Fractions and Decimals
In this chapter, students will deepen their understanding of fractions and decimals, learn how to perform operations with them, and apply these concepts to real-life situations.
1. Introduction to Fractions
- A fraction represents a part of a whole.
- It is written as a/b, where a is the numerator and b is the denominator.
- Examples: 1/2, 3/4, 7/8
2. Types of Fractions
- Proper Fractions: Numerator is less than denominator (e.g., 3/5).
- Improper Fractions: Numerator is greater than or equal to denominator (e.g., 7/4).
- Mixed Fractions: Combination of a whole number and a proper fraction (e.g., 2 1/3).
- Like Fractions: Fractions with the same denominator.
- Unlike Fractions: Fractions with different denominators.
- Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4).
3. Operations with Fractions
- Addition and Subtraction: Like fractions can be added/subtracted directly. Unlike fractions need to be converted to like fractions first.
- Multiplication: Multiply numerators and denominators.
- Division: Invert the second fraction and multiply.
Example: 2/3 + 1/6 = 4/6 + 1/6 = 5/6
4. Introduction to Decimals
- Decimals are another way to represent fractions, especially those with denominators of 10, 100, 1000, etc.
- Decimal point separates the whole number part from the fractional part.
- Examples: 0.5, 2.75, 3.06
5. Place Value in Decimals
- Tenths (0.1), Hundredths (0.01), Thousandths (0.001), etc.
- Example: In 3.47, 3 is the whole number, 4 is tenths, 7 is hundredths.
6. Operations with Decimals
- Addition and Subtraction: Line up the decimal points before performing the operation.
- Multiplication: Multiply as whole numbers, then place the decimal point in the product.
- Division: Move the decimal point to make the divisor a whole number, then divide.
Example: 0.6 + 0.25 = 0.85
7. Converting Between Fractions and Decimals
- To convert a fraction to a decimal, divide the numerator by the denominator.
- To convert a decimal to a fraction, write it as a fraction with denominator 10, 100, etc., and simplify.
- Example: 3/4 = 0.75; 0.2 = 2/10 = 1/5
8. Comparing and Ordering Fractions and Decimals
- Convert to like fractions or decimals to compare.
- Arrange in ascending or descending order as required.
9. Applications in Real Life
- Money (₹2.50, ₹0.75)
- Measurements (1.5 meters, 0.25 liters)
- Shopping, recipes, and more
10. Common Mistakes
- Not aligning decimal points in addition/subtraction.
- Forgetting to convert unlike fractions before adding/subtracting.
- Misplacing the decimal point in multiplication/division.
11. Practice Exercises
- Add: 3/8 + 1/8
- Subtract: 0.9 - 0.45
- Multiply: 2/5 × 3/4
- Convert 0.6 to a fraction.
- Arrange in ascending order: 0.7, 3/4, 0.65
12. Summary
- Fractions and decimals are ways to represent parts of a whole.
- Learn to convert, compare, and perform operations with them.
- They are used in daily life for money, measurement, and more.