Ratio & Proportion

Ratio & Proportion

🟢 Lite

Key Formula/Rule

a/b = c/d ⟹ ad = bc (cross-multiply to check or solve proportions)

Quick Memory Trick

“Ratios are just fractions in disguise — treat them the same way.” Cross-multiplication is the ultimate shortcut: top × bottom = bottom × top.

1-Sentence Summary

A ratio compares two quantities; a proportion states two ratios are equal, and you cross-multiply to solve unknowns.

Quick Example

Q: If 3/5 = x/20, find x. A: 3 × 20 = 5 × x → x = 60/5 = 12

Must Remember

• Duplicate ratio (a:b → a²:b²) and sub-duplicate ratio (a:b → √a:√b) are special types
• Compounded ratio of (a:b) and (c:d) = (ac:bd)
• In a proportion a:b = c:d, a and d are extremes, b and c are means → product of extremes = product of means

🟡 Standard

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)

🔴 Extended

Concept Deep Dive

Most students brush past ratio and proportion thinking it’s just simple fractions, but this topic is deceptively powerful. Every time you compare quantities — speeds, prices, concentrations, shares, wages — you’re working with ratios. GATE loves embedding ratio concepts inside larger problems where you’re not even told “this is a ratio question.” Spotting ratios hiding inside word problems is half the battle.

The cross-multiplication rule (ad = bc) isn’t just a mechanical trick — it’s algebraic truth. When you have a/b = c/d, you’re saying two fractions are equal. Multiply both sides by bd and you get ad = bc. That’s it. The reason this works every time is because you’re applying the same operation to both sides of an equation. Once you internalize this, you’ll never forget which terms to multiply.

The mean proportional is a concept that sneaks into GATE regularly. If a and b are in continued proportion, then a:b = b:c, which means b is the mean proportional between a and c. That gives us b² = ac. This shows up in geometry (similar triangles) and in work problems where rates form continued proportions. The third proportional follows the same logic: a:b = b:c, then c is the third proportional to a and b.

In mixture and alligation problems, ratio and proportion become the workhorse. When you mix two ingredients at different prices, the ratio of quantities is inversely proportional to the difference between each price and the mean price. This is the alligation rule: (Cheaper quantity)/(Dearer quantity) = (Mean price - Dearer price)/(Mean price - Cheaper quantity). It looks intimidating but it’s just a proportion dressed up in words.

Advanced Formula Derivation

Duplicate Ratio Derivation: If the ratio a:b is duplicated, we get a²:b². Proof: Duplicate ratio means the ratio of squares of the original terms. For a:b, duplicate = a²:b². This is useful when comparing areas or volumes where the ratio scales by a power factor.

Alligation Rule Derivation: When mixing two ingredients priced at p₁ and p₂ (p₁ < p₂) to get a mean price m: Let quantities be q₁ and q₂. Total cost = p₁q₁ + p₂q₂ = m(q₁ + q₂) p₁q₁ + p₂q₂ = mq₁ + mq₂ p₂q₂ - mq₂ = mq₁ - p₁q₁ q₂(p₂ - m) = q₁(m - p₁) q₁/q₂ = (p₂ - m)/(m - p₁) This is the alligation formula — the ratio of quantities is inversely related to their distance from the mean price.

GATE-Level Numerical Problems

Q1 (GATE 2020): A sum of ₹3,600 is to be divided among three persons A, B, and C in the ratio 2:3:5. B’s share is increased by 10% and C’s share is decreased by 10%. What is B’s new share?

• Working: Original shares — A = 2x, B = 3x, C = 5x where 10x = 3600, so x = 360. B’s original = 1080. New B = 1080 × 1.10 = 1188.
• Answer: ₹1,188
• Common error: Applying the percentage change to A or C instead of B, or forgetting that increasing one share requires decreasing another proportionally.

Q2 (GATE 2018): Two alloys A and B contain copper and zinc in the ratios 3:4 and 5:6 respectively. If 14 kg of alloy A is mixed with 22 kg of alloy B, what is the ratio of copper to zinc in the resulting mixture?

• Working: Alloy A: Cu = 14 × 3/7 = 6 kg, Zn = 14 × 4/7 = 8 kg. Alloy B: Cu = 22 × 5/11 = 10 kg, Zn = 22 × 6/11 = 12 kg. Total Cu = 16 kg, Total Zn = 20 kg. Ratio Cu:Zn = 16:20 = 4:5.
• Answer: 4:5
• Common error: Adding the ratios directly (3:4 + 5:6 ≠ 8:10) instead of computing actual quantities.

Q3: A container has wine and water in the ratio 7:5. If 12 liters of the mixture are drawn off and replaced by 12 liters of water, the ratio becomes 7:9. Find the total volume of the container.

• Working: Let total = V liters. Wine initially = 7V/12, Water = 5V/12. When 12L drawn: Wine removed = 12 × 7/12 = 7L. Remaining wine = 7V/12 - 7. Remaining water = 5V/12 - 5 + 12 (added water). New ratio = (7V/12 - 7) / (5V/12 + 7) = 7/9. Cross-multiply: 9(7V/12 - 7) = 7(5V/12 + 7). Simplify: (63V/12 - 63) = (35V/12 + 49). 63V/12 - 35V/12 = 49 + 63 = 112. 28V/12 = 112, V = 48.
• Answer: 48 liters

Multiple Approaches

Method A: Algebraic substitution Let the total be T. Set up wine = 7T/12, water = 5T/12. Draw off 12L and replace. Use the mixture removal formula (amount removed = fraction × total drawn). Solve the resulting equation.

Method B: Fractional approach Work entirely in fractions. The fraction of wine after replacement = (original wine fraction) × (fraction remaining after removal). This avoids setting up total T explicitly.

When to use: Method A is safer for complex problems with multiple replacements. Method B is faster for single-step replacements where total isn’t needed.

Tricky Cases

• Continued proportion traps: When asked for “mean proportional between a and b,” remember it’s √(ab), not (a+b)/2. Many students confuse arithmetic mean with geometric mean in proportion context.
• Inverse ratio confusion: If A:B = 3:4, then B:A (inverse ratio) = 4:3. Always flip when asked for the reverse comparison.
• Percentage change in ratio: When a ratio’s terms change by percentages, you cannot simply add/subtract percentages. A 20% increase in a and 20% decrease in b changes the ratio from a:b to 1.2a:0.8b = 3a:2b of original — the ratio doesn’t just shift by 40%.

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