Meaning of Fraction

A fraction is a way of representing a part of a whole or a group of items. It tells how many parts something is divided into and how many of those parts we are considering.

A fraction has two parts:

1. The numerator — the top number, which tells the number of parts taken.


2. The denominator — the bottom number, which tells the number of equal parts the whole is divided into.

Example:

³⁄₄ means 3 parts are taken out of 4 equal parts of a whole.

Parts of a Fraction

1. Numerator: shows how many parts are selected.


2. Denominator: shows the total number of equal parts the whole is divided into.

Example: ⁵⁄₈ → 5 (numerator), 8 (denominator).

Types of Fractions

1. Proper Fractions

These are fractions where the numerator is less than the denominator. They represent quantities less than one whole.

Examples: ²⁄₃, ³⁄₅, ⁷⁄₁₀, ⁴⁄₉

2. Improper Fractions

These are fractions where the numerator is equal to or greater than the denominator. They represent quantities that are equal to or more than one whole.

Examples: ⁵⁄₄, ⁹⁄₆, ⁷⁄₇, ⁸⁄₃

3. Mixed Numbers (or Mixed Fractions)

A mixed number consists of a whole number and a proper fraction.

Examples: 1¹⁄₂, 3²⁄₅, 4³⁄₈, 5¹⁄₄

4. Equivalent Fractions

These are fractions that look different but represent the same value or portion of a whole.

Examples:

¹⁄₂ = ²⁄₄ = ³⁄₆ = ⁴⁄₈

³⁄₅ = ⁶⁄₁₀ = ⁹⁄₁₅

To form equivalent fractions, multiply or divide both numerator and denominator by the same number.

5. Like and Unlike Fractions

1. Like fractions: have the same denominators.



Example: ²⁄₇, ⁵⁄₇, ⁶⁄₇

2. Unlike fractions: have different denominators.



Example: ³⁄₄, ²⁄₅, ⁵⁄₆

Converting Between Forms

(a) Improper Fraction → Mixed Number

Divide the numerator by the denominator.

1. Quotient = whole number part


2. Remainder = numerator of the fraction

Example: ¹¹⁄₄ = 2³⁄₄ (because 11 ÷ 4 = 2 remainder 3)

(b) Mixed Number → Improper Fraction

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

Example: 3²⁄₅ = (3 × 5) + 2 = ¹⁷⁄₅

Simplifying Fractions (Reducing to Lowest Terms)

A fraction is said to be in lowest terms when the numerator and denominator have no common factor other than 1.

1. Find the Highest Common Factor (HCF) of numerator and denominator.


2. Divide both by the HCF.

Examples:

¹²⁄₁₈ = ²⁄₃

⁴⁵⁄₆₀ = ³⁄₄

²⁷⁄₃₆ = ³⁄₄

Operations with Fractions

(a) Addition of Fractions

Case 1: Same denominator — add numerators; keep denominator.

Example: ³⁄₈ + ²⁄₈ = ⁵⁄₈

Case 2: Different denominators — find the LCM of denominators, then convert to like fractions.

Example: ³⁄₄ + ²⁄₅

LCM of 4 and 5 = 20

³⁄₄ = ¹⁵⁄₂₀, ²⁄₅ = ⁸⁄₂₀

Add: ¹⁵⁄₂₀ + ⁸⁄₂₀ = ²³⁄₂₀ = 1³⁄₂₀

(b) Subtraction of Fractions

Follow the same process as in addition.

Example: ⁷⁄₁₀ − ²⁄₅

²⁄₅ = ⁴⁄₁₀

So, ⁷⁄₁₀ − ⁴⁄₁₀ = ³⁄₁₀

(c) Multiplication of Fractions

Multiply numerators together and denominators together.

Example: ³⁄₄ × ⁵⁄₆ = ¹⁵⁄₂₄ = ⁵⁄₈

(d) Division of Fractions

To divide, multiply the first fraction by the reciprocal of the second.

Example: ⁴⁄₅ ÷ ²⁄₃ = ⁴⁄₅ × ³⁄₂ = ¹²⁄₁₀ = ⁶⁄₅ = 1¹⁄₅

Comparing and Ordering Fractions

Method 1 — Using Common Denominator

Convert all fractions to have the same denominator.

Example: Arrange ¹⁄₂, ³⁄₄, ²⁄₃ in ascending order.

LCM of 2, 4, and 3 = 12.

¹⁄₂ = ⁶⁄₁₂, ³⁄₄ = ⁹⁄₁₂, ²⁄₃ = ⁸⁄₁₂

Order: ¹⁄₂, ²⁄₃, ³⁄₄

Method 2 — Cross Multiplication (for two fractions)

Compare ³⁄₅ and ⁴⁄₇:

3 × 7 = 21, 4 × 5 = 20. Since 21 > 20, ³⁄₅ > ⁴⁄₇.

Applications of Fractions in Real Life

1. In sharing things equally (food, land, money).


2. In measurements (cooking, tailoring, carpentry).


3. In time (half hour, quarter hour).


4. In school marks and averages.


5. In business (discounts, profit sharing).


6. In probability and statistics.


7. In dividing work among people.


8. In understanding percentages and decimals.

Common Mistakes Students Make

1. Adding or subtracting denominators directly.


2. Forgetting to find LCM for unlike fractions.


3. Using wrong reciprocal in division.


4. Forgetting to convert mixed numbers before operations.


5. Ignoring simplification.


6. Confusing numerator and denominator.


7. Forgetting to check results with estimation.

Class Activity / Practice Questions

1. Simplify: ⁷²⁄₉₆


2. Add: ³⁄₇ + ²⁄₅


3. Subtract: ⁴⁄₉ − ¹⁄₆


4. Multiply: ⁵⁄₈ × ¹²⁄₁₅


5. Divide: ⁹⁄₁₄ ÷ ³⁄₇


6. Convert: 5³⁄₄ to improper fraction.


7. Order: ²⁄₅, ³⁄₄, ⁵⁄₈


8. A piece of land is shared equally among 5 people. If the total area is 4¹⁄₂ hectares, how much land does each person get?


9. Find ²⁄₃ of 81.


10. A man used ³⁄₅ of his salary for rent and ¹⁄₄ for food. What fraction of his salary was left?

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TOPIC: EXPRESSING FRACTIONS AS A RATIO, DECIMAL AND PERCENTAGE

Meaning of Fraction

A fraction represents a part of a whole. For example, 1/2 means one part out of two equal parts. Fractions can be written in different forms such as ratios, decimals, and percentages. Converting between these forms helps us to understand quantities in different ways and to compare them easily.

Relationship Between Fractions, Ratios, Decimals and Percentages

These forms show the same idea (part of a whole) in different ways:

Form	Example	Meaning
Fraction	1/2	One part out of two
Ratio	1 : 2	One to two
Decimal	0.5	Half
Percentage	50%	Fifty out of every hundred

A. EXPRESSING A FRACTION AS A RATIO

To write a fraction as a ratio, replace the division line ("/") with a colon ":" and simplify if possible.

B. EXPRESSING A FRACTION AS A DECIMAL

To convert a fraction to a decimal: divide the numerator (top) by the denominator (bottom).

1. Step 1: Write the fraction.


2. Step 2: Divide numerator by denominator.


3. Step 3: Write the result in decimal form.

Extra tip: If the division does not end, use long division and show repeating digits (e.g., 1/3 = 0.333...).

C. EXPRESSING A FRACTION AS A PERCENTAGE

To convert a fraction to a percentage, multiply the fraction by 100%.

Formula: Percentage = Fraction × 100%

1. Step 1: Write the fraction.


2. Step 2: Multiply by 100.


3. Step 3: Simplify to get the percentage value.

CONNECTION BETWEEN FRACTION, DECIMAL AND PERCENTAGE

A fraction, ratio, decimal, and percentage all represent the same idea — a comparison or part of a whole — but are written differently:

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
2/5	0.4	40%
1/5	0.2	20%
3/10	0.3	30%
7/10	0.7	70%
3/8	0.375	37.5%
9/20	0.45	45%
5/2	2.5	250%

Conversion formulas (quick)

1. Fraction → Decimal: divide numerator by denominator.


2. Decimal → Percentage: multiply decimal by 100.


3. Percentage → Fraction: write over 100 and simplify.


4. Fraction → Percentage: multiply fraction by 100.


5. Percentage → Decimal: divide by 100.

Key Points to Remember

1. Fraction, decimal and percentage represent the same value in different forms.


2. Multiply by 100 to convert decimal → percentage (or fraction → percentage).


3. Divide by 100 to convert percentage → decimal.


4. Always simplify fractions to lowest terms where possible.

Word Problems and Applications

1. Convert 3/5 to: (a) ratio (b) decimal (c) percentage.
   Ratio = 3:5, Decimal = 0.6, Percentage = 60%


2. Express 0.8 as: (a) fraction (b) ratio (c) percentage.
   Fraction = 4/5, Ratio = 4:5, Percentage = 80%


3. Express 45% as: (a) fraction (b) ratio (c) decimal.
   Fraction = 9/20, Ratio = 9:20, Decimal = 0.45


4. Express 7/8 as decimal and percentage.
   Decimal = 0.875, Percentage = 87.5%


5. Convert 3/2 to decimal and percentage.
   Decimal = 1.5, Percentage = 150%


6. Express 2:5 as fraction, decimal, and percentage.
   Fraction = 2/5, Decimal = 0.4, Percentage = 40%


7. Express 125% as fraction and decimal.
   Fraction = 5/4, Decimal = 1.25


8. Express 0.375 as fraction and percentage.
   Fraction = 3/8, Percentage = 37.5%


9. Convert 12/25 to decimal and percentage.
   Decimal = 0.48, Percentage = 48%


10. Express 7/20 as ratio, decimal, and percentage.
    Ratio = 7:20, Decimal = 0.35, Percentage = 35%

F. Applications in Real Life

1. Percentages are used to show discounts, profit, loss, or exam scores.


2. Decimals are used in money, weight, and length measurements.


3. Ratios are used in comparing quantities, like boys to girls in a class.


4. Fractions are used in sharing food, land, or time.


5. Converting between them helps us to interpret data correctly.

Common Mistakes Students Make

1. Forgetting to multiply by 100 when changing fraction to percentage.


2. Dividing denominator by numerator instead of numerator by denominator.


3. Writing the wrong place value when converting decimal to fraction.


4. Failing to simplify fractions.


5. Confusing the order of numerator and denominator when writing ratio.

Classwork / Practice Questions

1. Express 3/10 as a ratio, decimal, and percentage.


2. Convert 7/5 to decimal and percentage.


3. Write 0.65 as a fraction, ratio, and percentage.


4. Change 80% to fraction, ratio, and decimal.


5. Convert 9/25 to decimal and percentage.


6. Express 1 : 4 as fraction, decimal, and percentage.


7. Convert 11/20 to ratio, decimal, and percentage.


8. Express 0.2 as fraction, ratio, and percentage.


9. Write 125% as fraction and decimal.


10. Convert 5/8 to decimal and percentage.

Conclusion

Fractions, ratios, decimals, and percentages are different ways of expressing the same idea — part of a whole. It helps us to compare values more easily, solve everyday problems in trade, measurement, and statistics, and work with data in money, scores, and growth rates.