Chapter 2: Fractions and Decimals
Overview of the Chapter
Introduction to Fractions and Decimals
“Fractions and Decimals” in Class 7 Mathematics builds on the foundational concepts introduced in earlier classes. It delves deeper into operations, properties, and applications of fractions and decimals. Understanding these concepts is crucial for progressing in mathematics and solving real-world problems involving parts of a whole and decimal numbers.
Fractions
Types of Fractions
Proper Fractions: A fraction where the numerator (top number) is less than the denominator (bottom number). For example, in the fraction 3/4, 3 is less than 4, so it is a proper fraction.
Improper Fractions: A fraction where the numerator is greater than or equal to the denominator. For example, in the fraction 5/4, 5 is greater than 4, making it an improper fraction.
Mixed Fractions: A combination of a whole number and a proper fraction. For example, 1 1/2 is a mixed fraction, combining the whole number 1 and the proper fraction 1/2.
Simplifying Fractions
Equivalent Fractions
Definition: Fractions that represent the same value even though they have different numerators and denominators are called equivalent fractions.
Example: 1/2 is equivalent to 2/4, 3/6, etc., because when simplified, all these fractions equal 1/2.
Operations with Fractions
Addition and Subtraction: To add or subtract fractions, first convert them to like fractions by finding a common denominator. Then add or subtract the numerators while keeping the denominator the same.
Example: To add 1/4 and 1/6, find the common denominator (12). Convert each fraction: 1/4 = 3/12 and 1/6 = 2/12. Then add: 3/12 + 2/12 = 5/12.
Multiplication: Multiply the numerators to get the new numerator and the denominators to get the new denominator.
Example: Multiply 2/3 by 3/4: (2 × 3) / (3 × 4) = 6/12, which simplifies to 1/2.
Division: To divide by a fraction, multiply by its reciprocal (invert the numerator and denominator of the divisor).
Example: Divide 2/3 by 4/5: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12, which simplifies to 5/6.
Decimals
Understanding Decimals
Definition: Decimals are numbers with a whole part and a fractional part separated by a decimal point. Each position to the right of the decimal point represents a power of ten (tenths, hundredths, thousandths, etc.).
Place Value: The value of a digit depends on its position relative to the decimal point. For example, in the number 3.456, 4 is in the tenths place (0.4), 5 is in the hundredths place (0.05), and 6 is in the thousandths place (0.006).
Converting Fractions to Decimals
Method: To convert a fraction to a decimal, divide the numerator by the denominator using long division or a calculator.
Example: Convert 3/4 to a decimal: 3 ÷ 4 = 0.75.
Operations with Decimals
Addition and Subtraction: Align the decimal points of the numbers vertically and then add or subtract as you would with whole numbers.
Example: Add 2.5 and 3.75:
2.50
+ 3.75
------
6.25
Multiplication: Multiply the numbers ignoring the decimal points. Count the total number of decimal places in the factors. Place the decimal point in the product so that it has that many decimal places.
Example: Multiply 2.5 by 0.4:
2.5 (1 decimal place)
x 0.4 (1 decimal place)
------
1000
100
------
1.000
The product is 1.000, which simplifies to 1.0 or 1.
Division: Move the decimal point in the divisor to the right to make it a whole number, and move the decimal point in the dividend the same number of places to the right. Then divide as with whole numbers.
Example: Divide 6.4 by 0.2:
64 ÷ 2 = 32
The result is 32.
Converting Decimals to Fractions
Method: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places. Then simplify the fraction.
Example: Convert 0.75 to a fraction: 0.75 = 75/100. Simplify by dividing by the GCD, which is 25: 75 ÷ 25 / 100 ÷ 25 = 3/4.
Comparing Fractions and Decimals
Methods to Compare
Fractions: To compare fractions, convert them to like fractions with a common denominator or convert them to decimals and compare the decimal values.
Example: Compare 3/4 and 5/8 by converting to decimals: 3/4 = 0.75 and 5/8 = 0.625. Since 0.75 is greater than 0.625, 3/4 > 5/8.
Decimals: Align the decimal points and compare digit by digit from left to right.
Example: Compare 0.45 and 0.456. Since the tenths place is the same, compare the hundredths place. Since 5 is less than 6, 0.45 < 0.456.
Applications of Fractions and Decimals
Real-life Applications
Measurements: Fractions and decimals are used in measuring lengths, weights, and volumes. For example, a piece of fabric might be 1.5 meters long, and a recipe might require 0.75 cups of sugar.
Money: Financial transactions and pricing often involve decimals. For example, an item might cost $2.99, and a bank account might have a balance of $123.45.
Data Representation: Fractions and decimals are used in statistics, graphs, and probabilities. For example, a survey might show that 25% (0.25) of people prefer a certain product.
Conclusion
Summary of the Chapter
The chapter “Fractions and Decimals” provides a comprehensive understanding of fractions and decimals, their properties, and operations. It explains how to convert between fractions and decimals and perform basic arithmetic operations with them. The chapter emphasizes the importance of these concepts in real-life applications.
Additional Information
Key Terms and Concepts
Fraction: A part of a whole represented as a/b.
Decimal: A number with a whole part and a fractional part separated by a decimal point.
Proper Fraction: A fraction where the numerator is less than the denominator.
Improper Fraction: A fraction where the numerator is greater than or equal to the denominator.
Mixed Fraction: A combination of a whole number and a proper fraction.
Equivalent Fraction: Different fractions that represent the same value.
Simplest Form: A fraction reduced to its lowest terms.
Important Examples and Cases
Addition and Subtraction:
Example: 1/4 + 1/6 = 5/12, 0.25 + 0.75 = 1.00.
Multiplication and Division:
Example: 2/3 × 3/4 = 1/2, 2.5 × 0.4 = 1.00.
Notable Observations
Fractions and Decimals in Real Life: Understanding fractions and decimals is crucial for dealing with measurements, financial transactions, and data representation.
Properties of Fractions and Decimals: Knowing the properties helps in simplifying mathematical operations involving fractions and decimals.
FAQs
Proper fractions are fractions where the numerator is less than the denominator (e.g., 3/4).
Improper fractions are fractions where the numerator is greater than or equal to the denominator (e.g., 5/4).
Mixed fractions are combinations of a whole number and a proper fraction (e.g., 1 1/2).
Simplify a fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Equivalent fractions are different fractions that represent the same value (e.g., 1/2 is equivalent to 2/4).
Convert the fractions to like fractions with a common denominator, then add the numerators while keeping the denominator the same.
Multiply the numerators to get the new numerator and the denominators to get the new denominator (e.g., 2/3 × 3/4 = 6/12 = 1/2).
To divide by a fraction, multiply by its reciprocal (e.g., 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6).
A decimal is a number with a whole part and a fractional part separated by a decimal point (e.g., 2.5).
The position of a digit relative to the decimal point determines its value (e.g., in 3.456, 4 is in the tenths place, 5 is in the hundredths place, 6 is in the thousandths place).
Divide the numerator by the denominator (e.g., 3/4 = 0.75).
Align the decimal points of the numbers vertically and then add as you would with whole numbers (e.g., 2.5 + 3.75 = 6.25).
Multiply the numbers ignoring the decimal points, then place the decimal point in the product so that it has the same number of decimal places as the total in the factors (e.g., 2.5 × 0.4 = 1.00).
Move the decimal point in the divisor to the right to make it a whole number, and move the decimal point in the dividend the same number of places to the right. Then divide as with whole numbers (e.g., 6.4 ÷ 0.2 = 64 ÷ 2 = 32).
Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places, and then simplify (e.g., 0.75 = 75/100 = 3/4).
Convert them to like fractions with a common denominator or convert them to decimals and compare the decimal values (e.g., 3/4 = 0.75 and 5/8 = 0.625, so 3/4 > 5/8).
Align the decimal points and compare digit by digit from left to right (e.g., 0.45 < 0.456).
Fractions are used in measurements (e.g., 1.5 meters), recipes (e.g., 0.75 cups of sugar), and various other contexts.
Decimals are used in financial transactions (e.g., $2.99), measurements (e.g., 123.45 cm), and data representation.
Fractions are used to represent parts of a whole in lengths, weights, and volumes (e.g., 1/2 liter).
Decimals are used to represent money, showing parts of a dollar (e.g., $12.45).
Divide both the numerator and the denominator by their GCD, which is 4: 8 ÷ 4 / 12 ÷ 4 = 2/3.
The decimal form of 1/8 is 0.125.
Write 0.4 as 4/10 and then simplify: 4/10 = 2/5.
The fraction form of 0.125 is 125/1000, which simplifies to 1/8.
Chapter 2: Fractions and Decimals MCQs
1. Which of the following is a proper fraction?
2. How do you simplify the fraction 8/12?
3. What is 3/4 in decimal form?
4. What is the product of 2/3 and 3/4?
5. Which of the following is the decimal form of 1/8?
6. How do you convert 0.4 to a fraction?
7. What is the result of 2.5 × 0.4?
8. How do you divide 6.4 by 0.2?
9. What is the sum of 1/4 and 1/6?
10. Which of the following represents 0.75 as a fraction?
11. How do you compare 3/4 and 5/8?
12. What is the equivalent fraction of 2/3 with a denominator of 9?
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