Ratio and Proportion

Ratio and Proportion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ratio and Proportion — Key Facts for GAT Pakistan

Ratio Definition:

• Ratio of a to b is written as a:b = a/b
• a and b are called terms (a is antecedent, b is consequent)
• Ratio must be in simplest form (divide by GCD)

Proportion Definition:

• When two ratios are equal, they form a proportion
• a:b = c:d is written as a:b :: c:d
• a and d are called extremes; b and c are called means
• Property: a×d = b×c (cross multiplication)

Key Properties:

Property	Example
a:b = c:d ⟹ ad = bc	2:3 = 4:6 ⟹ 2×6 = 3×4 = 12
a:b = c:d ⟹ a:c = b:d	2:3 = 4:6 ⟹ 2:4 = 3:6
Duplicate ratio (a:b)² = a²:b²	(2:3)² = 4:9
Subduplicate ratio √a:√b	√2:√3
Compound ratio a/b × c/d = ac:bd	(2:3)(4:5) = 8:15

⚡ GAT Exam Tip: Remember “invertendo” property: if a:b = c:d, then b:a = d:c

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Ratio and Proportion — Detailed Study Guide

Types of Ratios

Type	Definition	Example
Simple Ratio	Two quantities	3:5
Compound Ratio	Product of ratios	(2:3)(4:5) = 8:15
Duplicate Ratio	Square of terms	(3:4)² = 9:16
Subduplicate	Square root	√3:√4 = √3:2
Triplicate	Cube of terms	(2:3)³ = 8:27
Inverse Ratio	Reciprocal	3:5 inverse is 5:3

Continued Ratio: When three or more ratios are equal: a:b:c = 2:3:5 means a/2 = b/3 = c/5 = k

Worked Example:

Example: If a:b = 2:3 and b:c = 4:5, find a:b:c

Solution:
Make 'b' common:
a:b = 2:3 = 8:12 (multiply by 4)
b:c = 4:5 = 12:15 (multiply by 3)

Therefore a:b:c = 8:12:15

Alternate using LCM method:
b = 3k = 4m, so 3k = 4m ⟹ k = 4m/3
a = 2k = 8m/3, c = 5m
a:b:c = 8m/3 : 4m : 5m = 8:12:15

Division in Given Ratios

Partitive Division:

Example: Divide Rs. 560 in the ratio 2:3:4

Solution:
Sum of ratio = 2+3+4 = 9
First part = (2/9) × 560 = Rs. 124.44
Second part = (3/9) × 560 = Rs. 186.67
Third part = (4/9) × 560 = Rs. 248.89

Verification: 124.44 + 186.67 + 248.89 ≈ 560 ✓

Worked Problem:

Example: Rs. 1200 is divided among A, B, C such that A gets 2/5 of total,
B gets 1/4 of total. Find C's share.

Solution:
A's share = (2/5) × 1200 = Rs. 480
B's share = (1/4) × 1200 = Rs. 300
C's share = 1200 - 480 - 300 = Rs. 420

As ratio: A:B:C = 480:300:420 = 48:30:42 = 8:5:7

⚡ Common Mistake: When dividing in ratios, always find the sum of ratio parts first, then multiply the fraction.

Direct and Inverse Proportion

Direct Proportion:

• When one increases, other also increases
• x ∝ y ⟹ x/y = constant ⟹ x₁/y₁ = x₂/y₂

Inverse Proportion:

• When one increases, other decreases
• x ∝ 1/y ⟹ xy = constant ⟹ x₁y₁ = x₂y₂

Worked Examples:

Example 1 (Direct): If 5 books cost Rs. 200, how much do 8 books cost?

Solution:
Books and cost are directly proportional
5:8 :: 200:x
5x = 8 × 200
x = 1600/5 = Rs. 320

Example 2 (Inverse): If 10 workers can complete a job in 20 days,
how many days will 25 workers take?

Solution:
Workers and days are inversely proportional
10 × 20 = 25 × x
x = 200/25 = 8 days

Example 3: If 3 pipes can fill a tank in 6 hours, how long will
5 pipes take? (pipes are equally efficient)

Solution:
3 × 6 = 5 × x
x = 18/5 = 3.6 hours = 3 hours 36 minutes

⚡ GAT PYQ: “If 8 men can dig a ditch in 9 days, how many days will 12 men take to dig the same ditch?” → Answer: 6 days

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Ratio and Proportion — Complete Notes for GAT

Proportionality Theorems

Third Proportional: If a:b = b:c, then c is the third proportional to a and b.

Example: Find third proportional to 3 and 6.
Solution: 3:6 = 6:x
         3x = 36
         x = 12

Mean Proportional: Mean proportional between a and b is √(ab).

Example: Find mean proportional between 4 and 9.
Solution: √(4 × 9) = √36 = 6

Fourth Proportional: If a:b = c:x, then x is the fourth proportional.

Example: Find fourth proportional to 2, 5, 6.
Solution: 2:5 :: 6:x
         2x = 30
         x = 15

Alligation Method

Alligation Rule (for mixing):

Question: In what ratio must two types of rice costing Rs. 30/kg and
Rs. 45/kg be mixed to get a mixture costing Rs. 36/kg?

Solution using alligation:
         Rs. 30        Mean = Rs. 36        Rs. 45
              |-----------36-45=9-----------|
              |-----------30-36=6-----------|
Ratio = 9:6 = 3:2

Answer: Mix in ratio 3:2

Alligation for Profit/Loss:

Example: A merchant mixes 10 kg of rice at Rs. 20/kg with 15 kg at
Rs. 30/kg. At what price per kg should he sell the mixture to make
20% profit?

Solution:
Cost price of mixture:
= (10×20 + 15×30)/(10+15) = (200+450)/25 = 650/25 = Rs. 26/kg

Selling price for 20% profit:
= 26 × 1.20 = Rs. 31.20/kg

⚡ Important: Alligation gives the ratio of quantities, not the final amounts!

GAT-Style Practice Questions

1. If a:b = 3:4 and b:c = 5:6, find a:b:c
   (a) 3:4:6 (b) 15:20:24 (c) 3:5:6 (d) 15:20:18

   Answer: (b) 15:20:24
   Solution: a:b = 3:4 = 15:20 (×5)
             b:c = 5:6 = 20:24 (×4)
             a:b:c = 15:20:24

2. The ratio of boys to girls in a class is 3:2. If there are 45 boys,
   how many girls are there?
   (a) 20 (b) 25 (c) 30 (d) 35

   Answer: (c) 30
   Solution: 3 parts = 45
             1 part = 15
             Girls (2 parts) = 30

3. Divide Rs. 4800 among A, B, C in ratio 3:4:5.
   How much does C get?
   (a) Rs. 1200 (b) Rs. 1600 (c) Rs. 2000 (d) Rs. 2400

   Answer: (c) Rs. 2000
   Solution: Sum of ratio = 12
             C's share = (5/12) × 4800 = Rs. 2000

4. A sum of money is divided among A, B, C in the ratio 1:2:3.
   If B gets Rs. 200 more than A, find the total sum.
   (a) Rs. 600 (b) Rs. 1200 (c) Rs. 1800 (d) Rs. 2400

   Answer: (b) Rs. 1200
   Solution: Let A = x, B = 2x, C = 3x
             B - A = 200
             2x - x = 200 ⟹ x = 200
             Total = x + 2x + 3x = 6x = Rs. 1200

5. The fourth proportional to 3, 5, 6 is:
   (a) 10 (b) 9 (c) 8 (d) 15

   Answer: (a) 10
   Solution: 3:5 :: 6:x
             3x = 30
             x = 10

⚡ GAT Strategy: For combined ratio problems, always make the common term equal using LCM before combining.

Chain Ratio Problems

To combine several linked ratios, equalise the shared term at each junction using the LCM, then read off the required ratio.

Example: If A:B = 2:3, B:C = 4:5, C:D = 6:7, find A:D

Solution:
Step 1 — Combine A:B and B:C by making B common.
The LCM of 3 and 4 is 12.
A:B = 2:3 = 8:12 (×4)
B:C = 4:5 = 12:15 (×3)
So A:B:C = 8:12:15

Step 2 — Bring in C:D by making C common.
A:B:C has C = 15 and C:D has C = 6; the LCM of 15 and 6 is 30.
A:B:C = 8:12:15 = 16:24:30 (×2)
C:D   = 6:7 = 30:35 (×5)
So A:B:C:D = 16:24:30:35

Step 3 — Read off A:D.
A:D = 16:35

Age Problems with Ratios

Age problems combine ratios with linear equations. Set each present age as a multiple of a common variable, then apply the future or past condition.

Example: The ratio of the ages of a father and his son is 7:3. After 6 years,
the ratio becomes 5:3. Find their present ages.

Solution:
Let father's age = 7x and son's age = 3x.
After 6 years: (7x + 6)/(3x + 6) = 5/3
Cross-multiply:
3(7x + 6) = 5(3x + 6)
21x + 18 = 15x + 30
6x = 12
x = 2

Present ages: Father = 7x = 14 years, Son = 3x = 6 years.

Verification:
After 6 years → Father = 20, Son = 12, ratio = 20:12 = 5:3 ✓

⚡ GAT Tip: After solving, always substitute the value back into the “after N years” (or “N years ago”) condition to confirm the ratio matches. A negative or impossible age means the chosen ratios are inconsistent, so recheck the equation setup before selecting an answer.

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