Decimal-Fractions

COMPLETE MATHEMATICS STUDY BOOK

DECIMAL & FRACTION

PAGE 1 — DECIMAL NUMBERS

A decimal number is a number that contains a decimal point. The decimal point separates the whole-number part from the fractional part.

Example:
25.75 = 25 + 0.75

Here, 25 is the whole-number part and 75 represents the fractional part.

Place Value of Decimals

Hundreds	Tens	Units	Decimal	Tenths	Hundredths	Thousandths
3	4	5	.	6	7	8

345.678

3 = Hundreds
4 = Tens
5 = Ones
6 = Tenths
7 = Hundredths
8 = Thousandths

Types of Decimals

1. Terminating Decimal

A decimal that ends after a finite number of digits.

Examples: 0.5, 2.75, 4.125

2. Non-Terminating Decimal

A decimal that continues indefinitely.

Examples: 0.333..., 1.272727...

Decimal to Fraction

To convert a terminating decimal into a fraction, remove the decimal point and put the number over 10, 100, 1000, etc., depending on the number of decimal places.

Example 1:
0.5 = 5/10 = 1/2

Example 2:
0.75 = 75/100 = 3/4

Example 3:
2.125 = 2125/1000 = 17/8

Important Rule

1 decimal place → denominator 10
2 decimal places → denominator 100
3 decimal places → denominator 1000
4 decimal places → denominator 10000

DECIMAL & FRACTION

OPERATIONS ON DECIMALS

PAGE 2 — ADDITION, SUBTRACTION & MULTIPLICATION

1. Addition of Decimals

For addition, write the numbers one below another and keep all decimal points in the same column.

25.75
+ 4.60
────────
30.35

Therefore, 25.75 + 4.60 = 30.35.

2. Subtraction of Decimals

45.80
− 12.35
────────
33.45

Therefore, 45.80 − 12.35 = 33.45.

3. Multiplication of Decimals

Multiply the numbers as ordinary whole numbers. Then count the total number of decimal places in both factors and put the decimal point in the answer accordingly.

Example:

2.5 × 1.2
25 × 12 = 300
There are 2 decimal places altogether.
Therefore, answer = 3.00 = 3

4. Multiplication by 10, 100 and 1000

Number	× 10	× 100	× 1000
4.25	42.5	425	4250
7.8	78	780	7800

Shortcut: When multiplying by 10, 100 or 1000, move the decimal point 1, 2 or 3 places to the right respectively.

5. Division of Decimals

When dividing a decimal by 10, 100 or 1000, move the decimal point to the left.

Number	÷ 10	÷ 100	÷ 1000
450	45	4.5	0.45
25.6	2.56	0.256	0.0256

6. Comparing Decimals

To compare decimals, first make the number of decimal places equal by adding zeros.

Compare 0.7 and 0.65.

0.7 = 0.70
0.70 > 0.65

Therefore, 0.7 > 0.65

Quick Practice

25.5 + 4.25 = ?
50.75 − 12.50 = ?
2.5 × 4 = ?
8.4 ÷ 2 = ?
Which is greater: 0.45 or 0.5?

DECIMAL & FRACTION

FRACTIONS

PAGE 3 — TYPES & BASIC CONCEPTS

A fraction represents a part of a whole. It is written in the form:

a / b

Here, a is called the numerator and b is called the denominator. The denominator cannot be zero.

Types of Fractions

1. Proper Fraction

Numerator is smaller than denominator.

Examples: 2/5, 3/7, 5/9

2. Improper Fraction

Numerator is greater than or equal to denominator.

Examples: 7/4, 9/5, 8/8

3. Mixed Fraction

A whole number together with a proper fraction.

Example: 2 1/3

4. Unit Fraction

A fraction whose numerator is 1.

Examples: 1/2, 1/5, 1/10

Equivalent Fractions

Fractions that have the same value are called equivalent fractions.

1/2 = 2/4 = 3/6 = 4/8

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

Simplest Form

A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.

Example:
12/18

HCF of 12 and 18 = 6

12 ÷ 6 / 18 ÷ 6 = 2/3

Mixed Fraction to Improper Fraction

Example:
2 3/5

(2 × 5 + 3) / 5
= 13/5

Therefore, 2 3/5 = 13/5

Improper Fraction to Mixed Fraction

Example:
17/4

17 ÷ 4 = 4 remainder 1

Therefore, 17/4 = 4 1/4

Fraction to Decimal

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

1/2 = 1 ÷ 2 = 0.5

3/4 = 3 ÷ 4 = 0.75

1/5 = 1 ÷ 5 = 0.2

DECIMAL & FRACTION

FRACTION OPERATIONS & PRACTICE

PAGE 4 — ADDITION, SUBTRACTION, MULTIPLICATION & DIVISION

1. Addition of Fractions

If the denominators are the same, add the numerators and keep the denominator unchanged.

2/7 + 3/7 = 5/7

For different denominators, first find the LCM of the denominators.

1/2 + 1/3

LCM of 2 and 3 = 6

1/2 = 3/6
1/3 = 2/6

Therefore,
3/6 + 2/6 = 5/6

2. Subtraction of Fractions

5/6 − 1/3

1/3 = 2/6

5/6 − 2/6 = 3/6 = 1/2

3. Multiplication of Fractions

Multiply numerator by numerator and denominator by denominator.

2/3 × 4/5

= (2 × 4)/(3 × 5)
= 8/15

Answer = 8/15

4. Division of Fractions

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

2/3 ÷ 4/5

= 2/3 × 5/4
= 10/12
= 5/6

Remember:
Division of fractions = Multiplication by reciprocal.

Decimal and Fraction Conversion Table

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
3/5	0.6	60%

Practice Questions

Convert 0.75 into a fraction.
Convert 3/5 into a decimal.
Simplify 18/24.
Convert 3 2/5 into an improper fraction.
Convert 19/4 into a mixed fraction.
Add: 2/5 + 1/5.
Add: 1/2 + 1/4.
Subtract: 7/8 − 1/4.
Multiply: 3/4 × 2/5.
Divide: 5/6 ÷ 2/3.
Calculate: 2.75 + 1.25.
Calculate: 5.50 − 2.75.
Calculate: 2.5 × 0.4.
Calculate: 8.4 ÷ 2.
Which is greater: 3/4 or 0.7?

Quick Revision

• Decimal point separates whole and fractional parts.
• 0.5 = 1/2.
• To simplify a fraction, divide numerator and denominator by their HCF.
• For addition/subtraction of unlike fractions, use LCM.
• For multiplication, multiply numerator and denominator directly.
• For division, multiply by the reciprocal.