SATHEE: Ratio & Proportion - All Formulas & Shortcuts

Ratio & Proportion - All Formulas & Shortcuts

Quick reference guide for all Ratio & Proportion formulas with instant solving techniques

📘 Basic Definitions

1. Ratio

Definition:

Ratio = Comparison of two quantities of same kind Written as: a:b or a/b

Important:

Both quantities must be in same units
a:b is read as “a is to b”
First term (a) = Antecedent
Second term (b) = Consequent

2. Types of Ratios

Duplicate Ratio:

If ratio = a:b Duplicate ratio = a²:b²

Triplicate Ratio:

If ratio = a:b Triplicate ratio = a³:b³

Sub-duplicate Ratio:

If ratio = a:b Sub-duplicate ratio = √a:√b

Sub-triplicate Ratio:

If ratio = a:b Sub-triplicate ratio = ∛a:∛b

Inverse/Reciprocal Ratio:

If ratio = a:b Inverse ratio = b:a = 1/a:1/b

3. Proportion

Definition:

When two ratios are equal a:b = c:d OR a/b = c/d Read as: “a is to b as c is to d”

Properties:

a:b = c:d ⟹ ad = bc (Cross multiplication) ⟹ a/c = b/d (Alternendo) ⟹ (a+b)/b = (c+d)/d (Componendo) ⟹ (a-b)/b = (c-d)/d (Dividendo)

Terms:

a, d = Extremes
b, c = Means
Product of extremes = Product of means

🎯 Key Formulas

4. Fourth Proportional

If a:b = c:d, then d is fourth proportional

d = bc/a

Example: If 3:4 = 6:x

x = (4 × 6)/3 = 8

5. Third Proportional

If a:b = b:c, then c is third proportional

c = b²/a

Example: If 4:6 = 6:x

x = 6²/4 = 36/4 = 9

6. Mean Proportional

If a:b = b:c, then b is mean proportional

b = √(ac)

Example: Mean proportional between 4 and 9

b = √(4 × 9) = √36 = 6

⚡ Componendo & Dividendo

7. Componendo Rule

If a/b = c/d, then:

(a + b)/b = (c + d)/d

Usage: When you need to find sum in numerator

8. Dividendo Rule

If a/b = c/d, then:

(a - b)/b = (c - d)/d

Usage: When you need to find difference in numerator

9. Componendo-Dividendo (Most Important!)

If a/b = c/d, then:

(a + b)/(a - b) = (c + d)/(c - d)

This is THE MOST POWERFUL tool for ratio problems!

Example: If (x+y)/(x-y) = 5/3, find x/y

Solution:

By componendo-dividendo: [(x+y) + (x-y)] / [(x+y) - (x-y)] = (5+3)/(5-3) 2x / 2y = 8/2 x/y = 4/1 = 4:1

💡 Ratio Shortcuts

10. Comparing Ratios

To compare a:b with c:d:

Method 1: Cross multiply

If ad > bc, then a:b > c:d If ad < bc, then a:b < c:d If ad = bc, then a:b = c:d

Method 2: Convert to decimal

a:b = a/b (decimal) c:d = c/d (decimal) Compare decimals

Example: Compare 3:4 and 5:7

3/4 = 0.75 5/7 = 0.714 Therefore, 3:4 > 5:7

11. Compounded Ratio

Ratio of products:

(a:b) compounded with (c:d) = ac:bd

Example:

(2:3) compounded with (4:5) = (2×4):(3×5) = 8:15

For three ratios:

(a:b), (c:d), (e:f) = ace:bdf

12. Adding/Subtracting Ratios

Important: You CANNOT directly add/subtract ratios!

If a:b and c:d need to be combined:

Step 1: Make second term same (LCM of b and d)

a:b = am:bm (where m = LCM/b) c:d = cn:dn (where n = LCM/d)

Step 2: Then add/subtract first terms

Example: Add 2:3 and 4:5

LCM(3,5) = 15 2:3 = 10:15 4:5 = 12:15 Sum = 22:15 (but this is unusual - mostly used in mixture problems)

🔥 Distribution in Ratios

13. Dividing Amount in Given Ratio

Divide amount A in ratio a:b:

First part = A × a/(a+b) Second part = A × b/(a+b)

Example: Divide 500 in ratio 2:3

First part = 500 × 2/5 = 200 Second part = 500 × 3/5 = 300

For three parts (a:b:c):

First = A × a/(a+b+c) Second = A × b/(a+b+c) Third = A × c/(a+b+c)

14. Finding Ratio from Shares

If A gets x, B gets y from total:

Ratio A:B = x:y

Simplify by dividing by HCF

Example: A gets 200, B gets 300

Ratio = 200:300 = 2:3 (divide by HCF 100)

🎓 Advanced Techniques

15. When Ratios Change

Initially a:b, becomes c:d:

Finding actual values:

Let initial be ax and bx
Final becomes cx and dx
Use given condition to find x

Example: Initial ratio 3:4, after adding 10 to each, ratio becomes 5:6

(3x + 10):(4x + 10) = 5:6 6(3x + 10) = 5(4x + 10) 18x + 60 = 20x + 50 2x = 10 x = 5 Initial numbers: 15 and 20

16. Ratio of Ratios

If a:b = 3:4 and b:c = 5:6:

Find a:b:c:

Method: Make b same in both

a:b = 3:4 = 15:20 (multiply by 5) b:c = 5:6 = 20:24 (multiply by 4) Therefore, a:b:c = 15:20:24

17. Inverse Ratio (Indirect Proportion)

If a is inversely proportional to b:

a × b = constant (k) a₁/a₂ = b₂/b₁

Example: If 5 workers complete in 10 days, how many days for 8 workers?

Workers × Days = constant 5 × 10 = 8 × x x = 50/8 = 6.25 days

💎 Special Ratios

18. Golden Ratio

φ = (1 + √5)/2 ≈ 1.618

19. Common Ratios to Memorize

Ratio Decimal Percentage

1:1 1.0 50%-50%

1:2 0.5 33%-67%

1:3 0.333 25%-75%

1:4 0.25 20%-80%

2:3 0.667 40%-60%

3:4 0.75 43%-57%

3:5 0.6 38%-62%

4:5 0.8 44%-56%

Ratio	Decimal	Percentage
1:1	1.0	50%-50%
1:2	0.5	33%-67%
1:3	0.333	25%-75%
1:4	0.25	20%-80%
2:3	0.667	40%-60%
3:4	0.75	43%-57%
3:5	0.6	38%-62%
4:5	0.8	44%-56%

📊 Exam Patterns

Pattern 1: Ages in Ratio

Present ages in ratio a:b, after n years ratio is c:d:

Set up: (ax + n)/(bx + n) = c/d Solve for x

Pattern 2: Income-Expenditure

Incomes in ratio a:b, expenditures in ratio c:d:

If savings are equal: a × income - c × exp = b × income - d × exp

Pattern 3: Mixture Problems

Mix two quantities in ratio a:b:

Quantity 1 = Total × a/(a+b) Quantity 2 = Total × b/(a+b)

Pattern 4: Partnership

Investment ratio = Profit ratio (if time is same)

A:B = Investment_A:Investment_B

🎯 Mental Math Tricks

Trick 1: Quick Division in Ratio

Divide 1000 in ratio 3:2:

Total parts = 3 + 2 = 5 Each part = 1000/5 = 200 First = 3 × 200 = 600 Second = 2 × 200 = 400

Trick 2: Percentage to Ratio

30% : 70% = ?

Remove % and simplify 30:70 = 3:7

Trick 3: Decimal to Ratio

0.6 : 0.4 = ?

Multiply by 10: 6:4 = 3:2

Trick 4: Mixed Number Ratio

2½ : 3⅓ = ?

Convert to improper: 5/2 : 10/3 Multiply by LCM(2,3) = 6 (5/2 × 6) : (10/3 × 6) = 15:20 = 3:4

🔍 Quick Reference

Ratio Properties

✅ a:b = ka:kb (multiply both by same number) ✅ a:b = a/k : b/k (divide both by same number) ✅ a:b:c can be simplified by dividing by HCF ✅ If a:b = c:d, then a+b:b = c+d:d (Componendo) ✅ If a:b = c:d, then a-b:b = c-d:d (Dividendo)

Common Mistakes

❌ Adding ratios directly: 2:3 + 1:2 ≠ 3:5 ❌ Forgetting to simplify: 10:15 should be 2:3 ❌ Mixing units: 2kg:500g should be 2000g:500g = 4:1 ✅ Always make units same before forming ratio ✅ Always simplify by dividing by HCF

Practice Questions: