio a:b:c with total T, each part is (ratio part × T)/(a+b+c)
Proportion means two ratios are equal: 2:3 = 4:6 is a valid proportion
When difference is given in ratio problems, subtract ratio terms to find multiplier

📌 Exam Facts

Cross multiplication formula: if a/b = c/d, then ad = bc
Golden ratio value is approximately 1.618:1
In direct proportion, if A doubles, B also doubles
In inverse proportion, if A becomes half, B becomes double
Compound ratio formula: (a:b) combined with (c:d) gives ac:bd
For ratio a:b, percentage share of first term is a/(a+b) × 100%
Mean proportional between a and b is √(ab)
If three numbers are in continued proportion a:b:c, then b² = ac

🚀 60-Second Revision

Remember: Use K method for all ratio problems - if ratio is a:b, quantities are aK and bK
Formula: Cross multiplication rule ad = bc when a:b = c:d
Formula: For total division, part = (ratio term × total)/sum of all ratio terms
Trap: Always check if proportion is direct (same direction) or inverse (opposite direction)
Shortcut: In difference problems, subtract ratio terms to find the multiplier K
Remember: Simplify ratios by HCF before starting calculations
Formula: Compound ratio of multiple ratios = multiply all first terms : multiply all second terms

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