Fractions: Operations, Rules & Practice

Fractions are simply another way of representing division, proportions, and parts of a whole. In aptitude tests, however, fractions are rarely tested only as basic arithmetic. You may need to compare them quickly, simplify complex expressions, convert between forms, or use them inside word problems.

Basic Fraction Rules

  • A fraction such as 524, in which the numerator is less than the denominator, is a proper fraction. Its value is less than 1.
  • A fraction such as 3012, in which the numerator is greater than the denominator, is an improper fraction. Its value is greater than 1.
  • A fraction such as 88, where numerator and denominator are equal, is also improper, but its value is exactly 1.

Every fraction can be written as a decimal (or whole number) by dividing the numerator by the denominator. For instance, 310 = 0.3, 4816 = 3, and 1008 = 12.5.

Contents
  1. Understanding fractions
  2. Equivalent fractions
  3. Greatest common factor (GCF)
  4. Simplifying fractions
  5. Least common multiple (LCM)
  6. Adding and subtracting fractions
  7. Multiplying fractions
  8. Dividing fractions
  9. Comparing fractions
  10. Improper fractions and mixed numbers
  11. Complex fractions
  12. Fractions, decimals and percentages
  13. Fractions in aptitude problems
  14. Aptitude-test shortcuts
  15. Practice questions

1. Understanding Fractions

A fraction has two parts: the numerator (top), the number being divided, and the denominator (bottom), the number doing the dividing. The denominator can never be zero. A fraction is simply a division: 35 = 3 ÷ 5.

Understanding the concept

One fraction can carry four meanings: a part of a whole (3 of 5 equal slices), a ratio (3 to 5), a division (3 ÷ 5) and a proportion (60% of something). Aptitude questions switch between these without warning, so ask yourself each time: what does this fraction represent?

Example 1Convert 35 and 74 to decimals.

Divide the numerator by the denominator.

35 = 3 ÷ 5 = 0.6    74 = 7 ÷ 4 = 1.75

Since 7 is larger than 4, the second fraction is improper and its value exceeds 1.

Example 2A student scores 18 marks out of 24. Express this as a fraction in lowest form and as a percentage.
1824 = 34 = 0.75 = 75%

The fraction is a part of a whole (marks gained out of total marks), and dividing 3 by 4 gives the decimal that converts directly to a percentage.

Example 3IBA levelIf ab = 37 and a + b = 50, find b − a.

Write a = 3k and b = 7k. Then 10k = 50, so k = 5.

a = 15,   b = 35,   b − a = 20
Example 4IBA levelWhat fraction of 3 hours 20 minutes is 50 minutes?

Use the same unit: 3 h 20 min = 200 minutes.

50200 = 14

2. Equivalent Fractions

Multiplying or dividing the numerator and denominator by the same non-zero number does not change the value: ab = a×kb×k. For example 34 = 68 = 1520.

Understanding the concept

Equivalent fractions describe the same amount cut into different numbers of pieces. They are the tool for creating common denominators and for simplifying. Do not automatically convert everything to decimals; building an equivalent fraction is often faster and exact.

Example 1Convert 56 into a fraction with denominator 24.

Since 6 × 4 = 24, multiply both numerator and denominator by 4.

56 = 5×46×4 = 2024
Example 2Find x if 79 = x63.

The denominator went from 9 to 63, a factor of 63 ÷ 9 = 7. Apply the same factor to the numerator.

x = 7 × 7 = 49   so   79 = 4963

Check with cross multiplication: 7 × 63 = 441 and 9 × 49 = 441. ✓

Example 3IBA levelFind the fraction equivalent to 38 whose numerator and denominator add up to 66.

Write the fraction as 3k/8k. Then 3k + 8k = 66, so k = 6.

3×68×6 = 1848
Example 4IBA levelA fraction equivalent to 59 has a denominator 28 more than its numerator. Find it.

Let it be 5k/9k. The difference is 9k − 5k = 4k = 28, so k = 7.

3563

3. Greatest Common Factor (GCF)

The greatest common factor (also called GCD or HCF) of two or more whole numbers is the largest number that divides all of them exactly. For 12 and 18 the common factors are 1, 2, 3 and 6, so the GCF is 6.

The logic

A factor is a number that fits into another exactly, with no remainder. The GCF is the biggest “building block” shared by every number. This is why it is the key to simplifying fractions (divide top and bottom by it) and to problems about cutting or grouping things into equal, largest-possible pieces. Two numbers whose GCF is 1 are called co-prime.

Methods of calculation

  • Listing factors: write all factors of each number and pick the largest common one. Fine for small numbers.
  • Prime factorisation: break each number into primes. Take every prime common to all numbers, each with its smallest power, and multiply.
  • Euclid’s (division) method: divide the larger number by the smaller, then divide the divisor by the remainder, and repeat until the remainder is 0. The last divisor is the GCF. Best for large numbers.
Useful facts: GCF × LCM = product of the two numbers (for two numbers). The GCF can never exceed the smallest number. If one number divides another, the smaller is the GCF.
Example 1Find the GCF of 36 and 60 by prime factorisation.
36 = 2² × 3²    60 = 2² × 3 × 5

Common primes are 2 and 3. Take the smallest powers: 2² and 3¹.

GCF = 2² × 3 = 12
Example 2Find the GCF of 84, 126 and 210.
84 = 2² × 3 × 7   126 = 2 × 3² × 7   210 = 2 × 3 × 5 × 7

The primes common to all three are 2, 3 and 7, each with smallest power 1.

GCF = 2 × 3 × 7 = 42
Example 3IBA levelUse Euclid’s method to find the GCF of 1071 and 462.
1071 = 2 × 462 + 147
462 = 3 × 147 + 21
147 = 7 × 21 + 0

The remainder reaches 0 when the divisor is 21, so the GCF is 21. No factorisation was needed.

Example 4IBA levelRibbons of length 144 cm, 180 cm and 252 cm are cut into equal pieces, as long as possible, with none left over. Find the piece length and the total number of pieces.
144 = 2⁴ × 3²   180 = 2² × 3² × 5   252 = 2² × 3² × 7
GCF = 2² × 3² = 36 cm

“As long as possible” signals the GCF. Pieces: 144 ÷ 36 = 4, 180 ÷ 36 = 5, 252 ÷ 36 = 7.

Total pieces = 4 + 5 + 7 = 16

4. Simplifying Fractions

A fraction is in lowest terms when the numerator and denominator share no common factor greater than 1. To simplify, find the greatest common factor (GCF) and divide both parts by it. Example: 1824 = 34 (GCF 6).

Understanding the concept

Simplifying early keeps numbers small and reduces mistakes later. When multiplying fractions, you can cancel a factor from any numerator with any denominator before multiplying, and look for cancellation before doing any arithmetic.

Example 1Simplify 84126.

Both numbers are divisible by 42 (84 = 2 × 42 and 126 = 3 × 42).

84126 = 23
Example 2Evaluate 1528 × 1425 by cancelling first.

15 and 25 share a factor of 5; 14 and 28 share a factor of 14.

1528 × 1425 = 32 × 15 = 310

Multiplying first would give 210/700, a harder fraction to reduce.

Example 3IBA levelSimplify 2³ × 3² × 52² × 3 × 5².

Cancel the common prime powers: 2³ ÷ 2² = 2, 3² ÷ 3 = 3, 5 ÷ 5² leaves 5 in the denominator.

2 × 35 = 65
Example 4IBA levelSimplify 10011430.

Factorise: 1001 = 7 × 11 × 13 and 1430 = 2 × 5 × 11 × 13.

7 × 11 × 132 × 5 × 11 × 13 = 710

5. Least Common Multiple (LCM)

The least common multiple of two or more whole numbers is the smallest positive number that is a multiple of all of them. Multiples of 4 are 4, 8, 12, 16, 20, 24 …; multiples of 6 are 6, 12, 18, 24 …; so the LCM of 4 and 6 is 12.

The logic

Multiples keep growing, so the lists of two numbers eventually meet. The LCM is the first meeting point. It is the tool for events that repeat in cycles (bells, traffic lights, buses) and for creating the least common denominator when adding or subtracting fractions.

Methods of calculation

  • Listing multiples: write multiples of each number until the first common one appears.
  • Prime factorisation: break each number into primes. Take every prime that appears in any number, each with its highest power, and multiply.
  • Division method: divide all numbers by a common prime at the same time, bring down numbers that do not divide, and repeat. Multiply all the divisors and the final remaining numbers.
  • Formula: LCM = (a × b) ÷ GCF for two numbers.
Contrast with GCF: for the GCF use the smallest powers of common primes; for the LCM use the highest powers of all primes. A number that leaves the same remainder r when divided by several numbers has the form LCM × k + r.
Example 1Find the LCM of 12 and 18.
12 = 2² × 3    18 = 2 × 3²

Highest powers: 2² and 3².

LCM = 2² × 3² = 36
Example 2Find the LCM of 8, 12 and 20 using the division method.
2 | 8, 12, 20 → 4, 6, 10
2 | 4, 6, 10 → 2, 3, 5

Now 2, 3 and 5 share no common prime, so stop.

LCM = 2 × 2 × 2 × 3 × 5 = 120
Example 3IBA levelTwo numbers have GCF 6 and LCM 180. If one number is 36, find the other.

Use GCF × LCM = product of the two numbers.

6 × 180 = 36 × N
N = 1080 ÷ 36 = 30

Check: GCF(36, 30) = 6 and LCM = 180. ✓

Example 4IBA levelThree bells ring every 12, 15 and 20 minutes. They ring together at 9:00 a.m. When will they next ring together?
12 = 2² × 3   15 = 3 × 5   20 = 2² × 5
LCM = 2² × 3 × 5 = 60 minutes

“Together again” signals the LCM. 60 minutes after 9:00 a.m. is 10:00 a.m.

6. Adding and Subtracting Fractions

Fractions can be added or subtracted directly only when they have the same denominator: 37 + 27 = 57. Otherwise, first rewrite them with a common denominator, preferably the least common denominator (LCD).

Never add the denominators: 23 + 58 is not 711. The denominator shows the size of the parts, which must match before numerators can be combined.

Understanding the concept

You cannot add thirds and eighths directly, just as you cannot add centimetres and inches. Converting both to twenty-fourths gives pieces of equal size that can be counted together.

Example 1Evaluate 23 + 58.

The LCD of 3 and 8 is 24.

23 = 1624    58 = 1524
1624 + 1524 = 3124 = 1724
Example 2Evaluate 712 − 38.

The LCD of 12 and 8 is 24.

1424 − 924 = 524

Subtract only the numerators; the common denominator stays unchanged.

Example 3IBA levelEvaluate 11×2 + 12×3 + 13×4 + 14×5.

Use 1n(n+1) = 1n − 1n+1. The middle terms cancel (telescoping).

(1 − 12) + (12 − 13) + (13 − 14) + (14 − 15) = 1 − 15 = 45
Example 4IBA levelPipe A fills a tank in 6 hours and pipe B in 4 hours. What fraction is filled in one hour if both work, and how long to fill the tank?
16 + 14 = 212 + 312 = 512

Time = 1 ÷ 512 = 125 hours = 225 hours (2 h 24 min).

7. Multiplying Fractions

Multiply the numerators, multiply the denominators, then simplify: ab × cd = acbd.

Understanding the concept

“Of” means multiply: 12 of 13 is 16. No common denominator is needed. The aptitude strategy is cancel first, multiply second, which keeps large-number problems small. Whole numbers and mixed numbers should first be written as fractions.

Example 1Evaluate 49 × 1516.

Cancel 4 with 16 (÷ 4) and 15 with 9 (÷ 3).

13 × 54 = 512
Example 2Evaluate 312 × 27.

Convert the mixed number first: 3½ = 7/2.

72 × 27 = 1

The 7s and the 2s cancel completely.

Example 3IBA levelEvaluate (1 − 12)(1 − 13)(1 − 14) … (1 − 110).

Each bracket is a fraction of the form (k − 1)/k.

12 × 23 × 34 × … × 910

Every numerator cancels the previous denominator, leaving 110.

Example 4IBA levelFind 23 of 34 of 56 of 72.
23 × 34 = 12   →   12 × 56 = 512

512 × 72 = 30.

8. Dividing Fractions

Dividing by a fraction equals multiplying by its reciprocal: ab ÷ cd = ab × dc. Only the divisor (the second fraction) is inverted.

Understanding the concept

“How many 14s fit in 3?” is 3 ÷ 14 = 12. Dividing by a fraction less than 1 gives a bigger answer, which is a quick check on your result.

Example 1Evaluate 56 ÷ 109.

Invert the second fraction and cancel.

56 × 910 = 12 × 32 = 34
Example 2Evaluate 38 ÷ 916.
38 × 169 = 11 × 23 = 23

Cancelling 3 with 9 and 8 with 16 gives 1 × 2 over 1 × 3.

Example 3IBA levelHow many packets of 38 kg can be made from 412 kg of sugar?
412 = 92   →   92 ÷ 38 = 92 × 83 = 12

12 packets can be made.

Example 4IBA levelIf 57 ÷ x = 1021, find x.

Rearrange: x = 57 ÷ 1021.

x = 57 × 2110 = 11 × 32 = 32

9. Comparing Fractions

Aptitude tests often ask which fraction is larger without needing full calculation.

  • Same denominator: the larger numerator wins, e.g. 712 > 512.
  • Same numerator: the smaller denominator wins, e.g. 57 > 59.
  • Cross multiplication: for ab vs cd, compare a × d with c × b; the side with the larger product has the larger fraction.
  • Benchmarking: compare with 0, 14, 12 or 1. For instance 49100 < 12 < 51100.

Understanding the concept

Choose the fastest tool: benchmark first, cross multiply if close, convert to decimals as a last resort.

Example 1Which is larger: 712 or 59?

Cross multiply.

7 × 9 = 63    5 × 12 = 60

63 > 60, so 712 > 59.

Example 2Arrange 23, 35, 58 from least to greatest.

Convert to decimals (or compare using benchmarks).

23 ≈ 0.667   35 = 0.6   58 = 0.625

Order: 35, 58, 23.

Example 3IBA levelWhich is the greatest: 1115, 710, 1725, 34?

Convert to decimals: 0.733, 0.70, 0.68, 0.75.

The greatest is 34.

Example 4IBA levelWithout full calculation, which is larger: 123124 or 124125?

Write each as 1 minus a small gap: 123124 = 1 − 1124 and 124125 = 1 − 1125.

1125 is the smaller gap, so 124125 is larger.

10. Improper Fractions and Mixed Numbers

An improper fraction has a numerator greater than or equal to its denominator. To convert to a mixed number, divide: the quotient is the whole part and the remainder is the new numerator. To convert back, use abc = ac + bc (whole × denominator + numerator).

Understanding the concept

A mixed number is a whole number plus a fraction, so 3⅖ means 3 + 25. Mixed numbers are easiest to read, but improper fractions are what you should use for multiplication and division.

Example 1Convert 175 to a mixed number.

17 = 5(3) + 2, so the whole part is 3 and the remainder is 2.

175 = 325
Example 2Convert 437 to an improper fraction.
437 = 4×7 + 37 = 317
Example 3IBA levelEvaluate 312 + 234 − 156.
72 + 114 − 116 = 4212 + 3312 − 2212 = 5312 = 4512
Example 4IBA levelAli walks 214 km in the morning and 123 km in the evening. How much farther must he walk to complete 6 km?
94 + 53 = 2712 + 2012 = 4712
6 − 4712 = 7212 − 4712 = 2512 = 2112 km

11. Complex Fractions

A complex fraction has one or more fractions in its numerator or denominator. The safest method: simplify the numerator, simplify the denominator, then divide the two results.

Understanding the concept

The main fraction bar means divide. So a complex fraction is a fraction ÷ fraction problem in disguise, and the reciprocal rule applies to it.

Example 1Simplify 23 + 1334.

Numerator: 23 + 13 = 1.

1 ÷ 34 = 1 × 43 = 43
Example 2Simplify 23 + 1654 − 12.

Numerator: 46 + 16 = 56. Denominator: 54 − 24 = 34.

56 ÷ 34 = 56 × 43 = 2018 = 109
Example 3IBA levelSimplify 11 + 11 + 12.

Work from the inside out. 1 + 12 = 32, so the inner fraction is 132 = 23.

1 + 23 = 53   →   153 = 35
Example 4IBA levelSimplify 13 + 1413 − 14.

Numerator: 412 + 312 = 712. Denominator: 412 − 312 = 112.

712 ÷ 112 = 7

12. Fractions, Decimals and Percentages

One quantity has three forms: fraction ↔ decimal ↔ percentage. For example 34 = 0.75 = 75%. Memorise the common ones:

FractionDecimalPercentage
120.550%
140.2525%
340.7575%
150.220%
180.12512.5%
130.333…33.3%
230.666…66.7%

Understanding the concept

Percent means “out of 100”, so a percentage is a fraction with denominator 100. These conversions save substantial time in multiple-choice questions.

Example 1Express 38 as a decimal and a percentage.
38 = 3 × 0.125 = 0.375 = 37.5%

Using 18 = 0.125 is faster than long division.

Example 2Express 62.5% as a fraction in lowest terms.
62.5% = 62.5100 = 6251000 = 58

Since 12.5% = 18, then 62.5% = 5 × 12.5% = 58.

Example 3IBA levelWhich is greater: 58 or 62%?

58 = 0.625 = 62.5%, which is larger than 62%.

Example 4IBA levelA price rises from Rs. 40 to Rs. 50. Express the rise as a fraction and a percentage of the original price.

Rise = 50 − 40 = 10. Always compare to the original.

1040 = 14 = 25%

13. Fractions in Aptitude Problems

Fractions usually appear inside word problems. Always ask: “What does the fraction represent?” It may be a portion of a quantity, a budget, time, distance, a population, a group, a probability or a rate. Understanding the context matters more than the arithmetic.

Understanding the concept

The whole is always 1. If parts are spent or used, add those fractions, then subtract the total from 1 to get what remains. Be careful: “of the remainder” means a fraction of what is left, not of the original.

Example 1A company spends 35 of its budget on operations and 14 on marketing. What fraction remains?
35 + 14 = 1220 + 520 = 1720
1 − 1720 = 320

The remaining portion is 320 of the budget.

Example 2A tank is 25 full. After adding water equal to 13 of its capacity, how full is it and what fraction is still empty?
25 + 13 = 615 + 515 = 1115
1 − 1115 = 415

The tank is 1115 full and 415 is empty.

Example 3IBA levelA man spends 13 of his money, then 14 of the remainder. If Rs. 1,200 is left, what was the original amount?

After the first spending, 23 remains. He then spends 14 of 23 = 16.

Left = 23 − 16 = 46 − 16 = 12

Half of the money is Rs. 1,200, so the original amount is Rs. 2,400.

Example 4IBA levelA can finish a job in 12 days and B in 18 days. A works alone for 3 days, then B joins him. How many more days to finish?

A completes 312 = 14 in 3 days, leaving 34.

Combined rate = 112 + 118 = 336 + 236 = 536 per day
Days = 34 ÷ 536 = 34 × 365 = 275 = 5.4

It takes 5.4 more days.

Aptitude-Test Shortcuts

  • Simplify early: reduce fractions before lengthy calculations.
  • Cancel before multiplying: turns large-number products into small ones.
  • Don’t add denominators: find a common denominator first.
  • Division means reciprocal: invert the divisor, then multiply.
  • Compare intelligently: use cross multiplication, benchmarks or decimals, whichever is fastest.
  • Recognise common equivalents: fraction, decimal and percentage forms save time.
  • Don’t calculate unnecessarily: in MCQs, estimation or comparison may be enough.

Practice Questions: Test Yourself

Choose an answer to see instantly whether it is right or wrong. If you pick a wrong option, the correct one is highlighted. Then press Show Solution to review the steps.

1. Which of the following is the least fraction?
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Correct answer: D. 58

Convert to decimals: 58 = 0.625, 711 ≈ 0.636, 1320 = 0.65, 23 ≈ 0.667.

The smallest decimal is 0.625, so the least fraction is 58.

2. Which of the following is NOT equivalent to 1830?
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Correct answer: A. 47

Simplify 1830 by dividing by 6: 35.

610 = 35 and 915 = 35.

47 cannot be reduced or scaled to 35 (cross products: 4 × 5 = 20 but 3 × 7 = 21), so it is not equivalent.

3. What is the value of 7!9!, where n! is the product of all integers from 1 to n?
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Correct answer: A. 172

Write 9! = 9 × 8 × 7!.

7!9×8×7! = 172 after cancelling 7!.

4. What is 34 + 56?
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Correct answer: A. 1912

The LCD of 4 and 6 is 12.

912 + 1012 = 1912

5. Evaluate 512 × 920.
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Correct answer: B. 316

Cancel 5 with 20 (÷ 5) and 9 with 12 (÷ 3).

14 × 34 = 316

6. Evaluate 79 ÷ 1427.
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Correct answer: D. 32

Invert the divisor: 79 × 2714.

Cancel 7 with 14 and 9 with 27: 11 × 32 = 32.

7. Which of the following is less than 59?
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Correct answer: C. 611

59 ≈ 0.556.

611 ≈ 0.545 is smaller. The others are 47 ≈ 0.571, 58 = 0.625 and 35 = 0.6, all larger.

Cross-check: 6 × 9 = 54 < 5 × 11 = 55, so 611 < 59.

8. Express 523 as an improper fraction.
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Correct answer: B. 173

Whole × denominator + numerator: 5 × 3 + 2 = 17.

523 = 173

9. Simplify 12 + 1413.
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Correct answer: D. 94

Numerator: 12 + 14 = 34.

34 ÷ 13 = 34 × 3 = 94

10. Sara spends 25 of her salary on rent and 14 on food. If Rs. 14,000 remains, what is her salary?
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Correct answer: B. Rs. 40,000

Spent: 25 + 14 = 820 + 520 = 1320.

Remaining: 1 − 1320 = 720.

720 of salary = 14,000, so salary = 14,000 × 207 = Rs. 40,000.

11. Evaluate (34 − 16) ÷ 712.
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Correct answer: D. 1

34 − 16 = 912 − 212 = 712.

712 ÷ 712 = 1 (any non-zero number divided by itself).

12. If 23 of a number is 18 more than 12 of the number, what is the number?
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Correct answer: C. 108

Let the number be x: 2x3 = x2 + 18.

2x3 − x2 = 18, so 4x − 3x6 = x6 = 18.

x = 18 × 6 = 108.