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Ratio & Proportion

Chapter Outline & Learning Goals

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

1. Understand what a ratio really means and simplify any ratio.
2. Distinguish between ratio, proportion, continued proportion, and variation.
3. Apply componendo, dividendo, and their combinations confidently.
4. Solve exam favourites: dividing an amount in a ratio, mixing ratios, income-expenditure problems, and coin problems.
5. Spot the traps examiners set — and beat them in under 30 seconds.

The Hook: A Cup of Tea That Fooled Everyone

Two friends order tea. The first mixes milk and water in the ratio 3:1. The second mixes them in the ratio 5:2. Both claim, "My tea is milkier!" Who is right?

Most students guess 5:2 because the numbers look bigger. But 3:1 means 75% milk, while 5:2 means only about 71% milk. The smaller-looking ratio wins!

This is exactly why Ratio & Proportion is dangerous in exams — it *looks* easy, so students rush, and examiners cash in on that overconfidence. In SSC CGL, IBPS PO, and RRB NTPC, this chapter hides inside percentage, profit-loss, mixture, partnership, and time-work questions. Master it once, and you unlock marks across five other chapters.

1. What Is a Ratio?

A ratio is a comparison of two quantities of the *same kind* by division. If a class has 40 boys and 30 girls, the ratio of boys to girls is:

$$40 : 30 = 4 : 3$$

Three golden rules:

• Same units only. You cannot compare 2 kg with 500 g until you convert: 2000 g : 500 g = 4 : 1.
• A ratio has no unit. It is a pure number.
• Order matters. 4:3 is not the same as 3:4.

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

🧠 Active Recall: Without looking back — can you compare 3 hours with 45 minutes directly as 3:45? Why or why not? What is the correct ratio?

2. Types of Ratios

For a ratio a : b (where a > b):

Compounding ratios: When ratios are multiplied term by term: (2:3) compounded with (4:5) = (2×4) : (3×5) = 8 : 15.

🧠 Active Recall: What is the duplicate ratio of 5 : 2? And the inverse ratio of 7 : 9?

3. Proportion — When Two Ratios Shake Hands

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

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

Written as a : b :: c : d. Here a and d are the extremes, b and c are the means. The magic rule:

> Product of extremes = Product of means

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

Fourth proportional: If a : b = c : d, then d = bc/a.

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

🧠 Active Recall: Find the third proportional to 9 and 12. (Hint: 9 : 12 = 12 : x.)

4. The Power Tools: Componendo & Dividendo

If a : b = c : d, then:

• Componendo: (a + b) : b = (c + d) : d
• Dividendo: (a − b) : b = (c − d) : d
• Componendo-Dividendo: (a + b) : (a − b) = (c + d) : (c − d)

Why it matters: Questions like "If (3x + 2y) : (3x − 2y) = 5 : 3, find x : y" take 60 seconds by expansion but only 10 seconds by componendo-dividendo:

$$\frac{3x}{2y} = \frac{5+3}{5-3} = \frac{8}{2} = 4 \implies \frac{x}{y} = \frac{8}{3}$$

🧠 Active Recall: If (a + b) : (a − b) = 7 : 1, what is a : b? Try it with componendo-dividendo before peeking at any formula.

5. The Exam Favourites

(a) Dividing an amount in a ratio

Divide ₹840 between A and B in the ratio 5 : 7.
A's share = 840 × 5/12 = ₹350; B's share = 840 × 7/12 = ₹490.

(b) Combining ratios (the A:B, B:C chain)

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Make B equal: LCM of 3 and 4 is 12.
A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.

(c) Coins problem

A bag has ₹1, 50p, and 25p coins in the ratio 3 : 4 : 5, totalling ₹225. Number of ₹1 coins?
Value ratio = 3×1 : 4×0.5 : 5×0.25 = 3 : 2 : 1.25 = 12 : 8 : 5.
₹1 coins' value = 225 × 12/25 = ₹108 → 108 coins.

(d) Income–expenditure trap

Incomes of A and B are in ratio 4 : 5, expenditures in ratio 3 : 4, and each saves ₹2000. Income of A?
4x − 3y = 2000 and 5x − 4y = 2000. Solving: x = 2000, so A's income = ₹8000.

🧠 Active Recall: In the coin problem, why do we multiply the coin ratio by the coin values before splitting the total?

6. The Memory Trick 🎭

"The Cross-Multiply Cricket Match" 🏏

Imagine proportion a : b :: c : d as a cricket match. The extremes (a and d) are the opening batsmen standing at the two far ends; the means (b and c) are the runners squeezed in the middle. The umpire's rule: *"The match is fair only when the openers' product equals the runners' product."*

So whenever you see "::", hear the umpire shout: "Outside × Outside = Inside × Inside!" — and you'll never forget ad = bc.

For componendo-dividendo, remember: "Add on top, subtract below" — the plus sign climbs up, the minus sign dives down.

7. Common Traps Examiners Set

1. Unit mismatch — 2 kg : 500 g is NOT 2 : 500.
2. "Increased in the ratio 3 : 5" means new value = 5/3 of old, not an increase of 5/3.
3. Ratio of A:B and B:C — you cannot write A:B:C until B is made equal in both.
4. Adding ratios directly — 1:2 + 2:3 is not 3:5 (unless compounding is asked, which means multiplying).
5. Saving-based income problems — savings = income − expenditure; set up two equations, don't guess.

🧠 Active Recall: A number is increased in the ratio 7 : 9. If the original number is 63, what is the new number? (Careful — read trap #2 again.)

8. Summary Table

📌 Review this chapter after 2 days. Redo the Active Recall prompts without looking at the solutions — if you can answer all of them cold, Ratio & Proportion is officially yours.