Ratio & Proportion

Ratio & Proportion

Concept Explanation

A ratio is simply a way to express how two numbers compare to each other. If you have 8 apples and 12 oranges, the ratio of apples to oranges is 8:12, which simplifies to 2:3. Think of it as a fraction — the order matters enormously. Writing apples to oranges as 3:2 would mean you have more oranges than apples, which flips the entire meaning.

Proportion comes in when you say two ratios are actually the same. If 2:3 equals 4:6, that’s a proportion — they’re equivalent fractions. The magic property of proportions is cross-multiplication: when a/b = c/d, you can multiply across diagonally (a × d = b × c) to get rid of fractions and solve for unknowns. This single move unlocks almost every ratio and proportion problem you’ll see.

Ratios can also be combined, split, and manipulated. The compounded ratio of (a:b) and (c:d) is simply (ac:bd). You can divide a ratio, multiply it, find its duplicate or sub-duplicate form. The key thing to remember is that multiplying both parts of a ratio by the same number doesn’t change the ratio — just like simplifying a fraction. That’s why 2:3, 4:6, and 100:150 all represent the same relationship.

Key Formulas

Symbol	Meaning
a : b	Ratio of a to b (a/b as a fraction)
a/b = c/d	Proportion statement
ad = bc	Cross-multiplication rule
a, d	Extremes (outer terms of a proportion)
b, c	Means (inner terms of a proportion)
Compounded ratio	(a:b) × (c:d) = (ac:bd)

Step-by-Step Example

Q: A mixture contains milk and water in the ratio 5:3. If 8 more liters of milk are added and the total mixture becomes 52 liters, find the original quantity of water.

Step 1: Let milk = 5x and water = 3x. Total = 5x + 3x = 8x = 52, so x = 6.5. Step 2: Original water = 3 × 6.5 = 19.5 liters. Answer: 19.5 liters

Common Mistakes

• Mixing up the order of a ratio → Always label which quantity is first (apples:oranges, not just “2:3”)
• Forgetting to simplify before solving → Always reduce ratios to lowest terms first
• Cross-multiplying in wrong direction → Remember: top × bottom = bottom × top

Quick Test (2 Qs)

1. Q: If 4:x = 12:27, what is x? Options: A) 6 B) 9 C) 12 D) 18. Ans: B) 9 (Reason: 4/x = 12/27 → 4×27 = 12×x → 108 = 12x → x = 9)
2. Q: The ratio of speeds of a car and bus is 7:4. If the car covers 280 km, how much does the bus cover in the same time? Options: A) 140 km B) 160 km C) 180 km D) 200 km. Ans: B) 160 km (Reason: 7:4 = 280:distance → 7/4 = 280/x → 7x = 1120 → x = 160)