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SubjectsMathematics
परिमेय संख्याओं का दशमलव प्रसार
In Class 10 Mathematics, this topic from the Real Numbers chapter explains how rational numbers are represented in decimal form. Students learn to distinguish terminating decimals from non-terminating recurring decimals and determine the type of expansion by reducing a fraction to its simplest form and examining the prime factors of its denominator. The topic also develops accuracy in long division, fraction-to-decimal conversion, and understanding the relationship between rational numbers and their decimal representations.
Correct answer: B. (\frac{1}{2})
Explanation: Step 1: (0.5) can be written as (\frac{5}{10}). Step 2: On simplifying, it becomes (\frac{1}{2}). Step 3: Exam tip: Convert a terminating decimal into a fraction using place value first.
Correct answer: C. (\frac{7}{8})
Explanation: Step 1: For a terminating decimal, the denominator must have only (2) and (5) as prime factors. Step 2: Since (8=2^3), (\frac{7}{8}) terminates. Step 3: Exam tip: Always prime-factorise the denominator first.
Correct answer: A. When (q) has only factors (2) and (5)
Explanation: Step 1: The rule is applied only after reducing the fraction to lowest form. Step 2: If the denominator (q) is of the form (2^m5^n), the decimal terminates. Step 3: Exam tip: Do not decide before simplifying the fraction.
Correct answer: C. (3)
Explanation: Step 1: (40=2^3\times5). Step 2: The larger power of (2) and (5) is (3), so the decimal ends after (3) places. Step 3: Exam tip: Use the larger exponent to count decimal places.
Correct answer: A. Terminating
Explanation: Step 1: (\frac{3}{6}) simplifies to (\frac{1}{2}). Step 2: The denominator is (2), so the decimal ends at (0.5). Step 3: Exam tip: Do not judge from the original denominator before reducing.
Correct answer: A. It is terminating
Explanation: Step 1: (\frac{7}{28}=\frac{1}{4}). Step 2: Since (4=2^2), the decimal terminates. Step 3: Exam tip: The lowest form gives the correct decision.
Correct answer: A. Because (12) has factor (3)
Explanation: Step 1: (\frac{5}{12}) is already in lowest form. Step 2: (12=2^2\times3), and the factor (3) prevents termination. Step 3: Exam tip: A factor other than (2) or (5) gives a recurring decimal.
Correct answer: A. Either terminating or non-terminating recurring
Explanation: Step 1: A rational number can be written as (\frac{p}{q}). Step 2: Its decimal either terminates or repeats a block of digits. Step 3: Exam tip: Non-terminating non-recurring decimals are linked with irrational numbers.
Correct answer: A. (\frac{1}{8})
Explanation: Step 1: (0.125=\frac{125}{1000}). Step 2: Simplifying (\frac{125}{1000}) gives (\frac{1}{8}). Step 3: Exam tip: For three decimal places, first use denominator (1000).
Correct answer: A. (0.08)
Explanation: Step 1: Multiply denominator (25) by (4) to make (100). Step 2: (\frac{2}{25}=\frac{8}{100}=0.08). Step 3: Exam tip: Make the denominator (10), (100), or (1000) for quick conversion.
Correct answer: B. (0.4)
Explanation: Step 1: (\frac{6}{15}) simplifies to (\frac{2}{5}). Step 2: (\frac{2}{5}=\frac{4}{10}=0.4). Step 3: Exam tip: Simplifying first makes decimal conversion faster.
Correct answer: B. (2)
Explanation: Step 1: (20=2^2\times5). Step 2: The larger exponent is (2), so the decimal terminates after (2) places. Step 3: Exam tip: The actual decimal (0.45) confirms the same result.
Correct answer: B. (0.056)
Explanation: Step 1: (125\times8=1000). Step 2: (\frac{7}{125}=\frac{56}{1000}=0.056). Step 3: Exam tip: When making the denominator (1000), keep three decimal places carefully.
Correct answer: C. (4)
Explanation: Step 1: (16=2^4). Step 2: The denominator has only (2), with exponent (4), so the decimal ends after (4) places. Step 3: Exam tip: (2^4) usually points to checking up to (4) decimal places.
Correct answer: C. (100)
Explanation: Step 1: For a terminating decimal, the denominator must have only (2) and (5). Step 2: (100=2^2\times5^2), so it is suitable. Step 3: Exam tip: Denominators like (10), (100), and (1000) give terminating decimals.
Correct answer: A. Because (6) has factor (3)
Explanation: Step 1: (\frac{1}{6}) is in lowest form. Step 2: (6=2\times3), and factor (3) makes the decimal non-terminating recurring. Step 3: Exam tip: An even denominator does not always mean termination.
Correct answer: B. (0.0064)
Explanation: Step 1: (625\times16=10000). Step 2: (\frac{4}{625}=\frac{64}{10000}=0.0064). Step 3: Exam tip: Count zeros carefully while placing the decimal point.
Correct answer: B. (0.74)
Explanation: Step 1: Multiply denominator (50) by (2) to make (100). Step 2: (\frac{37}{50}=\frac{74}{100}=0.74). Step 3: Exam tip: When changing the denominator, multiply the numerator by the same number.
Correct answer: B. It is a non-terminating recurring rational number
Explanation: Step 1: In (0.333\ldots), the digit (3) repeats. Step 2: A recurring decimal is rational, so it can be written as a fraction. Step 3: Exam tip: A repeating digit is a sign of a rational number.
Correct answer: B. (1.5)
Explanation: Step 1: Dividing (\frac{21}{14}) by (7) gives (\frac{3}{2}). Step 2: (\frac{3}{2}=1.5). Step 3: Exam tip: Reduce larger fractions before converting to decimals.
Correct answer: A. Terminating
Explanation: Step 1: (\frac{18}{45}) simplifies by (9) to (\frac{2}{5}). Step 2: The denominator is (5), so the decimal terminates. Step 3: Exam tip: Do not get misled by denominator (45); check the lowest form.
Correct answer: A. It terminates because the lowest form is (\frac{2}{5})
Explanation: Step 1: (\frac{22}{55}) simplifies to (\frac{2}{5}). Step 2: (\frac{2}{5}=0.4), so the decimal terminates. Step 3: Exam tip: Reduce the fraction before applying the denominator rule.
Correct answer: C. (4)
Explanation: Step 1: (80=2^4\times5). Step 2: The larger exponent is (4), so the decimal has (4) places. Step 3: Exam tip: You can also check by writing (\frac{3}{80}=0.0375).
Correct answer: B. (0.727272\ldots)
Explanation: Step 1: In (0.727272\ldots), the block (72) repeats. Step 2: A repeating decimal is rational. Step 3: Exam tip: Do not only see that a decimal is long; check whether a fixed pattern repeats.
Correct answer: B. (3)
Explanation: Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The exponents are (3) and (2), and the larger one is (3). Step 3: Exam tip: Maximum decimal places equal the larger exponent.
