Ratio: Combining, Comparing and Types of Ratio
Ratio: Combining, Comparing and Types of Ratio
A ratio compares two quantities of the same kind by division. In NTPC the ratio questions that come again and again are built on four skills: merging two ratios into one chain, reading a ratio out of "3A = 4B = 8C" type equations, naming duplicate and sub-duplicate ratios, and picking the largest ratio from options. Each is a 20 to 30 second question once the method is automatic.
1. Ratio Basics
a : b = a/b, where a is the antecedent and b is the consequent (b ≠ 0)
- Both quantities must be in the SAME unit before forming a ratio: 45 minutes : 2 hours = 45 : 120 = 3 : 8, not 45 : 2.
- A ratio does not change when both terms are multiplied or divided by the same non-zero number: 12 : 18 = 2 : 3 (divide by the HCF 6).
- A ratio DOES change when the same number is added to both terms (unless the terms are equal). This is the basis of the "add x to each term" questions in the next topic.
- A ratio has no unit; it is a pure number.
✗ 45 min : 2 h = 45 : 2 | ✓ Convert first: 45 min : 120 min = 3 : 8
2. Combining Ratios (the N-method)
If A : B = a : b and B : C = c : d, then A : B : C = (a × c) : (b × c) : (b × d)
Write the two ratios one below the other with B in the same column. Multiply crosswise in an N-shape: the first term is a × c, the middle term is b × c and the last term is b × d. Then A : C = a × c : b × d directly.
| Given | Combined | Check |
|---|---|---|
| A : B = 4 : 5, B : C = 6 : 7 | A : B : C = 24 : 30 : 35 | 24 : 30 = 4 : 5 and 30 : 35 = 6 : 7 |
| A : B = 2 : 3, B : C = 3 : 4 | A : B : C = 2 : 3 : 4 (B already equal) | No multiplication needed when B matches |
| A : B = 1 : 2, B : C = 3 : 5, C : D = 4 : 7 | A : B : C : D = 12 : 24 : 40 : 70 = 6 : 12 : 20 : 35 | Multiply the chain term by term |
For four quantities: A : B : C : D = (a c e) : (b c e) : (b d e) : (b d f) where A : B = a : b, B : C = c : d, C : D = e : f.
3. Ratio From Equal Products, Fractions or Percentages
| Given | Ratio | Example |
|---|---|---|
| pA = qB = rC | A : B : C = 1/p : 1/q : 1/r = qr : pr : pq | 2A = 3B = 4C gives 6 : 4 : 3 |
| A/p = B/q = C/r | A : B : C = p : q : r | A/3 = B/4 = C/7 gives 3 : 4 : 7 |
| x% of A = y% of B | A : B = y : x (inverse) | 20% of A = 30% of B gives A : B = 3 : 2 |
| (m/n) of A = (p/q) of B | A : B = (p/q) : (m/n) | 2/3 of A = 4/5 of B gives A : B = 6 : 5 |
The equal-product case flips the coefficients: the variable with the BIGGEST multiplier is the SMALLEST quantity.
हिंदी नोट: यदि pA = qB = rC हो तो A : B : C = 1/p : 1/q : 1/r होता है, यानी जिसका गुणांक सबसे बड़ा है वह राशि सबसे छोटी होगी। दो अनुपात जोड़ने के लिए बीच वाली राशि (B) को दोनों में बराबर कीजिए।
4. Types of Ratio
| Type | Of a : b | Example with 4 : 9 |
|---|---|---|
| Duplicate | a² : b² | 16 : 81 |
| Triplicate | a³ : b³ | 64 : 729 |
| Sub-duplicate | √a : √b | 2 : 3 |
| Sub-triplicate | ∛a : ∛b | ∛4 : ∛9 |
| Inverse (reciprocal) | b : a, that is 1/a : 1/b | 9 : 4 |
| Compound of a : b and c : d | ac : bd | Compound of 4 : 9 and 3 : 2 is 12 : 18 = 2 : 3 |
| Ratio of greater inequality | a > b | Decreases when the same number is added to both terms |
| Ratio of lesser inequality | a < b | Increases when the same number is added to both terms |
✗ Sub-duplicate of 64 : 25 is 32 : 12.5 | ✓ Sub-duplicate means square roots: √64 : √25 = 8 : 5
5. Comparing Ratios
| Method | When to use | Example |
|---|---|---|
| Decimal conversion | Options with small numbers | 5 : 8 = 0.625, 7 : 11 = 0.636 |
| Cross multiplication | Two ratios at a time | 7 : 9 vs 13 : 17: 7 × 17 = 119 > 9 × 13 = 117, so 7 : 9 is larger |
| Gap from 1 | Ratios close to 1 | 19 : 24 has gap 5/24 = 0.208, smallest gap is largest ratio |
| Common consequent | Denominators share factors | 3 : 4, 5 : 6, 7 : 12 → 9/12, 10/12, 7/12 |
Exam Pointer: Verified NTPC patterns from this topic: combining two or three ratios (December 2020, January 2021 and the March 2026 Graduate CBT-1), ratio from equal products (August 2025 UG CBT-1, March 2026 Graduate CBT-1), sub-duplicate ratio (February and March 2021) and picking the greatest ratio (June 2025 CBT-1, October 2025 Graduate CBT-2, March 2026 Graduate CBT-1). Traps: forgetting to equalise the middle term, and treating "3A = 4B" as A : B = 3 : 4.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Combining two or more ratios
[PYQ: NTPC Graduate CBT-1 19-Mar-2026 Shift-3 | NTPC CBT-1 19-Jan-2021 Shift-2 | NTPC CBT-1 29-Dec-2020 Shift-1]
EXAM LEVEL
Q. If A : B = 4 : 5 and B : C = 6 : 7, find A : B : C and A : C.
B is 5 in the first ratio and 6 in the second, so make it 30 in both: multiply the first ratio by 6 and the second by 5. A : B : C = 24 : 30 : 35, and A : C = 24 : 35. N-method in one line: (4 × 6) : (5 × 6) : (5 × 7).
Answer: 24 : 30 : 35; A : C = 24 : 35
EXAMATLAS LEVEL
Q. If A : B = 2 : 3, B : C = 4 : 5 and C : D = 6 : 7, find A : B : C : D and A : D.
Chain the numerators and denominators: A : B : C : D = (2 × 4 × 6) : (3 × 4 × 6) : (3 × 5 × 6) : (3 × 5 × 7) = 48 : 72 : 90 : 105. Divide by 3: 16 : 24 : 30 : 35. Check each link: 16 : 24 = 2 : 3, 24 : 30 = 4 : 5, 30 : 35 = 6 : 7. So A : D = 16 : 35. Multiplying only 2 × 4 × 6 against 3 × 5 × 7 gives 48 : 105 = 16 : 35 directly for A : D.
Answer: 16 : 24 : 30 : 35; A : D = 16 : 35
Pattern 2: Ratio from equal products, fractions or percentages
[PYQ: NTPC Graduate CBT-1 25-Mar-2026 Shift-3 | NTPC UG CBT-1 12-Aug-2025 Shift-1]
EXAM LEVEL
Q. If 4A = 5B = 6C, find A : B : C.
A : B : C = 1/4 : 1/5 : 1/6. Multiply by LCM 60: 15 : 12 : 10. Check: 4 × 15 = 5 × 12 = 6 × 10 = 60.
Answer: 15 : 12 : 10
EXAMATLAS LEVEL
Q. ₹5,550 is divided among A, B and C so that 3/4 of A's share = 60% of B's share = 0.9 of C's share. Find B's share.
Let each equal k. Then A = 4k/3, B = 5k/3 (because 60% = 3/5) and C = 10k/9. Multiply by 9 to clear fractions: A : B : C = 12 : 15 : 10, total 37 parts. One part = 5,550 ÷ 37 = 150, so B = 15 × 150 = ₹2,250. Check: 3/4 × 1,800 = 0.6 × 2,250 = 0.9 × 1,500 = 1,350.
Answer: ₹2,250
Pattern 3: Duplicate, sub-duplicate, compound and inverse ratios
[PYQ: NTPC CBT-1 19-Mar-2021 Shift-1 | NTPC CBT-1 23-Feb-2021 Shift-1]
EXAM LEVEL
Q. Find the sub-duplicate ratio of 196 : 441.
Sub-duplicate means the ratio of square roots: √196 : √441 = 14 : 21 = 2 : 3.
Answer: 2 : 3
EXAMATLAS LEVEL
Q. Find the ratio compounded of the duplicate ratio of 2 : 3, the sub-duplicate ratio of 81 : 16 and the inverse ratio of 3 : 4.
Duplicate of 2 : 3 is 4 : 9. Sub-duplicate of 81 : 16 is 9 : 4. Inverse of 3 : 4 is 4 : 3. Compound = product of antecedents : product of consequents = (4 × 9 × 4) : (9 × 4 × 3) = 144 : 108 = 4 : 3. Cancelling 9 and 4 before multiplying gets you there in one step.
Answer: 4 : 3
Pattern 4: Comparing ratios: the greatest or smallest
[PYQ: NTPC Graduate CBT-2 13-Oct-2025 Shift-1 | NTPC Graduate CBT-1 16-Mar-2026 Shift-2 | NTPC Graduate CBT-1 13-Jun-2025 Shift-1]
EXAM LEVEL
Q. Which is the greatest: 3 : 5, 5 : 8, 7 : 11, 2 : 3?
As decimals: 0.600, 0.625, 0.636, 0.667. The greatest is 2 : 3.
Answer: 2 : 3
EXAMATLAS LEVEL
Q. Arrange 7 : 9, 13 : 17, 25 : 32 and 19 : 24 in descending order.
All are close to 1, so compare gaps from 1: 2/9 = 0.222, 4/17 = 0.235, 7/32 = 0.219, 5/24 = 0.208. Smaller gap means larger ratio: 19 : 24 (0.792) > 25 : 32 (0.781) > 7 : 9 (0.778) > 13 : 17 (0.765). The "bigger numbers, bigger ratio" shortcut fails here because the gaps differ, which is exactly why 13 : 17 ends up last.
Answer: 19 : 24 > 25 : 32 > 7 : 9 > 13 : 17
CBT-2 LEVEL
Pattern 5: Expressions from a ratio and componendo-dividendo
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. If a : b = 5 : 3, find (4a − 3b) : (2a + 5b).
Put a = 5 and b = 3 (any common multiple works because the expression is of degree one in both parts): (20 − 9) : (10 + 15) = 11 : 25.
Answer: 11 : 25
EXAMATLAS LEVEL
Q. If (3x + 5y)/(3x − 5y) = 7/3, find x : y.
Componendo-dividendo: if p/q = r/s then (p + q)/(p − q) = (r + s)/(r − s). Applying it, (6x)/(10y) = (7 + 3)/(7 − 3) = 10/4. So x/y = (10/4) × (10/6) = 25/6. Check: x = 25, y = 6 gives 105/45 = 7/3.
Answer: 25 : 6
60-Second Revision
- Same units first; divide by HCF for simplest form; adding the same number changes a ratio.
- Combine with the N-method: (a × c) : (b × c) : (b × d); chain all links for four terms.
- pA = qB = rC gives 1/p : 1/q : 1/r; x% of A = y% of B gives A : B = y : x.
- Duplicate a² : b², sub-duplicate √a : √b, compound ac : bd, inverse b : a.
- Compare by decimals, cross multiplication or gap from 1 (only equal gaps allow the size shortcut).
- CBT-2: substitute ratio values; componendo-dividendo turns (p + q)/(p − q) into a clean ratio.