Fractions: Types, Operations and Comparison
Fractions: Types, Operations and Comparison
Fractions sit inside half the NTPC Maths paper: simplification, percentage, ratio, time and work. As a standalone topic NTPC asks ordering of fractions, chained fraction simplification, fraction-of-a-number and numerator-denominator change questions. All four are tested below with real-exam numbers.
1. Types of Fractions
| Type | Meaning | Example |
|---|---|---|
| Proper | Numerator < denominator, value < 1 | 5/8 |
| Improper | Numerator ≥ denominator, value ≥ 1 | 11/4 |
| Mixed | Whole number + proper fraction | 2 3/4 = 11/4 |
| Like | Same denominator | 3/11, 7/11 |
| Unlike | Different denominators | 2/3, 5/7 |
| Unit | Numerator 1 | 1/9 |
| Equivalent | Same value | 3/4 = 9/12 = 75/100 |
| Complex (compound) | Fraction inside a fraction | 1/(1 + 1/3) |
| Decimal fraction | Denominator a power of 10 | 7/100 = 0.07 |
Mixed to improper: a b/c = (a × c + b)/c e.g. 3 2/5 = 17/5
Lowest form: divide numerator and denominator by their HCF
2. Operations
| Operation | Rule | Example |
|---|---|---|
| Add or subtract | Bring to the LCM of denominators | 5/6 − 3/8 = 20/24 − 9/24 = 11/24 |
| Multiply | Numerator × numerator, denominator × denominator; cancel first | 4/9 × 15/8 = 5/6 |
| Divide | Multiply by the reciprocal | 3/4 ÷ 9/8 = 3/4 × 8/9 = 2/3 |
| Continued fraction | Solve from the bottom upwards | 1/(1 + 1/(2 + 1/3)) = 7/10 |
| "of" | Means multiply; in BODMAS it is done before division | 2/3 of 3/4 = 1/2 |
| Telescoping sum | 1/(a × (a + d)) = (1/d) × (1/a − 1/(a + d)) | 1/(2 × 5) + 1/(5 × 8) = (1/3)(1/2 − 1/8) = 1/8 |
3. Comparing Fractions: Five Fast Methods
| Situation | Method | Example |
|---|---|---|
| Two fractions | Cross multiply: a/b > c/d if ad > bc | 7/9 vs 4/5: 35 > 36? No, so 4/5 is larger |
| Same numerator | Smaller denominator is larger | 5/7 > 5/9 |
| Numerator and denominator differ by the same amount | Proper fractions: bigger numbers give the bigger fraction (closer to 1); improper fractions: bigger numbers give the smaller one | 7/9 < 11/13 < 15/17, but 9/7 > 13/11 |
| Fractions close to 1 | Compare the gap 1 − fraction; smaller gap is larger fraction | 23/28 gap 5/28 |
| Several mixed fractions | Convert to decimals to 3 places using the table below | 5/8 = 0.625 |
✗ 13/16 vs 17/21: bigger numbers, so 17/21 is bigger | ✓ The difference rule works only when the gap (den − num) is equal; here gaps are 3 and 4, so check: 13 × 21 = 273 > 17 × 16 = 272, 13/16 is bigger
4. Fraction to Decimal and Percent Table
| Fraction | Decimal | Percent | Fraction | Decimal | Percent |
|---|---|---|---|---|---|
| 1/2 | 0.5 | 50% | 1/11 | 0.0909... | 9.09% |
| 1/3 | 0.333... | 33.33% | 1/12 | 0.0833... | 8.33% |
| 1/4 | 0.25 | 25% | 1/13 | 0.076923... | 7.69% |
| 1/5 | 0.2 | 20% | 1/14 | 0.0714285... | 7.14% |
| 1/6 | 0.1666... | 16.67% | 1/15 | 0.0666... | 6.67% |
| 1/7 | 0.142857... | 14.29% (exactly 14 2/7%) | 1/16 | 0.0625 | 6.25% |
| 1/8 | 0.125 | 12.5% | 1/20 | 0.05 | 5% |
| 1/9 | 0.111... | 11.11% | 1/25 | 0.04 | 4% |
| 1/10 | 0.1 | 10% | 3/8 | 0.375 | 37.5% |
Multiples follow directly: 5/8 = 5 × 12.5% = 62.5%, 7/12 = 7 × 8⅓% = 58⅓% (about 58.33%).
हिंदी नोट: उचित (proper) भिन्नों की तुलना में यदि अंश और हर का अंतर समान हो तो बड़ी संख्याओं वाली भिन्न बड़ी होती है; विषम (improper) भिन्नों में उल्टा होता है। 1 के निकट की भिन्नों में 1 से अंतर निकालिए, जिसका अंतर सबसे कम हो वही सबसे बड़ी भिन्न है।
Exam Pointer: In the 2020-21 cycle NTPC shifts asked fraction ordering, chained fraction sums, "two-fifth of seven-seventeenth of..." style fraction of a number, numerator up by x% and denominator down by y%, and sum-of-numerator-and-denominator word problems; the 2025-26 cycle kept the chained simplification, and a telescoping sum of unit fractions came on 15 March 2021.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Ordering fractions and picking the largest or smallest
[PYQ: NTPC CBT-1 28-Apr-2016 Shift-2 | NTPC 2020-21 cycle]
EXAM LEVEL
Q. Arrange 5/8, 7/12, 9/16 and 2/3 in ascending order.
From the table: 5/8 = 0.625, 7/12 = 0.583, 9/16 = 0.5625, 2/3 = 0.667. Ascending: 9/16, 7/12, 5/8, 2/3.
Answer: 9/16 < 7/12 < 5/8 < 2/3
EXAMATLAS LEVEL
Q. Which of the following is the largest: 13/16, 17/21, 23/28, 31/38?
All are close to 1, so compare the gaps from 1: 3/16 = 0.1875, 4/21 = 0.190, 5/28 = 0.179, 7/38 = 0.184. The smallest gap is 5/28, so 23/28 is the largest. The equal-gap rule cannot be used because the gaps 3, 4, 5, 7 differ, which is exactly the trap.
Answer: 23/28
Pattern 2: Chained addition, subtraction and continued fractions
[PYQ: NTPC 2020-21 cycle | NTPC 2025-26 cycle]
EXAM LEVEL
Q. Find the value of 5/6 − 3/8 + 7/12 − 1/4.
LCM of 6, 8, 12, 4 is 24. Convert: 20/24 − 9/24 + 14/24 − 6/24 = (20 − 9 + 14 − 6)/24 = 19/24.
Answer: 19/24
EXAMATLAS LEVEL
Q. Find the value of 2/3 + 1/(1 + 1/(2 + 1/3)) − 3/4.
Start at the bottom: 2 + 1/3 = 7/3, its reciprocal 3/7, then 1 + 3/7 = 10/7, reciprocal 7/10. Now 2/3 + 7/10 − 3/4 with LCM 60: 40/60 + 42/60 − 45/60 = 37/60.
Answer: 37/60
Pattern 3: Telescoping sums of unit fractions
[PYQ: NTPC CBT-1 15-Mar-2021 Shift-1]
EXAM LEVEL
Q. Find the value of 1/(2 × 5) + 1/(5 × 8) + 1/(8 × 11) + 1/(11 × 14).
Each denominator is a product of two numbers that differ by 3, so 1/(a(a + 3)) = (1/3)(1/a − 1/(a + 3)). Writing all four this way, the middle terms cancel and only the first and last survive: (1/3)(1/2 − 1/14) = (1/3)(6/14) = 1/7. Quick check with the shortcut n/(first × last) = 4/(2 × 14) = 1/7.
Answer: 1/7
EXAMATLAS LEVEL
Q. Find the value of 1/6 + 1/12 + 1/20 + 1/30 + ... + 1/132.
The trick is to see the denominators as consecutive products: 6 = 2 × 3, 12 = 3 × 4, 20 = 4 × 5, ..., 132 = 11 × 12. Here the gap is 1, so each term is 1/a − 1/(a + 1) and the sum telescopes to 1/2 − 1/12 = 5/12. Students who try an LCM of ten denominators run out of time.
Answer: 5/12
Pattern 4: Fraction of a fraction of a number
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. 3/4 of 2/5 of 5/9 of a number is 30. Find the number.
Multiply the fractions: 3/4 × 2/5 × 5/9 = 30/180 = 1/6. So 1/6 of the number is 30 and the number is 180.
Answer: 180
EXAMATLAS LEVEL
Q. 2/7 of a number exceeds 3/11 of the same number by 8. Find 5/8 of the number.
2/7 − 3/11 = (22 − 21)/77 = 1/77. So 1/77 of the number is 8 and the number is 616. Then 5/8 × 616 = 385. Stopping at 616 is the trap option.
Answer: 385
Pattern 5: Numerator and denominator changed by a percent or a constant
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. If the numerator of a fraction is increased by 20% and the denominator is decreased by 10%, the new fraction is 16/21. Find the original fraction.
New fraction = original × 1.2/0.9 = original × 4/3. So original = 16/21 × 3/4 = 4/7.
Answer: 4/7
EXAMATLAS LEVEL
Q. If 3 is added to both the numerator and the denominator of a fraction, it becomes 5/6. If 2 is subtracted from both, it becomes 5/7. Find the fraction.
Let it be n/d. First condition: 6(n + 3) = 5(d + 3), so 6n − 5d = −3. Second: 7(n − 2) = 5(d − 2), so 7n − 5d = 4. Subtracting the first from the second gives n = 7, then 5d = 42 + 3 = 45 and d = 9. Check: 10/12 = 5/6 and 5/7.
Answer: 7/9
Pattern 6: Sum of numerator and denominator, and fraction plus reciprocal
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. The sum of the numerator and the denominator of a fraction is 13. If the denominator is increased by 3, the fraction becomes 1/3. Find the fraction.
n + d = 13 and 3n = d + 3. Substitute d = 13 − n: 3n = 16 − n, so n = 4 and d = 9.
Answer: 4/9
EXAMATLAS LEVEL
Q. The denominator of a fraction is 1 more than twice its numerator. The sum of the fraction and its reciprocal is 2 16/21. Find the fraction.
2 16/21 = 58/21, so the denominator 21 suggests the fraction's terms multiply to 21. With d = 2n + 1, try n = 3: d = 7 and 3/7 + 7/3 = (9 + 49)/21 = 58/21. Option-testing with the product clue is faster than solving the quadratic 11n² − 26n − 21 = 0, whose positive root is also 3.
Answer: 3/7
Pattern 7: Fraction word problems (salary, tank, distribution)
[PYQ: NTPC CBT-1 28-Dec-2020 Shift-1]
EXAM LEVEL
Q. A man spends 1/3 of his salary on rent and 1/4 of the remainder on food. He is left with ₹9,000. Find his salary.
After rent 2/3 remains; food takes 1/4 of that, leaving 3/4 × 2/3 = 1/2 of the salary. So half the salary is ₹9,000 and the salary is ₹18,000. Adding 1/3 + 1/4 directly is the classic error.
Answer: ₹18,000
EXAMATLAS LEVEL
Q. A tank is 3/5 full. When 40 litres are drawn out it becomes 7/15 full. How many litres must now be added to make it 5/6 full?
3/5 − 7/15 = 9/15 − 7/15 = 2/15 of the tank is 40 L, so the capacity is 300 L. Now it holds 7/15 × 300 = 140 L and 5/6 full means 250 L. Add 250 − 140 = 110 L.
Answer: 110 litres
60-Second Revision
- Add or subtract via LCM; multiply after cancelling; divide by multiplying with the reciprocal.
- Continued fractions: solve from the bottom up; telescoping: 1/(a(a + d)) = (1/d)(1/a − 1/(a + d)).
- Compare: cross multiply for two; gap-from-1 for fractions near 1; equal-gap rule only when gaps are equal.
- "Remainder" fractions multiply (1/4 of remainder after 1/3 leaves 3/4 × 2/3).
- Numerator up p% and denominator down q%: multiply by (100 + p)/(100 − q).
- Learn the 1/2 to 1/25 decimal-percent table; it powers percentage and ratio too.