One-Line Definition
A Ratio compares two quantities by division (a : b = a/b), while a Proportion states that two ratios are equal (a : b = c : d).
Recipe Scaling Model
A recipe calls for 2 cups flour to 3 cups sugar (Ratio 2 : 3):
- • Make 1 batch → 2 cups flour, 3 cups sugar
- • Double the recipe → 4 cups flour, 6 cups sugar
- • The ratio remains 2 : 3 regardless of total batch size!
Ratios form the core backbone of arithmetic. Master ratios, and you automatically unlock solutions for mixtures, partnerships, ages, and Data Interpretation:
Proportion & Extremes
Product of extremes equals product of means.
Compendo & Dividendo
Essential for simplifying ratio equations quickly.
Merging Ratios (LCM Shortcut)
Given A : B = 3 : 4 and B : C = 2 : 5. To find A : B : C:
SHORTCUT RULES & KEY FORMULAS
Alligation Rule (Cross Subtraction)
When mixing two ingredients of price P1 and P2 to obtain a mixture of price M.
Ratio Variations & Powers
Duplicate Ratio = a² : b²
Sub-duplicate Ratio = √a : √b
Used frequently in geometric scaling and area ratio questions.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: The ratio of boys to girls in a school is 3 : 5. If there are 40 girls, how many boys are there?
Step 1: 5 parts = 40 girls → 1 part = 40 ÷ 5 = 8.
Step 2: 3 parts (boys) = 3 × 8 = 24 boys.
Q: Divide a sum of ₹600 between A and B in the ratio 2 : 3.
Step 1: Total parts = 2 + 3 = 5 parts.
Step 2: 1 part = ₹600 ÷ 5 = ₹120.
Step 3: A = 2 × 120 = ₹240; B = 3 × 120 = ₹360.
Q: In what ratio should tea priced at ₹120/kg be mixed with tea priced at ₹180/kg to get a mixture worth ₹160/kg?
Step 1: Cheaper C = 120, Dearer D = 180, Mean M = 160.
Step 2: Ratio = (D - M) : (M - C) = (180 - 160) : (160 - 120) = 20 : 40 = 1 : 2.
Mistake 1
Merging Ratios Without Equalizing
Combining A:B and B:C directly without making B equal in both terms. Always calculate the LCM of the common variable first!
Mistake 2
Confusing Parts with Values
Treating a ratio of 2:3 as absolute numbers 2 and 3 instead of 2x and 3x.
Mistake 3
Wrong Alligation Position
Placing Cheaper and Dearer values in reversed positions, leading to inverted quantity ratios.
1. The ratio of two numbers is 4 : 5 and their sum is 135. Find the numbers.
Answer: 9 parts = 135 → 1 part = 15. Numbers are 4×15 = 60 and 5×15 = 75.
2. If A:B = 2:3 and B:C = 4:5, find A:C.
Answer: (A/B) × (B/C) = (2/3) × (4/5) = 8 / 15 → A:C = 8 : 15.
3. A milkman has 20 liters of a mixture of milk and water containing 10% water. How much water must be added to make it 25% water?
Answer: Milk is constant = 90% of 20 = 18L. In new mixture, milk = 75% = 18L → New Total = 18 / 0.75 = 24L. Water added = 24 - 20 = 4 liters.
FREQUENTLY ASKED QUESTIONS
❓ How do you combine three ratios A:B, B:C, and C:D into A:B:C:D?
Equalize the common terms across ratios using LCM. For example, if A:B = 3:4 and B:C = 2:5, multiply B:C by 2 so B becomes 4 in both. Then A:B:C = 3:4:10.
❓ What is the Alligation Rule and when should I use it?
Alligation is a visual cross-subtraction rule used when two ingredients of different prices/concentrations are mixed to produce a target mixture. Ratio Q1:Q2 = (Price2 - MeanPrice) / (MeanPrice - Price1).
❓ What is Componendo and Dividendo?
It is an algebraic theorem stating that if a/b = c/d, then (a + b) / (a - b) = (c + d) / (c - d). It speeds up solving complex ratio fractions.