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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. (80)
Explanation: Step 1: A reduced denominator must contain only (2) and (5). Step 2: (80=2^4\cdot 5), so it is possible. (48), (84), and (98) contain primes like (3) or (7). Step 3: If lowest form is given, check the prime factors of the denominator directly.
Correct answer: B. Non-terminating recurring
Explanation: Step 1: (\frac{99}{9900}) reduces to (\frac{1}{100}) because (9900\div 99=100). Step 2: The reduced denominator is (100=2^2\cdot 5^2), so the decimal terminates. Step 3: With large numbers, check reduction carefully by division.
Correct answer: B. Non-terminating recurring
Explanation: Step 1: (\frac{27}{990}=\frac{3}{110}). Step 2: (110=2\cdot 5\cdot 11), so (11) remains in the denominator. Hence the decimal is non-terminating recurring. Step 3: If a reduced denominator has a prime other than (2) or (5), it will not terminate.
Correct answer: A. (r)
Explanation: Step 1: The reduced denominator has only powers of (2) and (5). Step 2: The number of decimal places equals the larger exponent. Since (r>s), the larger exponent is (r). Step 3: Remember (\max(r,s)) for decimal places.
Correct answer: A. (8)
Explanation: Step 1: (0.375=\frac{375}{1000}). Step 2: Dividing numerator and denominator by (125) gives (\frac{3}{8}). So the denominator is (8). Step 3: Always reduce after converting a decimal to a fraction.
Correct answer: B. (0.\overline{25})
Explanation: Step 1: (0.\overline{25}) has a repeating block, so it is rational. Step 2: It is not terminating because the decimal does not end. (\sqrt{2}) and (\pi) are irrational. Step 3: Rational numbers can be terminating or non-terminating recurring.
Correct answer: C. (7)
Explanation: Step 1: (56=2^3\cdot 7). Step 2: For a terminating decimal, (7) must not remain in the reduced denominator. Therefore (m) must contain (7). Step 3: Cancel all denominator primes other than (2) and (5).
Correct answer: C. (5)
Explanation: Step 1: (3125=5^5). Step 2: The denominator has power (0) of (2) and power (5) of (5). So the decimal terminates after (5) places. Step 3: Remembering (3125=5^5) helps in quick factorisation.
Correct answer: A. (10x=2.1818\ldots), (1000x=218.1818\ldots)
Explanation: Step 1: In (x=0.21818\ldots), the non-repeating part is (2) and the repeating part is (18). Step 2: First use (10x=2.1818\ldots), then (1000x=218.1818\ldots) so the recurring parts align. Step 3: Choose powers of (10) based on the lengths of the non-repeating and repeating parts.
Correct answer: B. (6)
Explanation: Step 1: The denominator is (2^6\cdot 5^3), and the fraction is in lowest form because (37) does not cancel. Step 2: The larger exponent is (6), so the decimal terminates after (6) places. Step 3: Do not add the exponents for decimal places.
Correct answer: C. (\sqrt{5})
Explanation: Step 1: Rational numbers have either terminating or non-terminating recurring decimals. Step 2: (\sqrt{5}) is irrational, so its decimal is non-terminating non-recurring. Step 3: To identify non-terminating non-recurring decimals, look for irrational numbers.
Correct answer: B. Terminating because (3^0=1)
Explanation: Step 1: (3^0=1), so there is actually no factor (3) in the denominator. Step 2: The denominator is (2^2\cdot 5^3), containing only (2) and (5). Hence the decimal terminates. Step 3: Do not get confused by a zero exponent.
Correct answer: A. A denominator with six (9)'s
Explanation: Step 1: The repeating block (142857) has (6) digits. Step 2: For a purely recurring decimal, before reducing, the denominator has the same number of (9)'s. So it is (999999). Step 3: The number of repeating digits tells the number of (9)'s.
Correct answer: B. (4)
Explanation: Step 1: (16=2^4), so (2^4) cancels from the denominator. Step 2: The reduced denominator is (2^3\cdot 5^4). The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Include powers hidden in the numerator during cancellation.
Correct answer: B. Rational and non-terminating recurring
Explanation: Step 1: The block (36) repeats in the decimal. Step 2: A recurring decimal is always rational, but it is not terminating. So it is rational and non-terminating recurring. Step 3: When a repeating block appears, identify the number as rational.
Correct answer: C. It is non-terminating recurring
Explanation: Step 1: The denominator has (7), and the numerator (1) cannot cancel it. Step 2: The reduced denominator has (7) besides (2) and (5), so the decimal is non-terminating recurring. Step 3: Having (2) and (5) in the denominator does not guarantee termination.
Correct answer: B. (4p)
Explanation: Step 1: We need (10^5=2^5\cdot 5^5). Step 2: The denominator (2^3\cdot 5^5) lacks (2^2). So multiply numerator and denominator by (2^2=4). Hence (N=4p). Step 3: To make (10^k), multiply by the missing prime power.
Correct answer: B. Non-terminating recurring
Explanation: Step 1: (35=5\cdot 7). Step 2: The factor (5) and one (7) cancel, but one (7) remains. The reduced denominator is (2^2\cdot 7). So the decimal is non-terminating recurring. Step 3: After partial cancellation, check what factor remains.
Correct answer: B. (625)
Explanation: Step 1: (0.0048=\frac{48}{10000}). Step 2: The greatest common factor of (48) and (10000) is (16), so (\frac{48}{10000}=\frac{3}{625}). The denominator is (625). Step 3: Even for small decimals, reduce to lowest form.
Correct answer: C. The denominator has at least one prime factor other than (2) and (5)
Explanation: Step 1: A non-terminating decimal of a rational number is recurring. Step 2: This happens when the reduced denominator has at least one prime factor other than (2) and (5). So option (C) is correct. Step 3: (2) or (5) may also be present, but some other prime must remain.
Correct answer: B. (\max(a,b))
Explanation: Step 1: To make the denominator (10^k=2^k5^k), both exponents must be made equal. Step 2: The required (k) equals the larger exponent. So the number of decimal places is (\max(a,b)). Step 3: This rule is frequently tested in terminating decimal questions.
Correct answer: A. Terminating with (2) decimal places
Explanation: Step 1: (44=2^2\cdot 11). Step 2: After cancellation, the denominator becomes (2\cdot 5=10). So the decimal terminates after (1) place. Since that exact statement is not listed, the given options contain an issue. Step 3: Complete your calculation before trusting the options.
Correct answer: A. Terminating with (1) decimal place
Explanation: Step 1: (44=2^2\cdot 11). Step 2: Cancelling (2^2\cdot 11) from (2^3\cdot 5\cdot 11) leaves (2\cdot 5=10). So the decimal terminates after (1) place. Step 3: Count decimal places only after complete cancellation.
Correct answer: B. Rational and non-terminating recurring
Explanation: Step 1: The digit (6) repeats, so the decimal is recurring. Step 2: Every recurring decimal is rational, but this one does not terminate. Hence it is rational and non-terminating recurring. Step 3: A bar over digits shows the repeating part.
Correct answer: B. (4)
Explanation: Step 1: (81=3^4), so (3^4) cancels completely from the denominator. Step 2: The reduced denominator is (2^4\cdot 5^2). The larger exponent is (4), so the decimal terminates after (4) places. Step 3: First check cancellation of prime factors other than (2) and (5).
