Ratios & Proportions

Ratios and proportions are fundamental concepts used in mixture problems, partnership, and many real-world applications.

Core Concepts

What is a Ratio?

A ratio compares two quantities of the same kind. If a:b, it means a is to b.

Ratio 3:5 means for every 3 units of a, there are 5 units of b.

Important: Ratios are unitless and can be simplified like fractions.

Properties of Ratios

1. Multiplying/Dividing both terms by the same non-zero number doesn’t change the ratio.
   • 6:10 = 3:5 (divided by 2)
2. Ratios are order-sensitive — 3:5 ≠ 5:3
3. Compound ratio of a:b and c:d = ac:bd
4. Duplicate ratio of a:b = a²:b²
5. Triplicate ratio of a:b = a³:b³

What is a Proportion?

When two ratios are equal, they are in proportion.

a:b = c:d  →  a:b :: c:d  →  ad = bc (cross multiplication)

Example: 2:3 :: 4:6 → 2×6 = 3×4 → 12 = 12 ✓

Properties of Proportion

If a:b = c:d:

1. ad = bc (cross multiplication)
2. b:a = d:c (invertendo)
3. a:c = b:d (alternendo)
4. (a+b):b = (c+d):d (componendo)
5. (a-b):b = (c-d):d (dividendo)
6. (a+b):(a-b) = (c+d):(c-d) (componendo & dividendo)

Finding Ratios

Method 1: From Given Information

If A has ₹120 and B has ₹180:

A:B = 120:180 = 2:3

Method 2: From Fractions

If A gets 1/3 and B gets 1/5 of something:

A:B = 1/3 : 1/5 = 5:3 (cross multiply)

Method 3: Three-Way Ratios

If A:B = 2:3 and B:C = 4:5:

Make B common: A:B = 2:3 = 8:12, B:C = 4:5 = 12:15
A:B:C = 8:12:15

Dividing a Quantity in a Given Ratio

Basic Division

Divide ₹5000 in the ratio 3:2:

Total parts = 3 + 2 = 5
A's share = (3/5) × 5000 = ₹3000
B's share = (2/5) × 5000 = ₹2000

Three-Way Division

Divide ₹6000 among A, B, C in ratio 2:3:5:

Total parts = 2 + 3 + 5 = 10
A = (2/10) × 6000 = ₹1200
B = (3/10) × 6000 = ₹1800
C = (5/10) × 6000 = ₹3000

Mixtures

What is a Mixture?

A mixture combines two or more ingredients in a certain ratio.

Replacing in a Mixture

Problem: A container has 80 liters of milk. 8 liters are taken out and replaced with water. This is repeated 3 times. Find the final quantity of milk.

Formula:

Final milk = Initial × (1 - removal/capacity)^n
= 80 × (1 - 8/80)^3
= 80 × (0.9)^3
= 80 × 0.729
= 58.32 liters

General Replacement Formula

If Q liters of a mixture is replaced with water n times, and initial pure liquid = Q:

Pure liquid remaining = Q × (1 - R/Q)^n

Where R = amount removed each time.

Alligation

Alligation is a shortcut to find the ratio in which two ingredients at different prices/ concentrations must be mixed to get a desired price/concentration.

Alligation Rule

Cheaper                    Dearer
  C₂                        C₁
         Mean (M)
  
  Dearer - Mean : Mean - Cheaper
     (C₁ - M)      (M - C₂)

The ratio of Cheaper:Dearer = (C₁ - M):(M - C₂)

Example 1: Price Mixing

In what ratio should rice at ₹30/kg be mixed with rice at ₹50/kg to get a mixture worth ₹35/kg?

Cheaper = ₹30, Dearer = ₹50, Mean = ₹35

Ratio = (50 - 35):(35 - 30) = 15:5 = 3:1

Example 2: Concentration Mixing

How much water must be added to 60 liters of milk costing ₹20/liter to bring the cost to ₹15/liter?

Milk = ₹20, Water = ₹0, Mean = ₹15

Ratio of Milk:Water = (0 - 15):(15 - 20)
                    = -15:-5 (take absolute) = 3:1

Correction: the right formula uses the diagonal differences:
Ratio = (20-15):(15-0) = 5:15 = 1:3

Milk:Water = 1:3
If milk = 60 liters, water = 60 × 3 = 180 liters

Alligation on Percentage/Concentration

A 30% sugar solution is mixed with a 70% sugar solution to get a 45% solution. Find the ratio.

Ratio = (70-45):(45-30) = 25:15 = 5:3

Partnership

Simple Partnership

When partners invest for the same time period:

Profit share ratio = Investment ratio

Example: A invests ₹50,000, B invests ₹30,000. Total profit ₹40,000.

A:B = 50000:30000 = 5:3
A's profit = (5/8) × 40000 = ₹25,000
B's profit = (3/8) × 40000 = ₹15,000

Compound Partnership

When partners invest for different time periods:

Profit share ratio = Investment × Time ratio

Example: A invests ₹40,000 for 12 months, B invests ₹60,000 for 8 months.

A:B = (40000×12):(60000×8) = 480000:480000 = 1:1
Profits are shared equally.

Variations

Direct Proportion

If a increases when b increases: a ∝ b → a = kb

Example: Cost is directly proportional to quantity. If 5 kg costs ₹200, 8 kg costs:

Cost = (200/5) × 8 = ₹320

Inverse Proportion

If a increases when b decreases: a ∝ 1/b → ab = k

Example: If 12 workers complete a job in 10 days, 15 workers take:

12 × 10 = 15 × d → d = 8 days

Joint Proportion

If a varies jointly with b and c: a ∝ bc → a = kbc

Chain Rule

If:

• 5 men make 10 chairs in 6 days working 8 hours/day
• How many days for 8 men to make 20 chairs working 6 hours/day?

(Men₁ × Days₁ × Hours₁)/Work₁ = (Men₂ × Days₂ × Hours₂)/Work₂
(5 × 6 × 8)/10 = (8 × D₂ × 6)/20
24 = 48D₂/20
D₂ = 24 × 20/48 = 10 days

Tricks & Shortcuts

Trick 1: Ratio Manipulation

If A:B = 2:3, then:

• A/(A+B) = 2/5
• B/(A+B) = 3/5
• (A-B)/(A+B) = -1/5
• A/B = 2/3

Trick 2: Adding Same Number to Both Terms

If a:b is the original ratio and we add x to both:

New ratio = (a+x):(b+x)

Problem: The ratio of A’s and B’s ages is 3:5. After 5 years, the ratio becomes 3:4. Find present ages.

Let present ages be 3x and 5x.
(3x+5)/(5x+5) = 3/4
4(3x+5) = 3(5x+5)
12x + 20 = 15x + 15
3x = 5 → x = 5/3  (fractional — problem data is inconsistent; try a cleaner ratio below)

Redoing with cleaner numbers:

(3x+5)/(5x+5) = 3/4
12x + 20 = 15x + 15
5 = 3x → x = 5/3
A = 3(5/3) = 5, B = 5(5/3) = 25/3

Let’s work a cleaner example.

Problem: The ratio of A’s and B’s ages is 5:3. After 6 years, ratio becomes 3:2. Find present ages.

(5x+6)/(3x+6) = 3/2
2(5x+6) = 3(3x+6)
10x + 12 = 9x + 18
x = 6
A = 30, B = 18

Trick 3: Multiplicative Factor

If A:B = 2:3 and B:C = 4:7:

A:B = 2:3 → multiply by 4 → 8:12
B:C = 4:7 → multiply by 3 → 12:21
A:B:C = 8:12:21

Trick 4: Equal Ratios Property

If a/b = c/d = e/f = k, then:

(a+c+e)/(b+d+f) = k

Practice Questions

Q1: Division Problem

Divide ₹11,640 among A, B, C such that A gets 3/4 of what B gets, and B gets 2/5 of what C gets.

Solution:

A = 3/4 × B, B = 2/5 × C → C = 5/2 × B
A:B:C = 3/4 : 1 : 5/2 = 3:4:10
Total parts = 17
A = (3/17) × 11640 = ₹2053 (approx)
B = (4/17) × 11640 = ₹2739 (approx)
C = (10/17) × 11640 = ₹6847 (approx)

Q2: Mixture Problem

A vessel has 80 liters of pure milk. 8 liters are drawn and replaced with water. This is done 3 times. Find milk remaining.

Solution:

Milk = 80 × (1 - 8/80)³ = 80 × (0.9)³ = 80 × 0.729 = 58.32 liters

Q3: Alligation

In what ratio should milk at ₹40/liter be mixed with water (free) to get a mixture worth ₹32/liter?

Solution:

Milk:Water = (32-0):(40-32) = 32:8 = 4:1

Q4: Partnership

A invests ₹30,000 for 12 months. B joins after 4 months with ₹45,000. C joins after 6 months with ₹50,000. Total profit is ₹42,000. Find shares.

Solution:

A: 30000 × 12 = 360000
B: 45000 × 8 = 360000
C: 50000 × 6 = 300000
Ratio = 360:360:300 = 6:6:5
Total parts = 17
A = (6/17) × 42000 = ₹14,824 (approx)
B = (6/17) × 42000 = ₹14,824 (approx)
C = (5/17) × 42000 = ₹12,353 (approx)

Q5: Age Problem

The ratio of present ages of A and B is 7:9. After 4 years, the ratio will be 5:6. Find their present ages.

Solution:

Let ages be 7x and 9x.
(7x+4)/(9x+4) = 5/6
6(7x+4) = 5(9x+4)
42x + 24 = 45x + 20
3x = 4 → x = 4/3  (fractional — original problem data may be inconsistent; redo with cleaner numbers below)

Actually:

42x + 24 = 45x + 20
4 = 3x → x = 4/3
A = 28/3, B = 36/3 = 12

Let me use a cleaner problem:

Revised Q5: Ratio of A and B’s ages is 5:3. After 6 years, ratio is 3:2. Find ages.

Solution:

(5x+6)/(3x+6) = 3/2
10x + 12 = 9x + 18
x = 6
A = 30, B = 18

Q6: Chain Rule

If 6 men or 10 women can reap a field in 20 days, how many days will 12 men and 5 women take?

Solution:

6 men = 10 women → 1 man = 10/6 = 5/3 women
12 men + 5 women = 12(5/3) + 5 = 20 + 5 = 25 women
10 women take 20 days → work = 200 woman-days
25 women take = 200/25 = 8 days

Summary Table