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4.1 A Quick Recap of Decimals

Decimals are an extension of the Indian place value system to represent fractions whose denominators are powers of 10, such as 1/10, 1/100, 1/1000, etc. They allow us to express parts of a whole in a compact and precise way. For example, the decimal number 27.53 represents 2 tens, 7 units (ones), 5 tenths, and 3 hundredths. This means 27 whole units plus 53 hundredths. Decimals are closely related to fractions and can be converted back and forth. The chapter begins by recalling how decimals represent sums of decimal fractions and how these relate to place value. It also introduces the idea that operations like multiplication and division with decimals are natural extensions of operations with whole numbers. An activity is presented where one student says a fraction and the other converts it to its decimal equivalent, emphasizing denominators of 10, 100, 1000, etc. Another example involves converting weights of spices from grams to kilograms, expressing them both as fractions and decimals, reinforcing the connection between fractions and decimals. The section also introduces a simple rule for dividing any number by 10, 100, 1000, etc., by moving the decimal point to the left by as many places as there are zeros in the divisor. This is demonstrated through examples such as 123 ÷ 10 = 12.3 and 678 ÷ 1000 = 0.678. This foundational understanding sets the stage for learning decimal multiplication and division in the following sections.

• Decimals extend the Indian place value system to represent fractions with denominators 10, 100, 1000, etc.
• Example: 27.53 = 2 tens + 7 units + 5 tenths + 3 hundredths.
• Decimals can be expressed as sums of decimal fractions, e.g., 0.254 = 0.2 + 0.05 + 0.004.
• Dividing by 10, 100, 1000 moves the decimal point left by the number of zeros.
• Operations on decimals extend naturally from operations on whole numbers.
• Conversion between fractions with denominators 10, 100, 1000 and decimals is straightforward.
• 📌 Decimal: A number expressed in the scale of tens, hundreds, thousandths, etc., using a decimal point.
• 📌 Place Value System: A system where the position of a digit determines its value.
• 📌 Decimal Fraction: A fraction with denominator as a power of 10.

4.2 Decimal Multiplication

Decimal multiplication involves multiplying numbers that have digits after the decimal point. The process is an extension of multiplication of whole numbers and fractions. The key idea is to first ignore the decimal points and multiply the numbers as whole numbers, then place the decimal point in the product based on the total number of decimal places in the factors. For example, multiplying 9.5 by 5 can be done by converting 9.5 to the fraction 95/10 and 5 to 5/1, multiplying numerators and denominators to get 475/10 = 47.5. Another example is multiplying 12.5 km per litre by 7.5 litres to find distance covered: (125/10) × (75/10) = 9375/100 = 93.75 km. The chapter explains that the product of two decimals can be greater than, less than, or between the numbers multiplied depending on their values relative to 1. For instance, 0.25 × 0.8 = 0.2 is less than both numbers, while 2.25 × 8 = 18 is greater than both. The number of decimal places in the product equals the sum of decimal places in the multiplicand and multiplier. This is illustrated through examples such as 5.7 × 13.35 = 75.81. A general rule is framed: multiply the numbers ignoring decimals, then place the decimal point in the product so that the number of decimal digits equals the sum of decimal digits in the numbers multiplied. This section also includes real-world problems like calculating the area of a rectangle with decimal sides and distance walked by Ajay in a week, reinforcing practical applications of decimal multiplication.

• Multiply decimals by first ignoring decimal points and treating numbers as whole numbers.
• Place decimal point in product so that decimal places equal sum of decimal places in factors.
• Product of decimals can be greater than, less than, or between the numbers multiplied.
• Decimal multiplication corresponds to multiplication of fractions with denominators as powers of 10.
• Real-world problems include cost calculation, distance, and area using decimal multiplication.
• Number of decimal digits in product = sum of decimal digits in multiplicand and multiplier.
• 📌 Multiplier: The number by which another number is multiplied.
• 📌 Multiplicand: The number that is to be multiplied.
• 📌 Decimal Places: Number of digits to the right of the decimal point.