Ratio and Proportion; Mixture and Alligation

Ratio and Proportion

6.1 Definitions

Ratio a:b = a/b

• a = antecedent, b = consequent
• Ratio has NO units
• Can be simplified like fractions

Proportion: If a:b = c:d, then a, b, c, d are in proportion

• Written as a:b::c:d

Product of means = Product of extremes

• b × c = a × d (cross multiplication)

6.2 Types of Proportions

6.3 Compounded Ratio

• If a:b and c:d are two ratios, their compound ratio = ac:bd

6.4 Duplicate/Triplicate/Sub-duplicate Ratios

6.5 Working With Multiple Ratios

To find A:B:C when A:B and B:C are given:

• Make B's value common
• A:B = 3:4, B:C = 5:6
• Multiply: A:B:C = (3×5):(4×5):(4×6) = 15:20:24

Example Problem: If A:B = 3:4, B:C = 5:6, C:D = 7:8:

• A:B = 3:4 → multiply both by 5 → 15:20
• B:C = 5:6 → multiply both by 4 → 20:24
• C:D = 7:8 → multiply both by 3 → 21:24... adjust to 24:..
• (C in B:C = 24) and (C in C:D should match)
• A:B:C = 15:20:24 and D = 24 × 8/7 → adjust
• Final: A:B:C:D = 2×4×6 : 3×4×6 : 3×5×6 : 3×5×7

6.6 Proportion in Direct and Inverse

Direct Proportion: If A increases, B increases proportionally

• A ∝ B → A = kB → A₁/A₂ = B₁/B₂

Inverse Proportion: If A increases, B decreases

• A ∝ 1/B → A₁/A₂ = B₂/B₁

6.7 Dividing a Number in a Given Ratio

• To divide N in ratio a:b:c → parts are Na/(a+b+c), Nb/(a+b+c), Nc/(a+b+c)

Mixture and Alligation

7.1 Alligation Rule (Cross Method)

Used when two ingredients at different prices are mixed to give a mixture at a mean price.

      Cheaper (C)          Dearer (D)
              \              /
               \            /
           Mean Price (M)
              /            \
             /              \
       (D - M)           (M - C)

Ratio of mixing = (D - M) : (M - C)

7.2 Rule of Alligation Formula

(Quantity of Cheaper) / (Quantity of Dearer) = (D - M) / (M - C)

7.3 Understanding Alligation

• Cheaper Component Amount = D - M (dearer price minus mean price)
• Dearer Component Amount = M - C (mean price minus cheaper price)

Example: A mixture of Rs. 8/kg and Rs. 12/kg tea sells at Rs. 10/kg. Find ratio of mixing.

• Cheaper = 8, Dearer = 12, Mean = 10
• Ratio = (12-10) : (10-8) = 2:2 = 1:1

7.4 Repeated Dilution Formula

When from a vessel containing V litres of pure liquid, x litres are taken out and replaced by water, after n operations:

Quantity of pure liquid = V × (1 - x/V)ⁿ

Example: A cask contains 40 litres of milk. 4 litres removed and 4 litres water added 3 times. Milk remaining:

• = 40 × (1 - 4/40)³ = 40 × (9/10)³ = 40 × 729/1000 = 29.16 litres

7.5 Water and Milk Mixture Problems

If mixture has milk:water = a:b and we add x litres of water:

• New ratio of water = (Original water amount + x) : original milk amount