Basic MathMath

Fractions: Definition, Types & Examples Guide

1 year ago
Last Update on April 4, 2026
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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 represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number).

Basic Definition:

Numerator
Denominator

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

3
4
, 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:

1
2

One-half

3
4

Three-fourths

5
8

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:

5
3

Five-thirds

7
4

Seven-fourths

11
8

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:

1
1
2

One and a half

2
3
4

Two and three-fourths

3
5
8

Three and five-eighths

Unit Fractions:

A unit fraction has a numerator of 1.

Examples of unit fractions:

1
2

One-half

1
4

One-fourth

1
10

One-tenth

Like and Unlike Fractions:

Like fractions have the same denominator.

Unlike fractions have different denominators.

Examples of like fractions:

3
8
and
5
8

Examples of unlike fractions:

2
3
and
3
5

Equivalent Fractions:

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

Examples of equivalent fractions:

1
2
=
2
4
=
3
6
=
4
8

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

2
3
:

  • Multiply by 2:
    2 × 2
    3 × 2
    =
    4
    6
  • Multiply by 3:
    2 × 3
    3 × 3
    =
    6
    9
  • Multiply by 4:
    2 × 4
    3 × 4
    =
    8
    12

Simplifying Fractions:

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

To simplify

12
18
:

  1. Find the GCD of 12 and 18: GCD(12, 18) = 6
  2. Divide both numbers by the GCD:
    12 ÷ 6
    18 ÷ 6
    =
    2
    3

Therefore,

12
18
=
2
3
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

2
3
and
3
5
:

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

Method 2: Cross Multiplication

Another method to compare fractions is to cross multiply.

Compare

2
3
and
3
5
:

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

Method 3: Converting to Decimals

Convert the fractions to decimals and then compare the decimal values.

Compare

2
3
and
3
5
:

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

Method 4: Using a Number Line

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

Compare

2
3
and
3
5
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

2
3
(≈ 0.667) is to the right of
3
5
(= 0.6).

Therefore,

2
3
>
3
5

Method 5: Using Benchmark Fractions

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

Compare

3
8
and
5
9
using benchmarks:

  1. Compare each fraction to 1/2:
    • 3
      8
      <
      4
      8
      =
      1
      2
      (since 3 < 4)
    • 5
      9
      >
      4.5
      9
      =
      1
      2
      (since 5 > 4.5)
  2. Since
    3
    8
    <
    1
    2
    and
    5
    9
    >
    1
    2
    , we can conclude that
    3
    8
    <
    5
    9

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

17
4
to a mixed number:

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

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

3
2
5
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:
    17
    5

Converting Fractions to Decimals

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

Convert

3
4
to a decimal:

3 ÷ 4 = 0.75

Convert

2
3
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:
    75
    100
  3. Simplify:
    75
    100
    =
    3
    4

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

3
4
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:
    25
    100
  2. Simplify:
    25
    100
    =
    1
    4

6. Operations with Fractions

Addition
Subtraction
Multiplication
Division

Adding Fractions

Adding Fractions with Like Denominators

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

Add

3
8
+
2
8
:

  1. 3
    8
    +
    2
    8
    =
    3 + 2
    8
    =
    5
    8

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

2
3
+
1
4
:

  1. Find the LCM of 3 and 4: LCM(3, 4) = 12
  2. Convert to equivalent fractions with denominator 12:
    • 2
      3
      =
      2 × 4
      3 × 4
      =
      8
      12
    • 1
      4
      =
      1 × 3
      4 × 3
      =
      3
      12
  3. Add the numerators:
    8
    12
    +
    3
    12
    =
    11
    12

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

2
3
5
+
1
1
2
:

Method 1: Add parts separately

  1. Add whole numbers: 2 + 1 = 3
  2. Add fractions:
    3
    5
    +
    1
    2
    • Find LCM of denominators: LCM(5, 2) = 10
    • Convert to equivalent fractions:
      3
      5
      =
      6
      10
      and
      1
      2
      =
      5
      10
    • Add:
      6
      10
      +
      5
      10
      =
      11
      10
      =
      1
      1
      10
  3. Combine: 3 +
    1
    1
    10
    =
    4
    1
    10

Method 2: Convert to improper fractions

  1. Convert to improper fractions:
    • 2
      3
      5
      =
      2 × 5 + 3
      5
      =
      13
      5
    • 1
      1
      2
      =
      1 × 2 + 1
      2
      =
      3
      2
  2. Add the fractions:
    • Find LCM of denominators: LCM(5, 2) = 10
    • Convert to equivalent fractions:
      13
      5
      =
      26
      10
      and
      3
      2
      =
      15
      10
    • Add:
      26
      10
      +
      15
      10
      =
      41
      10
  3. Convert back to a mixed number:
    • 41
      10
      =
      4
      1
      10

Subtracting Fractions

Subtracting Fractions with Like Denominators

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

Subtract

7
8
-
3
8
:

  1. 7
    8
    -
    3
    8
    =
    7 - 3
    8
    =
    4
    8
    =
    1
    2

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

5
6
-
1
3
:

  1. Find the LCM of 6 and 3: LCM(6, 3) = 6
  2. Convert to equivalent fractions with denominator 6:
    • 5
      6
      is already in terms of 6
    • 1
      3
      =
      1 × 2
      3 × 2
      =
      2
      6
  3. Subtract the numerators:
    5
    6
    -
    2
    6
    =
    3
    6
    =
    1
    2

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

5
3
4
-
2
1
3
:

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

  1. Compare the fractions:
    3
    4
    >
    1
    3
    , so no regrouping is needed
  2. Subtract whole numbers: 5 - 2 = 3
  3. Subtract fractions:
    3
    4
    -
    1
    3
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions:
      3
      4
      =
      9
      12
      and
      1
      3
      =
      4
      12
    • Subtract:
      9
      12
      -
      4
      12
      =
      5
      12
  4. Combine: 3 +
    5
    12
    =
    3
    5
    12

Method 2: Convert to improper fractions

  1. Convert to improper fractions:
    • 5
      3
      4
      =
      5 × 4 + 3
      4
      =
      23
      4
    • 2
      1
      3
      =
      2 × 3 + 1
      3
      =
      7
      3
  2. Subtract the fractions:
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions:
      23
      4
      =
      69
      12
      and
      7
      3
      =
      28
      12
    • Subtract:
      69
      12
      -
      28
      12
      =
      41
      12
  3. Convert back to a mixed number:
    • 41
      12
      =
      3
      5
      12

Subtract

3
1
4
-
1
2
3
:

Method 1: Subtract parts separately (with regrouping)

  1. Compare the fractions:
    1
    4
    <
    2
    3
    , so regrouping is needed
  2. Rewrite
    3
    1
    4
    as
    2
    5
    4
    (borrowed 1 from the whole number)
  3. Subtract whole numbers: 2 - 1 = 1
  4. Subtract fractions:
    5
    4
    -
    2
    3
    • Find LCM of denominators: LCM(4, 3) = 12
    • Convert to equivalent fractions:
      5
      4
      =
      15
      12
      and
      2
      3
      =
      8
      12
    • Subtract:
      15
      12
      -
      8
      12
      =
      7
      12
  5. Combine: 1 +
    7
    12
    =
    1
    7
    12

Multiplying Fractions

Multiplying Fractions

To multiply fractions:

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

Multiply

2
3
×
3
5
:

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

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

1
1
2
×
2
1
3
:

  1. Convert to improper fractions:
    • 1
      1
      2
      =
      3
      2
    • 2
      1
      3
      =
      7
      3
  2. Multiply the fractions:
    • 3
      2
      ×
      7
      3
      =
      21
      6
  3. Convert to a mixed number:
    • 21
      6
      =
      3
      3
      6
      =
      3
      1
      2

Cancellation Method

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

Multiply

3
4
×
8
15
:

  1. Identify common factors:
    • Common factor between 3 and 15: 3
    • Common factor between 4 and 8: 4
  2. Cancel these factors:
    • 3
      4
      ×
      8
      15
      =
      1
      4
      ×
      8
      5
      =
      1
      1
      ×
      2
      5
      =
      2
      5

Dividing Fractions

Dividing Fractions

To divide fractions:

  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

3
4
÷
2
3
:

  1. Take the reciprocal of
    2
    3
    :
    3
    2
  2. Multiply:
    3
    4
    ×
    3
    2
    =
    9
    8
    =
    1
    1
    8

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

3
1
2
÷
1
1
4
:

  1. Convert to improper fractions:
    • 3
      1
      2
      =
      7
      2
    • 1
      1
      4
      =
      5
      4
  2. Take the reciprocal of the second fraction:
    • Reciprocal of
      5
      4
      is
      4
      5
  3. Multiply:
    • 7
      2
      ×
      4
      5
      =
      28
      10
      =
      14
      5
  4. Convert to a mixed number:
    • 14
      5
      =
      2
      4
      5

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 ÷

2
3
:

  1. Convert 6 to a fraction:
    6
    1
  2. Take the reciprocal of
    2
    3
    :
    3
    2
  3. Multiply:
    6
    1
    ×
    3
    2
    =
    18
    2
    = 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.

3
4
×
1
2
=
3
8
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:

2
3
=
8
12
and
3
4
=
9
12

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.

8
1
2
-
3
3
4

Convert to improper fractions:

17
2
-
15
4

Find a common denominator:

34
4
-
15
4
=
19
4
=
4
3
4

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.

2
3
×
1
1
2

Convert the mixed number to an improper fraction:

2
3
×
3
2

Multiply:

2
3
×
3
2
=
6
6
= 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.

3
1
4
÷
1
2

Convert the mixed number to an improper fraction:

13
4
÷
1
2

Multiply by the reciprocal:

13
4
×
2
1
=
26
4
=
13
2
= 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:

3
4
× 3 =
9
4
= 2 1/4 cups

Sugar:

1
2
× 3 =
3
2
= 1 1/2 cups

Flour:

1
1
4
× 3 =
5
4
× 3 =
15
4
= 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?

8
1
2
÷ 4 =
17
2
÷
4
1
=
17
2
×
1
4
=
17
8
= 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?

3
8
÷
1
16
=
3
8
×
16
1
=
48
8
= 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 =

1
3
× $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 =

3
4
× $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 =

3
10
= 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 =

3
5
× 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?

1
4
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)?

3
4
-
5
8
=
6
8
-
5
8
=
1
8
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 =
    1
    2
    of a whole note
  • A quarter note =
    1
    4
    of a whole note
  • An eighth note =
    1
    8
    of a whole note
  • A sixteenth note =
    1
    16
    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:

1
2
+
1
3
=
2
5

CORRECT:

1
2
+
1
3
=
3
6
+
2
6
=
5
6

Multiplying Numerators and Denominators in Addition:

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

INCORRECT:

1
2
+
1
3
=
1 × 1
2 × 3
=
1
6

CORRECT:

1
2
+
1
3
=
3
6
+
2
6
=
5
6

Adding Whole Numbers and Fractions Separately:

When adding mixed numbers, a common mistake is to add the whole numbers and fractions separately without handling the case where the sum of fractions is greater than 1.

INCORRECT:

3
2
5
+
2
4
5
=
5
6
5

CORRECT:

3
2
5
+
2
4
5
=
5
6
5
=
6
1
5

Forgetting to Find the Least Common Multiple:

When adding or subtracting fractions, you need to find the least common multiple (LCM) of the denominators, not just any common multiple.

INEFFICIENT:

1
2
+
1
4
=
2
4
+
1
4
=
3
4

This is correct but using the LCM is more efficient when dealing with fractions with larger denominators.

Incorrect Simplification:

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

INCORRECT:

4
8
=
4
4
= 1

CORRECT:

4
8
=
4 ÷ 4
8 ÷ 4
=
1
2

10. Interactive Fractions Quiz

Test Your Fractions Skills

Try these problems and check your answers:

Basic
Intermediate
Advanced
Word Problems
Score: 0/0

Fractions Quiz

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