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

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

Solution:
A:B = 2:3 = (2×4):(3×4) = 8:12
B:C = 4:5 = (4×3):(5×3) = 12:15
C:D = 6:7 = (6×2.5):(7×2.5) - wait, 12/15 = 4/5

Actually: Let me use the LCM approach properly
A:B = 2:3
B:C = 4:5 ⟹ B common = 3×4 = 12
So A:B = 2×4 : 3×4 = 8:12
And B:C = 4×3 : 5×3 = 12:15
So A:B:C = 8:12:15

C:D = 6:7
C common = 15 = 6×2.5
So C:D = 6×2.5 : 7×2.5 = 15:17.5

Therefore A:D = 8:17.5 = 16:35

Age Problems with Ratios

Example: The ratio of ages of father and son is 5:2. After 10 years,
the ratio becomes 3:1. Find their present ages.

Solution:
Let father's age = 5x, son's age = 2x
After 10 years: (5x+10)/(2x+10) = 3/1
5x + 10 = 3(2x + 10)
5x + 10 = 6x + 30
x = -20 (This is wrong, let me recheck)

Wait, if x = -20, Father = -100... impossible

Let me retry:
(5x+10)/(2x+10) = 3/1
5x + 10 = 3(2x + 10)
5x + 10 = 6x + 30
10 - 30 = 6x - 5x
-20 = x

Hmm, still -20. The ratio setup seems wrong for positive ages.

Actually for father:son = 5:2, and after 10 years becomes 3:1,
this means the ratio increases (from 2.5 to 3), which is possible!

Let me set it up again:
5x + 10 = 3(2x + 10)
5x + 10 = 6x + 30
x = -20

This suggests the numbers don't work with positive ages.
Let me check: if x=20, Father=100, Son=40, ratio=2.5
After 10 years: 110:50 = 2.2 (not 3!)

So the problem setup might be wrong or ages aren't possible.
For GAT, such impossible age problems do appear - just find the answer
that satisfies the equation.

If x = 10, Father=50, Son=20, ratio=2.5
After 10 years: 60:30 = 2:1 (ratio is 2, not 3)

If x = 20, Father=100, Son=40, ratio=2.5
After 10 years: 110:50 = 11:5 (ratio is 2.2)

So no solution with these ratios for positive ages.
The ages don't work for a ratio change from 5:2 to 3:1 after 10 years.

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