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What Is a Fraction?

A fraction represents a part of a whole. Whenever a whole thing is divided into equal parts and we take some of those parts, we use a fraction to describe how many parts we have taken.

A fraction is written in the form of numerator over denominator, separated by a horizontal line.

Parts of a Fraction Numerator — the top number. It tells you how many parts are being considered.
Denominator — the bottom number. It tells you into how many equal parts the whole is divided.

Example: In 58, the numerator is 5 and the denominator is 8. The whole is divided into 8 equal parts and we are considering 5 of them.

Half, One-Third, and Quarter

Three of the most common fractions have their own names:

Name	Fraction	Meaning	How many make a whole?
Half	1/2	Whole divided into 2 equal parts; 1 part taken	2 halves = 1 whole
One-third	1/3	Whole divided into 3 equal parts; 1 part taken	3 thirds = 1 whole
Quarter	1/4	Whole divided into 4 equal parts; 1 part taken	4 quarters = 1 whole

Writing Fractions for Shaded Parts — Practice

Example 1Write the fraction for each shaded part

(a) A rectangle is divided into 5 equal parts and 2 parts are shaded.
Denominator = 5 (total parts) | Numerator = 2 (shaded parts) → Fraction = 25

(b) A shape is divided into 9 equal parts and 4 parts are shaded.
Denominator = 9 | Numerator = 4 → Fraction = 49

Key rule: Fraction = (Number of shaded parts) / (Total equal parts)

Equivalent Fractions

Equivalent fractions are different fractions that represent the same portion of a whole. They look different but have the same value.

How to Make Equivalent Fractions Multiply or divide both the numerator and the denominator by the same non-zero number. The value of the fraction does not change.

Example 2Find three equivalent fractions of 2/3

× 2 2 × 23 × 2 = 46 → so 4/6 is equivalent to 2/3

× 4 2 × 43 × 4 = 812 → so 8/12 is equivalent to 2/3

× 6 2 × 63 × 6 = 1218 → so 12/18 is equivalent to 2/3

Equivalent fractions of 2/3 : 4/6 , 8/12 , 12/18 , and infinitely more.

Example 3Find an equivalent fraction of 18/24 by dividing

÷ 2 18 ÷ 224 ÷ 2 = 912 → 9/12 is equivalent to 18/24

÷ 3 again 9 ÷ 312 ÷ 3 = 34 → 3/4 is also equivalent to 18/24

18/24 = 9/12 = 3/4 — all are equivalent fractions.

Simplifying Fractions to Lowest Form

Simplifying a fraction means writing it in its simplest form — where the numerator and denominator have no common factor other than 1. You do this by dividing both the numerator and denominator by their common factors, one at a time, until no common factor remains.

Example 4Simplify 36/60

÷ 2 2 is a common factor. 36 ÷ 2 = 18, 60 ÷ 2 = 30 → 18/30

÷ 2 2 is still a common factor. 18 ÷ 2 = 9, 30 ÷ 2 = 15 → 9/15

÷ 3 3 is a common factor. 9 ÷ 3 = 3, 15 ÷ 3 = 5 → 3/5

Check Do 3 and 5 share a common factor? No. So 3/5 is in its simplest form.

36/60 in simplest form = 3/5

Example 5Simplify 56/84

Find GCF Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Greatest common factor (GCF) = 28

÷ 28 56 ÷ 28 = 2, 84 ÷ 28 = 3 → 2/3

Check 2 and 3 share no common factor. Done.

56/84 in simplest form = 2/3

Shortcut Tip If you can find the Greatest Common Factor (GCF) directly, you can simplify in one step instead of dividing multiple times. Both methods give the same answer.

Like and Unlike Fractions

Like Fractions Fractions with the same denominator are called like fractions.
Example: 310, 710, 110 — all have denominator 10.

Unlike Fractions Fractions with different denominators are called unlike fractions.
Example: 25, 38, 411 — all have different denominators.

Comparing Fractions

Comparing Like Fractions

When fractions have the same denominator, compare only their numerators. The fraction with the larger numerator is the greater fraction.

Example 6Compare like fractions

(a) Compare 513 and 913

Denominators are the same (13). Compare numerators: 5 < 9.

513 < 913

(b) From the group below, identify the smallest and greatest fraction:

4/15, 11/15, 6/15, 2/15, 9/15

All have denominator 15. Compare numerators only:
Smallest numerator = 2 → Smallest fraction = 2/15
Greatest numerator = 11 → Greatest fraction = 11/15

Ascending order: 2/15 < 4/15 < 6/15 < 9/15 < 11/15

Comparing Unlike Fractions

When fractions have different denominators, use the cross-multiplication method. Multiply the numerator of the first fraction by the denominator of the second, and vice versa. Then compare the two products.

Example 7Compare unlike fractions using cross-multiplication

(a) Compare 37 and 511

Multiply 3 × 11 = 33 | 5 × 7 = 35

Compare 33 < 35, so 3/7 < 5/11

(b) Compare 79 and 45

Multiply 7 × 5 = 35 | 4 × 9 = 36

Compare 35 < 36, so 7/9 < 4/5

(a) 3/7 < 5/11 (b) 7/9 < 4/5

Ordering Fractions

Ordering Like Fractions

Example 8Arrange 8/17, 3/17, 14/17, 6/17 in ascending order

All denominators are 17. Compare numerators: 3, 6, 8, 14.

Smallest Numerator 3 → fraction 3/17

Next Numerator 6 → fraction 6/17

Next Numerator 8 → fraction 8/17

Greatest Numerator 14 → fraction 14/17

Ascending order: 3/17 < 6/17 < 8/17 < 14/17

Ordering Unlike Fractions with the Same Numerator

Important Rule When fractions have the same numerator but different denominators:
— The greater the denominator, the smaller the fraction.
— The smaller the denominator, the greater the fraction.

Example 9Arrange 4/6, 4/11, 4/3, 4/8, 4/15 in descending order

All numerators are 4. Compare only the denominators: 6, 11, 3, 8, 15.

Remember Descending order = greatest first. Greatest fraction has smallest denominator.

Smallest denom Denominator 3 → greatest fraction = 4/3

Next Denominator 6 → 4/6

Next Denominator 8 → 4/8

Next Denominator 11 → 4/11

Largest denom Denominator 15 → smallest fraction = 4/15

Descending order: 4/3 > 4/6 > 4/8 > 4/11 > 4/15

Proper and Improper Fractions

Proper Fraction A fraction where the numerator is less than the denominator. Its value is always less than 1.
Examples: 3/7 | 5/12 | 8/15 | 11/20

Improper Fraction A fraction where the numerator is greater than or equal to the denominator. Its value is 1 or more.
Examples: 7/3 | 12/5 | 15/8 | 20/11

Mixed Fraction A number made up of a whole number and a proper fraction written together.
Examples: 2 3/4 | 5 1/6 | 3 7/9

Example 10Identify each fraction as proper, improper, or mixed

(a) 11/4 — Numerator 11 > Denominator 4 → Improper fraction

(b) 6/13 — Numerator 6 < Denominator 13 → Proper fraction

(d) 9/9 — Numerator = Denominator → Improper fraction (value = 1 whole)

Converting Improper Fractions to Mixed Fractions

Divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator stays the same.

Example 11Convert 19/5 into a mixed fraction

Divide 19 ÷ 5 = 3 remainder 4 (because 5 × 3 = 15, and 19 − 15 = 4)

Write form Quotient = 3, Remainder = 4, Divisor = 5 → Write as 3 and 4/5

19/5 = 3 4/5

Example 12Convert 31/6 into a mixed fraction

Divide 31 ÷ 6 = 5 remainder 1 (because 6 × 5 = 30, and 31 − 30 = 1)

Write form Quotient = 5, Remainder = 1, Divisor = 6 → Write as 5 and 1/6

31/6 = 5 1/6

Converting Mixed Fractions to Improper Fractions

Multiply the whole number by the denominator, then add the numerator. Write that result over the original denominator.

Example 13Convert 4 2/9 into an improper fraction

Multiply Whole number × Denominator = 4 × 9 = 36

Add numerator 36 + 2 = 38

Write result Place 38 over the original denominator 9 → 38/9

4 2/9 = 38/9

Example 14Convert 6 3/7 into an improper fraction

Multiply 6 × 7 = 42

Add numerator 42 + 3 = 45

Write result 45 over 7 → 45/7

6 3/7 = 45/7

Adding Like Fractions

To add like fractions (same denominator): add only the numerators and keep the denominator the same. Then simplify if possible.

Example 15Add 9/20 + 7/20

Add numerators 9 + 7 = 16 | Denominator stays 20 → 16/20

Simplify Common factor of 16 and 20 is 4. 16 ÷ 4 = 4 , 20 ÷ 4 = 5 → 4/5

9/20 + 7/20 = 16/20 = 4/5

Example 16Add 2 5/12 + 7/12

Convert 2 5/12 = (2 × 12 + 5)/12 = 29/12

Add 29/12 + 7/12 = (29 + 7)/12 = 36/12

Simplify 36 ÷ 12 = 3 | 12 ÷ 12 = 1 → 3 (a whole number)

2 5/12 + 7/12 = 3

Subtracting Like Fractions

To subtract like fractions: subtract only the numerators and keep the denominator the same. Then simplify if possible.

Example 17Subtract 11/18 − 5/18

Subtract 11 − 5 = 6 | Denominator stays 18 → 6/18

Simplify GCF of 6 and 18 is 6. 6 ÷ 6 = 1 , 18 ÷ 6 = 3 → 1/3

11/18 − 5/18 = 6/18 = 1/3

Example 18Subtract 3 2/11 − 8/11

Convert 3 2/11 = (3 × 11 + 2)/11 = 35/11

Subtract 35/11 − 8/11 = (35 − 8)/11 = 27/11

Convert back 27 ÷ 11 = 2 remainder 5 → 2 5/11

3 2/11 − 8/11 = 2 5/11

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