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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. It is non-terminating recurring because the reduced denominator still has (7) with (5)
Explanation: Step 1: (\frac{63}{245}) simplifies by (7) to (\frac{9}{35}). Step 2: Since (35=5\times7), factor (7) remains and the decimal will not terminate. Step 3: Exam tip: Having (5) in the denominator is not enough; only (2) and (5) should remain.
Correct answer: A. (0.09375)
Explanation: Step 1: (32=2^5), so the decimal ends after (5) places. Step 2: (32\times3125=100000), so (\frac{3}{32}=\frac{9375}{100000}=0.09375). Step 3: Exam tip: Count the total decimal places carefully while placing the decimal point.
Correct answer: A. (\frac{1}{16})
Explanation: Step 1: (0.0625=\frac{625}{10000}). Step 2: Simplifying by (625) gives (\frac{1}{16}). Step 3: Exam tip: Remembering common decimals like (0.0625), (0.125), and (0.25) is useful.
Correct answer: B. At the fourth place
Explanation: Step 1: The denominator (2^4\times5^4=(2\times5)^4=10^4). Step 2: So the decimal terminates after (4) places. Step 3: Exam tip: Equal powers of (2) and (5) directly form a power of (10).
Correct answer: C. (4)
Explanation: Step 1: (2000=2^4\times5^3). Step 2: The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Exam tip: Imagine converting the denominator to (10000); the number of places becomes clear.
Correct answer: B. (0.135135135\ldots)
Explanation: Step 1: In (0.135135135\ldots), the block (135) repeats. Step 2: A recurring decimal is rational, but it does not terminate. Step 3: Exam tip: If a non-terminating decimal has a regular repeated block, treat it as rational.
Correct answer: B. (2)
Explanation: Step 1: (\frac{18}{225}) simplifies by (9) to (\frac{2}{25}). Step 2: Since (25=5^2), the decimal terminates after (2) places. Step 3: Exam tip: The original denominator had (3^2), but it disappeared after simplification.
Correct answer: C. (\frac{44}{242})
Explanation: Step 1: (\frac{44}{242}) simplifies by (22) to (\frac{2}{11}). Step 2: The denominator (11) is not made of (2) and (5), so the decimal is non-terminating recurring. Step 3: Exam tip: Simplify every option before making the final choice.
Correct answer: C. (0.48) is terminating and (0.\overline{48}) is non-terminating recurring
Explanation: Step 1: (0.48) stops after two decimal places. Step 2: In (0.\overline{48}), the block (48) repeats, so it is non-terminating recurring. Step 3: Exam tip: Keep the difference between stopping decimals and repeating decimals clear.
Correct answer: B. Non-terminating recurring
Explanation: Step 1: The numerator (5) cancels one factor of (5) from (5^2) in the denominator. Step 2: The reduced denominator still has (2^3\times3\times5), so factor (3) remains. Step 3: Exam tip: If (3) remains after cancellation, the decimal will be recurring.
Correct answer: B. (5)
Explanation: Step 1: The denominator has only factors (2) and (5). Step 2: The exponent of (2) is (5), and the exponent of (5) is (3), so the larger exponent is (5). Step 3: Exam tip: Think of making the denominator like (10^5).
Correct answer: A. (0.0875)
Explanation: Step 1: (80\times125=10000). Step 2: (\frac{7}{80}=\frac{875}{10000}=0.0875). Step 3: Exam tip: Missing the zero after the decimal point changes the answer.
Correct answer: A. Both assertion and reason are true and the reason explains the assertion
Explanation: Step 1: (\frac{27}{375}) simplifies by (3) to (\frac{9}{125}). Step 2: Since (125=5^3), the decimal terminates. Step 3: Exam tip: In assertion-reason questions, check whether the reason is true and also explains the assertion.
Correct answer: A. (\frac{3}{11})
Explanation: Step 1: (0.\overline{27}=\frac{27}{99}). Step 2: Simplifying (\frac{27}{99}) by (9) gives (\frac{3}{11}). Step 3: Exam tip: For a two-digit repeating part, denominator (99) quickly gives the answer.
Correct answer: B. (4)
Explanation: Step 1: (\frac{16}{1250}) simplifies by (2) to (\frac{8}{625}). Step 2: Since (625=5^4), the decimal terminates after (4) places. Step 3: Exam tip: Simplification can change the number of decimal places.
Correct answer: A. (0.25)
Explanation: Step 1: (\frac{77}{308}) simplifies by (77) to (\frac{1}{4}). Step 2: (\frac{1}{4}=0.25). Step 3: Exam tip: Spotting the common factor in large numbers saves the most time.
Correct answer: B. It will terminate after (3) places
Explanation: Step 1: The denominator has only (2) and (5), so the decimal terminates. Step 2: The powers are (2) for (2) and (3) for (5), so the larger exponent is (3). Step 3: Exam tip: Since numerator (23) is coprime with the denominator, count places directly from the denominator.
Correct answer: A. (0.35)
Explanation: Step 1: Multiply denominator (20) by (5) to make (100). Step 2: (\frac{7}{20}=\frac{35}{100}=0.35). Step 3: Exam tip: With denominator (20), make (100) for a quick answer.
Correct answer: B. Non-terminating recurring
Explanation: Step 1: (36=2^2\times3^2). Step 2: Since the fraction is in lowest form, (3^2) remains in the denominator. Step 3: Exam tip: If factor (3) remains in the reduced denominator, the decimal does not terminate.
Correct answer: A. (\frac{1}{80})
Explanation: Step 1: (0.0125=\frac{125}{10000}). Step 2: Simplifying by (125), we get (\frac{1}{80}). Step 3: Exam tip: Count zeros very carefully in small decimals.
Correct answer: B. (4)
Explanation: Step 1: (125=5^3), so (5^3) cancels from (5^5) in the denominator. Step 2: The reduced denominator is (2^4\times5^2), whose larger exponent is (4). Step 3: Exam tip: Factors (2) or (5) in the numerator can reduce the denominator powers.
Correct answer: D. (\frac{51}{119})
Explanation: Step 1: (\frac{91}{182}=\frac{1}{2}), (\frac{39}{156}=\frac{1}{4}), and (\frac{68}{170}=\frac{2}{5}). Step 2: (\frac{51}{119}) does not reduce, and (119=7\times17). Step 3: Exam tip: In multi-option questions, reduce each fraction before choosing the non-terminating one.
Correct answer: B. (5)
Explanation: Step 1: (6250=2\times5^5). Step 2: The fraction is in lowest form, and the larger exponent is (5). Step 3: Exam tip: For denominators like (6250), identifying the power of (5) quickly gives the answer.
Correct answer: B. Both are non-terminating recurring
Explanation: Step 1: (14=2\times7) and (28=2^2\times7). Step 2: Factor (7) remains in both denominators, so both decimals do not terminate. Step 3: Exam tip: In comparison questions, prime-factorise both denominators separately.
Correct answer: A. (\frac{37}{300})
Explanation: Step 1: In (0.12\overline{3}=0.123333\ldots), (12) is the non-repeating part and (3) is the repeating part. Step 2: The fraction is (\frac{123-12}{900}=\frac{111}{900}=\frac{37}{300}). Step 3: Exam tip: Identify the non-repeating and repeating parts before placing (9) and (0) in the denominator.
