Fractions and Decimals Class 7 PDF: Complete NCERT Guide

Understanding Fractions: Basics and Types

Fractions represent a part of a whole or a collection. They are written as $\frac{a}{b}$, where:

• $a$ is the numerator (parts considered)
• $b$ is the denominator (total parts)

Types of Fractions:

• Proper fractions: Numerator < Denominator (e.g., $\frac{3}{5}$)
• Improper fractions: Numerator ≥ Denominator (e.g., $\frac{7}{4}$)
• Mixed fractions: Combination of whole number and fraction (e.g., $2 \frac{1}{3}$)
• Equivalent fractions: Different fractions representing the same value (e.g., $\frac{1}{2} = \frac{2}{4}$)

Key points:

• Fractions can be simplified by dividing numerator and denominator by their HCF.
• Fractions help in dividing quantities into equal parts.

Decimals: Definition and Place Value System

Decimals are numbers expressed in the base-10 system, showing values less than one using a decimal point.

Decimal parts:

• Tenths (1 digit after decimal)
• Hundredths (2 digits after decimal)
• Thousandths (3 digits after decimal)

For example, in 3.456:

• 3 is the whole number part
• 4 is tenths
• 5 is hundredths
• 6 is thousandths

Converting fractions to decimals:

Divide numerator by denominator. For example:

$$\frac{3}{4} = 3 \div 4 = 0.75$$

Decimals are easier for calculations and measurements in daily life.

Converting Between Fractions and Decimals

Conversion between fractions and decimals is essential for solving problems.

Fraction to decimal: Divide numerator by denominator.

Decimal to fraction:

• Count digits after decimal point (say $n$).
• Write decimal as $\frac{\text{number without decimal}}{10^n}$.
• Simplify the fraction.

Example: Convert 0.625 to fraction:

• 0.625 has 3 digits after decimal.
• Write as $\frac{625}{1000}$.
• Simplify by dividing numerator and denominator by 125:

$$\frac{625}{1000} = \frac{5}{8}$$

This skill helps in comparing and performing operations easily.

Operations on Fractions and Decimals

Mastering operations is key to solving Class 7 problems.

Addition and Subtraction of Fractions:

• Find LCM of denominators.
• Convert fractions to equivalent fractions with LCM as denominator.
• Add or subtract numerators.

Example:

$$\frac{2}{3} + \frac{1}{4}$$

LCM of 3 and 4 is 12.

$$= \frac{2 \times 4}{12} + \frac{1 \times 3}{12} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$

Multiplication of Fractions: Multiply numerators and denominators directly.

$$\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$$

Division of Fractions: Multiply first fraction by reciprocal of second.

$$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$$

Operations on Decimals:

• Align decimal points.
• Perform addition, subtraction as with whole numbers.
• For multiplication, multiply ignoring decimals, then place decimal in product.
• For division, convert divisor to whole number by multiplying numerator and denominator by 10, 100, etc.

These operations form the foundation for solving word problems.

Terminating and Recurring Decimals Explained

Decimals can be classified based on their decimal expansion:

Identifying recurring decimals:

• When division of numerator by denominator does not end, digits repeat.

Converting recurring decimals to fractions:

Example: Convert $0.333...$ to fraction.

Let $x = 0.333...$

Multiply both sides by 10:

$$10x = 3.333...$$

Subtract original equation:

$$10x - x = 3.333... - 0.333...$$ $$9x = 3$$ $$x = \frac{3}{9} = \frac{1}{3}$$

Understanding these decimals is important for precise calculations.

Summary Table: Fractions vs Decimals

Here is a quick comparison to help you remember the differences and similarities:

This table helps clarify when to use each form.