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Ratio & Proportion

Learning Goals

By the end of this chapter, you will be able to:

• Understand what a ratio really means (and what it does NOT mean)
• Simplify, compare, and combine ratios with confidence
• Master proportion, continued proportion, and the componendo–dividendo family of rules
• Solve partnership, mixture, and income–expenditure problems using ratios
• Spot the traps examiners set in SSC CGL Tier-II, IBPS PO Mains, CAT, and CSAT

The Hook: Two Cups of Tea

Imagine two cups of tea. Cup A has 2 spoons of sugar in 100 ml of tea. Cup B has 3 spoons of sugar in 150 ml of tea. Which one is sweeter?

Most people instantly say "Cup B — it has more sugar!" But wait. Cup B also has more water. When you compare 2/100 and 3/150, both simplify to 1/50. Both cups are equally sweet.

That tiny moment of confusion you just felt? That is exactly where examiners make their money. Ratio questions are rarely hard to calculate — they are hard to *interpret*. This chapter trains you to never fall for the "more sugar" trap again.

1. What Is a Ratio, Really?

A ratio is a comparison of two quantities of the same kind, expressed as a fraction. If a class has 40 boys and 60 girls, the ratio of boys to girls is:

$$40 : 60 = \frac{40}{60} = \frac{2}{3} = 2 : 3$$

Three golden rules:

1. Same units only. You cannot compare 2 kg with 500 grams until you convert: 2000 g : 500 g = 4 : 1.
2. A ratio has no units. It is a pure number — 2:3, not "2:3 kg".
3. Order matters. Boys : Girls = 2:3 is different from Girls : Boys = 3:2.

In the ratio a : b, a is the antecedent (first term) and b is the consequent (second term).

🧠 Active Recall: Without looking back — can a ratio compare 5 metres with 3 kilograms? Why or why not? What are the two terms of a ratio called?

2. Types of Ratios

Examiners love testing these definitions directly:

Mnemonic 🧠 — "Du-Tri-Sub squares it": *Du*plicate squares, *Tri*plicate cubes, and *Sub* just undoes them (root). Think: "The Duke squares, the Tribes cube, the Sub dives to the root."

🧠 Active Recall: What is the duplicate ratio of 3:4? What is the sub-triplicate ratio of 64:125?

3. Comparing Ratios

Which is bigger: 3:5 or 4:7? Convert to fractions and cross-multiply:

$$\frac{3}{5} \text{ vs } \frac{4}{7} \Rightarrow 3 \times 7 = 21 \text{ vs } 4 \times 5 = 20$$

Since 21 > 20, 3:5 is greater. Cross-multiplication is your fastest comparison tool — no LCM needed.

🧠 Active Recall: Compare 5:8 and 7:11 using cross-multiplication. Which is larger?

4. Proportion: When Two Ratios Shake Hands

Four quantities a, b, c, d are in proportion if:

$$a : b = c : d \quad \Longleftrightarrow \quad \frac{a}{b} = \frac{c}{d} \quad \Longleftrightarrow \quad ad = bc$$

Written as a : b :: c : d. Here:

• a and d are the extremes (outer terms)
• b and c are the means (inner terms)
• Product of extremes = Product of means (ad = bc)
• d is called the fourth proportional to a, b, c

Continued proportion: a : b = b : c means b² = ac. Here b is the mean proportional between a and c, and c is the third proportional to a and b.

Example: Find the mean proportional between 4 and 25.
b² = 4 × 25 = 100, so b = 10. ✔

🧠 Active Recall: In a : b :: c : d, which terms are the extremes? If 2, 6, x are in continued proportion, find x.

5. The Componendo–Dividendo Family

This is the Advanced-level heart of the chapter. If a/b = c/d, then:

Why it matters: Questions like "If (3x + 2y)/(3x − 2y) = 5/3, find x : y" collapse in seconds:

Apply C–D in reverse: 3x/2y = (5+3)/(5−3) = 8/2 = 4, so x/y = 8/3. Done.

Mnemonic 🧠 — "C adds, D subtracts, CD does both": Componendo adds, Dividendo subtracts, and Componendo-Dividendo puts the sum on top and the difference below.

🧠 Active Recall: If a/b = c/d, what does alternendo give? What sits in the numerator and denominator after componendo-dividendo?

6. The Equal-Ratios Power Tool

If a/b = c/d = e/f = k, then each ratio also equals:

$$\frac{a + c + e}{b + d + f} = k$$

More generally, any linear combination works: (pa + qc + re)/(pb + qd + rf) = k.

Example: If x/2 = y/3 = z/5 and x + y + z = 40, then x = 2k, y = 3k, z = 5k, so 10k = 40, k = 4. Hence x = 8, y = 12, z = 20.

This "let each ratio equal k" trick solves the large majority of proportion problems in exams. Make it your default move.

🧠 Active Recall: If a/3 = b/4 = c/5 and a + b + c = 60, what is b?

7. Dividing an Amount in a Given Ratio

To divide ₹N in the ratio a : b : c:

• Total parts = a + b + c
• Shares = (a/(a+b+c)) × N, (b/(a+b+c)) × N, (c/(a+b+c)) × N

Example: Divide ₹1,260 among A, B, C in the ratio 2 : 3 : 4.
Total parts = 9. A gets 2/9 × 1260 = ₹280, B gets ₹420, C gets ₹560.

Trap alert 🪤: If the ratio is given as fractions like 1/2 : 1/3 : 1/4, first clear denominators by multiplying by the LCM (12): you get 6 : 4 : 3. Never divide directly by fractions.

🧠 Active Recall: Divide ₹900 in the ratio 1/2 : 1/3 : 1/6. What is the largest share?

8. Combining Ratios: The A:B and B:C Bridge

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.

B is the bridge — make its value equal in both ratios. LCM of 3 and 4 is 12:

• A : B = 2 : 3 = 8 : 12
• B : C = 4 : 5 = 12 : 15

So A : B : C = 8 : 12 : 15.

This bridging technique powers partnership problems, chain ratios (A:B, B:C, C:D), and mixture questions across every exam.

🧠 Active Recall: If P : Q = 3 : 4 and Q : R = 2 : 5, find P : Q : R.

9. Income, Expenditure and the "Don't Assume the Salary" Trap

Classic exam question: "The incomes of A and B are in the ratio 3:2 and their expenditures in the ratio 5:3. Each saves ₹1,000. Find their incomes."

Let incomes be 3x and 2x; expenditures be 5y and 3y.

• 3x − 5y = 1000
• 2x − 3y = 1000

Solving: multiply the first equation by 2 and the second by 3 → 6x − 10y = 2000 and 6x − 9y = 3000. Subtracting the first from the second: y = 1000. Substituting back: 2x − 3000 = 1000, so x = 2000.

Incomes: A = 3x = ₹6,000, B = 2x = ₹4,000.

Check: A spends 5y = ₹5,000 and saves ₹1,000 ✔; B spends 3y = ₹3,000 and saves ₹1,000 ✔.

Trap alert 🪤: The income ratio and expenditure ratio use *different* unknowns (x and y). Never use the same variable for both — that is the single most common error in this chapter.

🧠 Active Recall: Why can't we take income as 3x, 2x and expenditure as 5x, 3x in the same problem?

10. Coins, Mixtures and Partnerships

Coins: A bag has 50p, 25p and 10p coins in the ratio 5 : 9 : 4, totalling ₹2,060. Value per "ratio unit" = 5(0.50) + 9(0.25) + 4(0.10) = 2.5 + 2.25 + 0.4 = ₹5.15. Number of units = 2060/5.15 = 400. So 50p coins = 2000, 25p coins = 3600, 10p coins = 1600.

Partnership: Profit is divided in the ratio of (investment × time). If A invests ₹5,000 for 12 months and B invests ₹8,000 for 6 months, the profit ratio is (5000×12) : (8000×6) = 60000 : 48000 = 5 : 4.

Mixtures: If two mixtures with milk:water ratios 2:1 and 4:1 are mixed equally, convert to a common total (3 and 5 → LCM 15): first gives 10:5, second gives 12:3. Combined milk : water = 22 : 8 = 11 : 4.

🧠 Active Recall: In partnership, profit is shared in the ratio of what two factors multiplied together?

11. Variation: Direct and Inverse

• Direct variation: y ∝ x means y = kx. More workers → more work done.
• Inverse variation: y ∝ 1/x means xy = k. More workers → less time.

Example: If 12 men build a wall in 10 days, how long for 15 men? (Inverse) 12 × 10 = 15 × d, so d = 8 days.

🧠 Active Recall: Speed and time for a fixed distance — direct or inverse variation?

Summary Table

🔁 Review This After 2 Days

Close the book and answer from memory:

1. What is the componendo-dividendo rule?
2. How do you combine A:B = 2:3 and B:C = 4:5?
3. In the income–expenditure problem, why do we need two different variables?
4. What is the mean proportional between 9 and 16?

If any answer hesitates, re-read only that section — not the whole chapter. Then revise again after 7 days. Spaced repetition is what moves ratios from "I studied this" to "I can't get this wrong."