CBSE Class 6 Maths: Decimals Exercise 8.2 - Deep Dive and Solutions

Understanding Decimal Place Values (up to Thousandths)

Just like whole numbers have place values like Ones, Tens, Hundreds, and so on, decimal numbers also have place values for the digits after the decimal point. These represent parts of a whole. The first digit to the right of the decimal point is the 'tenths' place (1/10). The second digit is the 'hundredths' place (1/100). And the third digit is the 'thousandths' place (1/1000).

Let's take an example: Consider the decimal number 123.456.

• 1 is in the Hundreds place (100)
• 2 is in the Tens place (10)
• 3 is in the Ones place (1)
• The decimal point separates the whole number part from the fractional part.
• 4 is in the Tenths place (1/10 or 0.1)
• 5 is in the Hundredths place (1/100 or 0.01)
• 6 is in the Thousandths place (1/1000 or 0.001)

So, 123.456 can be read as "One hundred twenty-three and four hundred fifty-six thousandths." Understanding these place values is the foundation for comparing decimals and converting them to fractions, as you'll see in Exercise 8.2. Remember, as you move further right from the decimal point, the value of each place becomes smaller.

Converting Decimals to Fractions and Vice-Versa

• Example 1: Convert 0.75 to a fraction in its simplest form. Step 1: Count the number of digits after the decimal point. Here, there are two digits (7 and 5). Step 2: Write the decimal number without the decimal point as the numerator. So, 75. Step 3: Write '1' followed by as many zeros as there were decimal places as the denominator. Since there are two decimal places, the denominator is 100. So, the fraction is 75/100. Step 4: Simplify the fraction. Both 75 and 100 are divisible by 25. 75 ÷ 25 = 3, and 100 ÷ 25 = 4. Final answer: 0.75 = 3/4.
• Example 2: Convert 3/20 to a decimal. Step 1: To convert a fraction to a decimal, we need to make the denominator 10, 100, 1000, or a power of 10. Here, the denominator is 20. Step 2: To make 20 into 100, we multiply it by 5. We must do the same to the numerator to keep the fraction equivalent. So, (3 × 5) / (20 × 5) = 15/100. Step 3: Now, write the fraction as a decimal. Since there are two zeros in the denominator (100), there will be two digits after the decimal point. So, 15/100 = 0.15. Final answer: 3/20 = 0.15.
• Example 3: Write 2.008 as a mixed fraction. Step 1: Separate the whole number part and the decimal part. The whole number part is 2. Step 2: Convert the decimal part (0.008) to a fraction. There are three digits after the decimal point (0, 0, 8). So, the denominator will be 1000. The numerator will be 8. So, 0.008 = 8/1000. Step 3: Simplify the fraction 8/1000. Both are divisible by 8. 8 ÷ 8 = 1, and 1000 ÷ 8 = 125. So, 8/1000 = 1/125. Step 4: Combine the whole number and the simplified fraction. The mixed fraction is 2 1/125. Final answer: 2.008 = 2 1/125.

Comparing Decimal Numbers

Comparing decimals is similar to comparing whole numbers, but with an important twist! You need to compare digits based on their place value, starting from the leftmost digit.

Here's how to compare two decimal numbers, step-by-step:

1. Compare the Whole Number Parts: First, look at the digits to the left of the decimal point. The number with the larger whole number part is the greater decimal. For example, 5.23 is greater than 3.98 because 5 is greater than 3.

1. If Whole Parts are Equal, Compare the Tenths: If the whole number parts are the same, move to the digit in the tenths place (the first digit after the decimal point). The number with the larger digit in the tenths place is greater. For example, 4.75 is greater than 4.38 because 7 (in tenths place) is greater than 3.

1. If Tenths are Equal, Compare the Hundredths: If both the whole parts and the tenths are equal, move to the hundredths place (the second digit after the decimal point) and compare them. For example, 0.125 is smaller than 0.130 because 2 (in hundredths place) is smaller than 3.

1. Continue this process: If the hundredths are also equal, compare the thousandths, and so on. A helpful trick is to add zeros to the end of a decimal so that both numbers have the same number of decimal places. For example, to compare 0.5 and 0.45, think of 0.5 as 0.50. Now it's easy to see that 0.50 is greater than 0.45.

Exam Tip: Avoiding Common Decimal Mistakes

When working with decimals, especially in exams, students often make a few common errors. Here are some tips to avoid them:

1. Aligning Decimal Points: Always align the decimal points when adding or subtracting decimals. This ensures that you are adding or subtracting digits with the same place value. Think of 2.5 + 0.15 as 2.50 + 0.15.
2. Comparing Decimals: Do not just compare the 'length' of the decimal part. For instance, 0.5 is NOT smaller than 0.15 just because 5 is a single digit and 15 has two digits. Always compare place by place from left to right, adding trailing zeros if needed (0.50 vs 0.15).
3. Reading Decimals: Make sure you read the decimal part correctly. 4.305 is "four and three hundred five thousandths," not "four point three zero five" or "four point three hundred five." Understanding this helps with conversion.
4. Simplifying Fractions: When converting decimals to fractions, always simplify the resulting fraction to its lowest terms. This is often required and can prevent loss of marks.

Practice Questions with Solutions

• Q: Write the following decimals in words and as a fraction (simplified): a) 0.32 b) 5.009 A: a) For 0.32: Step 1: Write in words. It has two digits after the decimal point, representing hundredths. So, "Thirty-two hundredths." Step 2: Convert to fraction. The digits after the decimal are 32, and there are two decimal places, so the denominator is 100. Fraction = 32/100. Step 3: Simplify the fraction. Both 32 and 100 are divisible by 4. 32 ÷ 4 = 8, and 100 ÷ 4 = 25. Simplified fraction = 8/25. Final answer: a) Thirty-two hundredths; 8/25 b) For 5.009: Step 1: Write in words. The whole part is 5. The decimal part is .009, which is nine thousandths. So, "Five and nine thousandths." Step 2: Convert to fraction. The whole part is 5. For the decimal part, 0.009, the digits are 9, and there are three decimal places, so the denominator is 1000. Fraction = 5 9/1000. Step 3: The fraction 9/1000 cannot be simplified further as 9 (3x3) and 1000 (2x2x2x5x5x5) share no common factors other than 1. Final answer: b) Five and nine thousandths; 5 9/1000
• Q: Convert the following fractions to decimals: a) 7/25 b) 13/8 A: a) For 7/25: Step 1: Make the denominator a power of 10 (10, 100, 1000, etc.). To change 25 to 100, multiply by 4. Step 2: Multiply both numerator and denominator by 4: (7 × 4) / (25 × 4) = 28/100. Step 3: Write as a decimal. Since the denominator is 100 (two zeros), there will be two decimal places. So, 28/100 = 0.28. Final answer: a) 0.28 b) For 13/8: Step 1: To convert 8 to a power of 10, 100 or 1000, we can multiply by 125 to get 1000 (since 8 × 125 = 1000). Step 2: Multiply both numerator and denominator by 125: (13 × 125) / (8 × 125) = 1625/1000. Step 3: Write as a decimal. Since the denominator is 1000 (three zeros), there will be three decimal places. So, 1625/1000 = 1.625. Final answer: b) 1.625
• Q: Compare the following decimal numbers using <, >, or =: a) 0.67 ___ 0.7 b) 2.050 ___ 2.05 A: a) For 0.67 ___ 0.7: Step 1: Align the decimal points and add trailing zeros to make the number of decimal places equal. 0.7 becomes 0.70. Step 2: Compare the whole number parts: Both are 0. Step 3: Compare the tenths place: 6 in 0.67 and 7 in 0.70. Since 6 < 7, 0.67 is less than 0.70. Final answer: a) 0.67 < 0.7 b) For 2.050 ___ 2.05: Step 1: Align the decimal points and add trailing zeros to make the number of decimal places equal. 2.05 can be written as 2.050. Step 2: Compare the whole number parts: Both are 2. Step 3: Compare the tenths place: Both are 0. Step 4: Compare the hundredths place: Both are 5. Step 5: Compare the thousandths place: Both are 0 (after adding a trailing zero to 2.05). Since all digits are equal, the numbers are equal. Final answer: b) 2.050 = 2.05
• Q: Write the place value of the underlined digit in each number: a) 12.345 b) 0.007 A: a) For 12.345: Step 1: Identify the position of the underlined digit '4' relative to the decimal point. Step 2: The first digit after the decimal is tenths, the second is hundredths. So, '4' is in the hundredths place. Final answer: a) Hundredths (or 4/100 or 0.04) b) For 0.007: Step 1: Identify the position of the underlined digit '7' relative to the decimal point. Step 2: The first digit after the decimal is tenths, the second is hundredths, and the third is thousandths. So, '7' is in the thousandths place. Final answer: b) Thousandths (or 7/1000 or 0.007)

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