Chapter 2: Fractions and Decimals

Introduction to Fractions and Decimals

Fractions and decimals are two different ways of representing numbers that are not whole. This chapter introduces students to the concepts of fractions and decimals, their types, how to perform operations (addition, subtraction, multiplication, division) with them, and how to convert between fractions and decimals. These concepts are foundational for understanding more advanced topics in mathematics.

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1. Fractions

A fraction represents a part of a whole. It is written in the form numerator/denominator, where:

• Numerator: The number of parts we have.
• Denominator: The total number of equal parts the whole is divided into.

Types of Fractions

1. Proper Fraction: The numerator is less than the denominator (e.g., 3/5, 1/2).
2. Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 7/4, 5/5).
3. Mixed Fraction: A combination of a whole number and a proper fraction (e.g., 2 ⅓, 1 ½).
4. Like Fractions: Fractions with the same denominator (e.g., 2/7 and 5/7).
5. Unlike Fractions: Fractions with different denominators (e.g., 3/4 and 2/5).

Equivalent Fractions

Fractions that represent the same value but look different are called equivalent fractions. To find them:

• Multiply or divide both the numerator and denominator by the same number.
• Example: 1/2 = 2/4 = 3/6 (multiply numerator and denominator by 2, then 3).

Simplest Form of a Fraction

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

• Example: 6/8 → Divide by 2 → 3/4 (simplest form).

Comparing Fractions

• Like Fractions: Compare the numerators directly (e.g., 3/7 < 5/7).
• Unlike Fractions: Convert to a common denominator using the Least Common Multiple (LCM), then compare.
  • Example: Compare 2/3 and 3/4.
    • LCM of 3 and 4 is 12.
    • 2/3 = 8/12, 3/4 = 9/12.
    • 8/12 < 9/12, so 2/3 < 3/4.

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2. Operations on Fractions

Addition and Subtraction

• Like Fractions: Add or subtract the numerators and keep the denominator the same.
  • Example: 2/5 + 3/5 = 5/5 = 1.
  • Example: 7/8 – 3/8 = 4/8 = 1/2 (simplified).
• Unlike Fractions: Convert to like fractions using LCM, then add or subtract.
  • Example: 1/3 + 1/4.
    • LCM of 3 and 4 is 12.
    • 1/3 = 4/12, 1/4 = 3/12.
    • 4/12 + 3/12 = 7/12.

Multiplication

• Multiply the numerators together and the denominators together.
• Simplify the result if possible.
• Example: 2/3 × 3/5 = (2 × 3)/(3 × 5) = 6/15 = 2/5 (simplified).

Division

• Multiply the first fraction by the reciprocal of the second fraction.
• Reciprocal: Flip the numerator and denominator (e.g., reciprocal of 2/3 is 3/2).
• Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = (3 × 5)/(4 × 2) = 15/8 = 1 7/8.

Mixed Fractions in Operations

• Convert mixed fractions to improper fractions before performing operations.
• Example: 2 ⅓ + 1 ½.
  • 2 ⅓ = 7/3, 1 ½ = 3/2.
  • LCM of 3 and 2 is 6.
  • 7/3 = 14/6, 3/2 = 9/6.
  • 14/6 + 9/6 = 23/6 = 3 5/6.

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3. Decimals

A decimal number has a whole number part and a fractional part separated by a decimal point (e.g., 4.75).

• The digits after the decimal point represent fractions of 10, 100, 1000, etc.

Place Value in Decimals

• Example: In 5.342:
  • 5: Units (whole number).
  • 3: Tenths (1/10).
  • 4: Hundredths (1/100).
  • 2: Thousandths (1/1000).

Types of Decimals

1. Terminating Decimals: Finite digits after the decimal point (e.g., 0.25, 1.5).
2. Recurring Decimals: Digits repeat after the decimal point (e.g., 0.333…, 0.1212…).

Comparing Decimals

• Compare the whole number part first.
• If equal, compare the decimal digits step-by-step.
• Example: 0.45 < 0.5 (5 > 4 in tenths place).

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4. Operations on Decimals

Addition and Subtraction

• Align the decimal points and add/subtract as with whole numbers.
• Example: 2.34 + 1.6.
  • Write as: 2.34 + 1.60.
  • Add: 3.94.
• Example: 5.7 – 2.38.
  • Write as: 5.70 – 2.38.
  • Subtract: 3.32.

Multiplication

• Multiply as whole numbers, then adjust the decimal point based on the total number of decimal places in the factors.
• Example: 2.5 × 1.2.
  • 25 × 12 = 300.
  • Total decimal places = 2 (1 from 2.5, 1 from 1.2).
  • 300 → 3.00 = 3.0.

Multiplication by Powers of 10

• Move the decimal point to the right by the number of zeros.
• Example: 3.45 × 100 = 345 (decimal moves 2 places right).

Division

• Convert to fractions or adjust decimals to whole numbers by shifting the decimal point.
• Example: 5.6 ÷ 2.
  • 5.6/2 = 2.8 (direct division).
• Example: 0.75 ÷ 0.25.
  • Shift decimals: 75 ÷ 25 = 3.

Division by Powers of 10

• Move the decimal point to the left by the number of zeros.
• Example: 45.6 ÷ 10 = 4.56 (decimal moves 1 place left).

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5. Conversion Between Fractions and Decimals

Fraction to Decimal

• Divide the numerator by the denominator.
• Example: 3/4 = 3 ÷ 4 = 0.75 (terminating).
• Example: 1/3 = 1 ÷ 3 = 0.333… (recurring).

Decimal to Fraction

• Write the decimal as a fraction with a denominator as a power of 10, then simplify.
• Example: 0.25 = 25/100 = 1/4.
• Example: 0.6 = 6/10 = 3/5.

Recurring Decimal to Fraction

• Use algebra to convert.
• Example: 0.333… = x.
  • 10x = 3.333…
  • 10x – x = 3.333… – 0.333…
  • 9x = 3 → x = 3/9 = 1/3.

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Key Points to Remember

• Always simplify fractions after operations.
• Align decimal points for addition and subtraction.
• Adjust decimal places correctly during multiplication and division.
• Practice converting between fractions and decimals for better understanding.

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Examples for Practice

1. Add: 2/5 + 3/10.
   • LCM = 10, 2/5 = 4/10.
   • 4/10 + 3/10 = 7/10.
2. Multiply: 1.5 × 2.4.
   • 15 × 24 = 360.
   • Decimal places = 2, so 3.60 = 3.6.
3. Convert 0.625 to a fraction.
   • 0.625 = 625/1000 = 5/8.

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