Basic MathMath

Fractions: Definition, Types & Examples Guide

2 years ago
Last Update on October 17, 2026
12 mins
Learn fractions with definitions, types, properties, and examples. Master basics with clear explanations and practical math problems.

Complete Guide to Fractions

1. Introduction to Fractions

A fraction a/b represents division of a by b, where b is not zero. It can describe equal parts of a whole, a ratio, or a number on a number line. The numerator is the top number and the denominator is the bottom number. The proper, improper, and mixed-number examples below use non-negative numerators and positive denominators.

Basic Definition:

NumeratorDenominator

The numerator represents how many parts we have.

The denominator represents the total number of equal parts that make up a whole.

For example, in the fraction 34, the numerator is 3 and the denominator is 4. This means we have 3 out of 4 equal parts.

3/4

1/2

Fractions are used to represent parts of a whole, to compare quantities, and to perform calculations in many real-world scenarios like cooking, measurement, time, and probability.

2. Types of Fractions

Proper Fractions:

A proper fraction has a numerator that is less than its denominator. The value of a proper fraction is always less than 1.

Examples of proper fractions:

12

One-half

34

Three-fourths

58

Five-eighths

Improper Fractions:

An improper fraction has a numerator that is greater than or equal to its denominator. The value of an improper fraction is always greater than or equal to 1.

Examples of improper fractions:

53

Five-thirds

74

Seven-fourths

118

Eleven-eighths

Mixed Numbers:

A mixed number consists of a whole number and a proper fraction. It represents a value that is greater than 1.

Examples of mixed numbers:

112

One and a half

234

Two and three-fourths

358

Three and five-eighths

Unit Fractions:

A unit fraction has a numerator of 1.

Examples of unit fractions:

12

One-half

14

One-fourth

110

One-tenth

Like and Unlike Fractions:

Like fractions have the same denominator.

Unlike fractions have different denominators.

Examples of like fractions:

38 and 58

Examples of unlike fractions:

23 and 35

Equivalent Fractions:

Equivalent fractions represent the same value even though they have different numerators and denominators.

Examples of equivalent fractions:

12 = 24 = 36 = 48

3. Equivalent Fractions

Creating Equivalent Fractions:

To create an equivalent fraction, multiply or divide both the numerator and denominator by the same non-zero number.

To find equivalent fractions for 23:

  • Multiply by 2: 2 × 23 × 2 = 46
  • Multiply by 3: 2 × 33 × 3 = 69
  • Multiply by 4: 2 × 43 × 4 = 812

Simplifying Fractions:

To simplify a fraction to its lowest terms, divide both the numerator and denominator by their greatest common divisor (GCD).

To simplify 1218:

  1. Find the GCD of 12 and 18: GCD(12, 18) = 6
  2. Divide both numbers by the GCD: 12 ÷ 618 ÷ 6 = 23

Therefore, 1218 = 23 in lowest terms.

Finding the Greatest Common Divisor (GCD):

There are several methods to find the GCD:

Method 1: Listing Factors

To find the GCD of 24 and 36:

  1. List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  2. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. Identify the common factors: 1, 2, 3, 4, 6, 12
  4. The greatest common factor is 12

Method 2: Prime Factorization

To find the GCD of 24 and 36:

  1. Prime factorization of 24: 23 × 3
  2. Prime factorization of 36: 22 × 32
  3. Take the common prime factors with the lowest exponents: 22 × 3
  4. Calculate: 22 × 3 = 4 × 3 = 12

Method 3: Euclidean Algorithm

To find the GCD of 24 and 36:

  1. Divide the larger number by the smaller: 36 ÷ 24 = 1 remainder 12
  2. Replace the larger number with the smaller, and the smaller with the remainder: GCD(24, 12)
  3. Repeat: 24 ÷ 12 = 2 remainder 0
  4. When the remainder is 0, the divisor is the GCD: GCD(24, 36) = 12

4. Comparing Fractions

Method 1: Using Common Denominators

To compare fractions with different denominators, convert them to equivalent fractions with a common denominator.

Compare 23 and 35:

  1. Find the least common multiple (LCM) of the denominators: LCM(3, 5) = 15
  2. Convert the fractions to equivalent fractions with denominator 15:
    • 23 = 2 × 53 × 5 = 1015
    • 35 = 3 × 35 × 3 = 915
  3. Compare the numerators: 10 > 9
  4. Therefore, 23 > 35

Method 2: Cross Multiplication

With positive denominators, another method to compare fractions is to cross multiply. Normalize negative denominators first; otherwise the inequality direction may change.

Compare 23 and 35:

  1. Cross multiply:
    • 2 × 5 = 10
    • 3 × 3 = 9
  2. Compare the products: 10 > 9
  3. Therefore, 23 > 35

Method 3: Converting to Decimals

Convert the fractions to decimals and then compare their values. Rounded decimals can conceal a small difference; use exact fractions or enough digits to establish the order. In this example, 0.666… is clearly greater than 0.6.

Compare 23 and 35:

  1. 23 = 2 ÷ 3 ≈ 0.667
  2. 35 = 3 ÷ 5 = 0.6
  3. Compare: 0.667 > 0.6
  4. Therefore, 23 > 35

Method 4: Using a Number Line

Locate the fractions on a number line to compare their values.

Compare 23 and 35 on a number line:

0
1/5
1/4
1/3
1/2
3/5
2/3
3/4
4/5
1

On the number line, we can see that 23 (≈ 0.667) is to the right of 35 (= 0.6).

Therefore, 23 > 35

Method 5: Using Benchmark Fractions

Compare fractions to common benchmark fractions like 0, 1/2, and 1.

Compare 38 and 59 using benchmarks:

  1. Compare each fraction to 1/2:
    • 38 < 48 = 12 (since 3 < 4)
    • 59 > 4.59 = 12 (since 5 > 4.5)
  2. Since 38 < 12 and 59 > 12, we can conclude that 38 < 59

5. Converting Fractions

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction to a mixed number:

  1. Divide the numerator by the denominator
  2. The quotient becomes the whole number part
  3. The remainder becomes the numerator of the fractional part
  4. The denominator remains the same

Convert 174 to a mixed number:

  1. Divide: 17 ÷ 4 = 4 remainder 1
  2. Whole number part: 4
  3. Fractional part: 14
  4. Mixed number: 414

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction:

  1. Multiply the whole number by the denominator
  2. Add the numerator to the product
  3. Use this sum as the new numerator over the original denominator

Convert 325 to an improper fraction:

  1. Multiply the whole number by the denominator: 3 × 5 = 15
  2. Add the numerator: 15 + 2 = 17
  3. Use this sum as the new numerator: 175

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

Convert 34 to a decimal:

3 ÷ 4 = 0.75

Convert 23 to a decimal:

2 ÷ 3 = 0.666... = 0.6̅ (the bar indicates that the digit repeats indefinitely)

Converting Decimals to Fractions

To convert a decimal to a fraction:

  1. For terminating decimals:
    • Count the number of decimal places
    • Multiply by the appropriate power of 10 to remove the decimal point
    • Write as a fraction and simplify if possible
  2. For repeating decimals:
    • Write as an equation involving variables
    • Manipulate the equation to isolate the repeating part
    • Solve for the variable to find the fraction

Convert 0.75 to a fraction:

  1. There are 2 decimal places, so multiply by 100: 0.75 × 100 = 75
  2. Write as a fraction: 75100
  3. Simplify: 75100 = 34

Convert 0.333... to a fraction:

  1. Let x = 0.333...
  2. Multiply by 10: 10x = 3.333...
  3. Subtract the original equation: 10x - x = 3.333... - 0.333...
  4. Simplify: 9x = 3
  5. Solve for x: x = 3/9 = 1/3

Converting Fractions to Percentages

To convert a fraction to a percentage:

  1. Convert the fraction to a decimal
  2. Multiply by 100
  3. Add the % symbol

Convert 34 to a percentage:

  1. Convert to a decimal: 3 ÷ 4 = 0.75
  2. Multiply by 100: 0.75 × 100 = 75
  3. Add the % symbol: 75%

Converting Percentages to Fractions

To convert a percentage to a fraction:

  1. Remove the % symbol and divide by 100
  2. Simplify the fraction if possible

Convert 25% to a fraction:

  1. Remove the % symbol and divide by 100: 25100
  2. Simplify: 25100 = 14

6. Operations with Fractions

Adding Fractions

Adding Fractions with Like Denominators

To add fractions with the same denominator, add the numerators and keep the denominator the same.

Add 38 + 28:

  1. 38 + 28 = 3 + 28 = 58

Adding Fractions with Unlike Denominators

To add fractions with different denominators:

  1. Find the least common multiple (LCM) of the denominators
  2. Convert each fraction to an equivalent fraction with the LCM as the denominator
  3. Add the numerators and keep the common denominator
  4. Simplify the result if possible

Add 23 + 14:

  1. Find the LCM of 3 and 4: LCM(3, 4) = 12
  2. Convert to equivalent fractions with denominator 12:
    • 23 = 2 × 43 × 4 = 812
    • 14 = 1 × 34 × 3 = 312
  3. Add the numerators: 812 + 312 = 1112

Adding Mixed Numbers

To add mixed numbers, you can either:

  1. Add the whole numbers and fractions separately, then combine.
  2. Convert mixed numbers to improper fractions, add, then convert back.

Add 235 + 112:

Method 1: Add parts separately

  1. Add whole numbers: 2 + 1 = 3
  2. Add fractions: 35 + 12
    • Find LCM of denominators: LCM(5, 2) = 10
    • Convert to equivalent fractions: 35 = 610 and 12 = 510
    • Add: 610 + 510 = 1110 = 1110
  3. Combine: 3 + 1110 = 4110

Method 2: Convert to improper fractions

  1. Convert to improper fractions:
    • 235 = 2 × 5 + 35 = 135
    • 112 = 1 × 2 + 12 = 32
  2. Add the fractions:
    • Find LCM of denominators: LCM(5, 2) = 10
    • Convert to equivalent fractions: 135 = 2610 and 32 = 1510
    • Add: 2610 + 1510 = 4110
  3. Convert back to a mixed number:
    • 4110 = 4110

Subtracting Fractions

Subtracting Fractions with Like Denominators

To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same.

Subtract 78 - 38:

  1. 78 - 38 = 7 - 38 = 48 = 12

Subtracting Fractions with Unlike Denominators

To subtract fractions with different denominators:

  1. Find the least common multiple (LCM) of the denominators
  2. Convert each fraction to an equivalent fraction with the LCM as the denominator
  3. Subtract the numerators and keep the common denominator
  4. Simplify the result if possible

Subtract 56 - 13:

  1. Find the LCM of 6 and 3: LCM(6, 3) = 6
  2. Convert to equivalent fractions with denominator 6:
    • 56 is already in terms of 6
    • 13 = 1 × 23 × 2 = 26
  3. Subtract the numerators: 56 - 26 = 36 = 12

Subtracting Mixed Numbers

To subtract mixed numbers, you can either:

  1. Subtract the whole numbers and fractions separately, then combine.
  2. Convert mixed numbers to improper fractions, subtract, then convert back.

Subtract 534 - 213:

Method 1: Subtract parts separately (with regrouping if needed)

  1. Compare the fractions: 34 > 13, so no regrouping is needed
  2. Subtract whole numbers: 5 - 2 = 3
  3. Subtract fractions: 34 - 13
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions: 34 = 912 and 13 = 412
    • Subtract: 912 - 412 = 512
  4. Combine: 3 + 512 = 3512

Method 2: Convert to improper fractions

  1. Convert to improper fractions:
    • 534 = 5 × 4 + 34 = 234
    • 213 = 2 × 3 + 13 = 73
  2. Subtract the fractions:
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions: 234 = 6912 and 73 = 2812
    • Subtract: 6912 - 2812 = 4112
  3. Convert back to a mixed number:
    • 4112 = 3512

Subtract 314 - 123:

Method 1: Subtract parts separately (with regrouping)

  1. Compare the fractions: 14 < 23, so regrouping is needed
  2. Rewrite 314 as 254 (borrowed 1 from the whole number)
  3. Subtract whole numbers: 2 - 1 = 1
  4. Subtract fractions: 54 - 23
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions: 54 = 1512 and 23 = 812
    • Subtract: 1512 - 812 = 712
  5. Combine: 1 + 712 = 1712

Multiplying Fractions

Multiplying Fractions

To multiply fractions:

  1. Multiply the numerators together
  2. Multiply the denominators together
  3. Simplify the result if possible

Multiply 23 × 35:

  1. Multiply numerators: 2 × 3 = 6
  2. Multiply denominators: 3 × 5 = 15
  3. Result: 615 = 25

Multiplying Mixed Numbers

To multiply mixed numbers:

  1. Convert mixed numbers to improper fractions
  2. Multiply the fractions as usual
  3. Convert the result back to a mixed number if appropriate

Multiply 112 × 213:

  1. Convert to improper fractions:
    • 112 = 32
    • 213 = 73
  2. Multiply the fractions:
    • 32 × 73 = 216
  3. Convert to a mixed number:
    • 216 = 336 = 312

Cancellation Method

To simplify multiplication of fractions, you can cancel common factors before multiplying.

Multiply 34 × 815:

  1. Identify common factors:
    • Common factor between 3 and 15: 3
    • Common factor between 4 and 8: 4
  2. Cancel these factors:
    • 34 × 815 = 14 × 85 = 11 × 25 = 25

Dividing Fractions

Dividing Fractions

To divide fractions, the divisor must be nonzero. A zero numerator in the divisor gives division by zero, which is undefined:

  1. Take the reciprocal (flip) of the second fraction
  2. Multiply the first fraction by the reciprocal of the second fraction
  3. Simplify the result if possible

Divide 34 ÷ 23:

  1. Take the reciprocal of 23: 32
  2. Multiply: 34 × 32 = 98 = 118

Dividing Mixed Numbers

To divide mixed numbers:

  1. Convert mixed numbers to improper fractions
  2. Take the reciprocal of the second fraction
  3. Multiply the fractions
  4. Convert the result back to a mixed number if appropriate

Divide 312 ÷ 114:

  1. Convert to improper fractions:
    • 312 = 72
    • 114 = 54
  2. Take the reciprocal of the second fraction:
    • Reciprocal of 54 is 45
  3. Multiply:
    • 72 × 45 = 2810 = 145
  4. Convert to a mixed number:
    • 145 = 245

Dividing a Whole Number by a Fraction

To divide a whole number by a fraction:

  1. Convert the whole number to a fraction with denominator 1
  2. Take the reciprocal of the second fraction
  3. Multiply the fractions

Divide 6 ÷ 23:

  1. Convert 6 to a fraction: 61
  2. Take the reciprocal of 23: 32
  3. Multiply: 61 × 32 = 182 = 9

7. Fraction Word Problems

Part of a Whole:

A recipe calls for 3/4 cup of flour. If you want to make 1/2 of the recipe, how much flour do you need?

Solution: To find half of 3/4 cup, multiply 3/4 by 1/2.

34 × 12 = 38 cup of flour

Comparing Fractions:

Sarah ate 2/3 of her pizza, and Michael ate 3/4 of his pizza. Who ate more of their pizza?

Solution: Compare the fractions 2/3 and 3/4.

Convert to equivalent fractions with a common denominator:

23 = 812 and 34 = 912

Since 9/12 > 8/12, Michael ate more of his pizza.

Adding and Subtracting Fractions:

A carpenter has a board that is 8 1/2 feet long. He cuts off a piece that is 3 3/4 feet long. How long is the remaining piece?

Solution: Subtract 3 3/4 from 8 1/2.

812 - 334

Convert to improper fractions:

172 - 154

Find a common denominator:

344 - 154 = 194 = 434

The remaining piece is 4 3/4 feet long.

Multiplying Fractions:

A recipe calls for 2/3 cup of sugar. If you want to make 1 1/2 batches of the recipe, how much sugar do you need?

Solution: Multiply 2/3 by 1 1/2.

23 × 112

Convert the mixed number to an improper fraction:

23 × 32

Multiply:

23 × 32 = 66 = 1

You need 1 cup of sugar.

Dividing Fractions:

A pitcher contains 3 1/4 gallons of water. If each glass holds 1/2 gallon, how many glasses can be filled?

Solution: Divide 3 1/4 by 1/2.

314 ÷ 12

Convert the mixed number to an improper fraction:

134 ÷ 12

Multiply by the reciprocal:

134 × 21 = 264 = 132 = 6.5

6 full glasses can be filled, with half a glass remaining.

8. Real-World Applications

Cooking and Recipes:

Fractions are commonly used in recipes to specify ingredient quantities and to scale recipes up or down.

A recipe calls for 3/4 cup of milk, 1/2 cup of sugar, and 1 1/4 cups of flour. How much of each ingredient is needed to make 3 batches?

Milk: 34 × 3 = 94 = 2 1/4 cups

Sugar: 12 × 3 = 32 = 1 1/2 cups

Flour: 114 × 3 = 54 × 3 = 154 = 3 3/4 cups

Measurement:

Fractions are used in measurements for length, weight, volume, and time.

A board is 8 1/2 feet long. If it is cut into 4 equal pieces, how long is each piece?

812 ÷ 4 = 172 ÷ 41 = 172 × 14 = 178 = 2 1/8 feet

A carpenter needs to make a cut at 3/8 of an inch from the edge of a board. If the ruler shows markings for 1/16 of an inch, how many markings should the carpenter count?

38 ÷ 116 = 38 × 161 = 488 = 6

The carpenter should count 6 markings from the edge.

Finance:

Fractions are used in finance for interest rates, discounts, and investments.

A store offers a 1/3 discount on all items. If a shirt originally costs $36, what is the discounted price?

Original price = $36

Discount amount = 13 × $36 = $12

Discounted price = $36 - $12 = $24

If 3/4 of your monthly salary goes to expenses, and your salary is $2,800, how much money do you have left for savings?

Amount for expenses = 34 × $2,800 = $2,100

Amount for savings = $2,800 - $2,100 = $700

Probability and Statistics:

Fractions are used to express probabilities and statistical data.

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a blue marble?

Total number of marbles = 5 + 3 + 2 = 10

Probability of drawing a blue marble = 310 = 0.3 or 30%

In a survey, 3/5 of respondents preferred product A, while the rest preferred product B. If there were 200 respondents, how many preferred each product?

Number preferring product A = 35 × 200 = 120

Number preferring product B = 200 - 120 = 80

Construction and Engineering:

Fractions are essential in construction, carpentry, and engineering for precise measurements and calculations.

A blueprint uses a scale where 1/4 inch represents 1 foot. If a wall is 20 feet long, how long will it be on the blueprint?

14 inch × 20 = 5 inches

A bolt with a diameter of 5/8 inch needs to fit through a hole. If the hole diameter is 3/4 inch, what is the clearance (difference between hole and bolt diameter)?

34 - 58 = 68 - 58 = 18 inch

Music:

Fractions are used to represent different note durations in music.

In music notation, notes are represented as fractions of a whole note:

  • A whole note = 1
  • A half note = 12 of a whole note
  • A quarter note = 14 of a whole note
  • An eighth note = 18 of a whole note
  • A sixteenth note = 116 of a whole note

If a song has 12 measures, and each measure contains 4 quarter notes, how many eighth notes would the entire song contain?

Number of quarter notes = 12 × 4 = 48

Number of eighth notes = 48 × 2 = 96 (since each quarter note equals 2 eighth notes)

9. Common Mistakes with Fractions

Adding or Subtracting Denominators:

A common mistake is to add or subtract the denominators when adding or subtracting fractions.

INCORRECT: 12 + 13 = 25

CORRECT: 12 + 13 = 36 + 26 = 56

Multiplying Numerators and Denominators in Addition:

Another common mistake is to multiply the numerators and denominators when adding fractions.

INCORRECT: 12 + 13 = 1 × 12 × 3 = 16

CORRECT: 12 + 13 = 36 + 26 = 56

Adding Whole Numbers and Fractions Separately:

Adding the whole numbers and fractions separately is valid. If the fractional sum is at least 1, regroup it to write a conventional mixed number. The intermediate value 5 + 6/5 is correct, but 5 6/5 is not a mixed number in standard form.

NOT YET REGROUPED: 325 + 245 = 565

CORRECT: 325 + 245 = 565 = 615

Forgetting to Find the Least Common Multiple:

When adding or subtracting fractions, any nonzero common denominator works. The least common multiple (LCM) is usually the most efficient positive choice because it keeps the numbers smaller.

USING THE LCM: 12 + 14 = 24 + 14 = 34

This example already uses the LCM, 4. A denominator of 8 would also work: 4/8 + 2/8 = 6/8 = 3/4, but it needs an extra simplification step.

Incorrect Simplification:

Simplifying fractions incorrectly by not dividing both the numerator and denominator by the same common factor.

INCORRECT: 48 = 44 = 1

CORRECT: 48 = 4 ÷ 48 ÷ 4 = 12

10. Interactive Fractions Quiz

Test Your Fractions Skills

Enter an integer, exact decimal, fraction (such as 3/4), or mixed number (such as 1 3/4). Equivalent values are accepted unless the question requests a specific form. Lowest-terms questions require a reduced fraction with a positive denominator; the decimal question requires decimal notation. Use at most 100 characters, with no units, expressions, or exponent notation. Invalid entries and equivalent answers needing a format correction remain editable and do not count. Valid answers count once across all four levels. Reset clears the score; the bar shows accuracy among answered questions. Print Quiz includes all 20 questions.

Score: 0/0

Fractions Quiz

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