Ratio, Proportion & Mixtures

Section A · Futures First International — Market Trading Intern OA

📌 What Has Been Asked (OA & Rounds)

Format	Typical Question Style
OA (MCQ)	"Two quantities are mixed in ratio 3:2. If price of first is ₹40/kg and second is ₹60/kg, find avg price per kg."
OA	"A:B = 3:4, B:C = 5:7. Find A:B:C."
OA	"In what ratio should water be mixed with milk worth ₹60/L to make it worth ₹45/L?"
OA	"A mixture has milk and water in 7:3. If 10L water is added, ratio becomes 7:5. Find original quantity."
Buzzer	"If 2:3 = 4:x, find x" → 6, instant
Interview	"A trader blends two commodities..." — tests proportional thinking

🧠 Core Concepts

1. Ratio Basics

a:b = ka:kb        (ratios can be scaled)
a:b and b:c → a:b:c   (make b common, then merge)

Combining two ratios:

A:B = 3:4, B:C = 5:7
Make B common: A:B = 15:20, B:C = 20:28
A:B:C = 15:20:28

2. Proportion

a:b = c:d  (or a/b = c/d)

Componendo:     (a+b)/b = (c+d)/d
Dividendo:      (a−b)/b = (c−d)/d
Componendo-Dividendo: (a+b)/(a−b) = (c+d)/(c−d)

3. Alligation / Mixture Rule (CRITICAL for FF)

When mixing two things at different prices/concentrations to get a mixture:

Cheaper ────── Mean ────── Dearer
   c              m            d

Ratio of Cheaper : Dearer = (d − m) : (m − c)

Visual:

    c           d
     \         /
      \       /
        m
     (d-m) : (m-c)

4. Replacement Formula

If a container has V litres of pure liquid, and x litres are replaced with water each time, after n replacements:

Pure liquid remaining = V × (1 − x/V)^n

⚡ Speed Tricks for Futures First

Trick 1: Alligation Cross (covers 80% of mixture questions)

"Mix ₹40/kg and ₹60/kg to get ₹45/kg"

40 ─── 45 ─── 60
    |       |
   60-45  45-40
   = 15  :  5
   = 3   :  1

→ Mix cheaper:dearer = 3:1

Trick 2: "Total quantity from ratio change"

"Milk:Water = 7:3 in 100L. How much water to add so ratio becomes 7:5?"

Milk = 70L (doesn't change). New ratio: Milk:Total water = 7:5
70/new_water = 7/5 → new_water = 50. Originally 30L water → add 20L

Trick 3: Quick ratio merging

A:B = 2:3, B:C = 4:5

LCM of B values (3 and 4) = 12
A:B = 8:12, B:C = 12:15
A:B:C = 8:12:15

Trick 4: "Partnership profit division"

If A invests ₹x for t₁ months and B invests ₹y for t₂ months:

A's share : B's share = x×t₁ : y×t₂

Trick 5: Successive dilution shortcut

20L milk, 4L replaced with water each time, done 3 times:

Milk remaining = 20 × (1 − 4/20)³ = 20 × (4/5)³ = 20 × 64/125 = 10.24L

🎯 What Can Be Asked (Variations)

"A vessel contains 60L of milk. 12L removed and replaced with water. Done twice. Find milk remaining." → 60 × (1 − 12/60)² = 60 × (4/5)² = 60 × 16/25 = 38.4L
"Two alloys A (gold:silver = 5:3) and B (gold:silver = 5:11) are melted together. Find ratio to get alloy with gold:silver = 1:1." → Gold fraction in A = 5/8, B = 5/16, desired = 1/2 = 8/16. → Alligation: A:B = (8/16 − 5/16):(5/8 − 8/16) = 3/16 : 2/16 = 3:2
"₹4200 divided among A, B, C such that A:B = 2:3 and B:C = 4:5. Find C's share." → A:B:C = 8:12:15. Total parts = 35. C = (15/35) × 4200 = ₹1800
"A and B invest ₹3000 and ₹4000 respectively. A withdraws after 4 months. If total profit is ₹2100, find A's share." → A:B = 3000×4 : 4000×12 = 12000:48000 = 1:4. A = 2100/5 = ₹420

📝 Practice Questions (with Answers)

Q1. In what ratio should water (free) be mixed with sugar solution at ₹30/L to get solution worth ₹20/L?

A: Alligation: Water(0):Mean(20):Sugar(30). Ratio = (30−20):(20−0) = 10:20 = 1:2

Q2. A:B = 5:7, B:C = 3:4. Find A:C.

A: A:B = 5:7, B:C = 3:4. A:C = (5/7) × (3/4) ... easier: A:B:C = 15:21:28. A:C = 15:28

Q3. A mixture of 80L has milk and water in ratio 3:1. How much water should be added to make ratio 2:3?

A: Milk = 60L, Water = 20L. Let x litres water added: 60/(20+x) = 2/3 → 180 = 40+2x → x = 70L

Q4. ₹7500 is divided among A, B, C in ratio 3:4:5. Find B's share.

A: B = (4/12) × 7500 = ₹2500

Q5. A container has 50L of pure milk. 10L is removed and replaced with water. This is done twice. Find concentration of milk.

A: Milk = 50 × (40/50)² = 50 × 16/25 = 32L → Concentration = 32/50 = 64%

🔑 Exam Strategy

Alligation is your best friend — it turns 3-step problems into 1-step
When you see "two things mixed at different prices/concentrations" → immediately think alligation cross
Ratio merging (make common term equal) takes <10 seconds once automatic
Replacement formula V(1 − x/V)^n is asked almost every year — memorize it
In trading context: mixing/blending is analogous to portfolio weighting — interviewers may frame it that way

05_ratio_proportion_mixtures.md