CBSE • Class 9 • Mathematics • Chapter 1 (Number Systems) • Exercise 1.2

Exercise 1.2: Number Systems — NCERT Solutions

Decimal expansions of rationals — terminating and non-terminating recurring forms.

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What this exercise covers

Terminating decimalsRecurring decimalsp/q form

Step-by-step solutions — Exercise 1.2

2 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 1.2 Q1 • 2 marks

Write the decimal expansion of 38\frac{3}{8} and state whether it is terminating or non-terminating recurring.
Hint (Socratic — try this first)▾
What are the prime factors of the denominator, and does that tell you whether the decimal stops?
Step-by-step solution▾

Long division of 3÷83 \div 8:

3.000÷8=0.3753.000 \div 8 = 0.375

Steps: 30÷8=330 \div 8 = 3 remainder 66; 60÷8=760 \div 8 = 7 remainder 44; 40÷8=540 \div 8 = 5 remainder 00.

Since the remainder becomes 00, the expansion ends: 38=0.375\frac{3}{8} = 0.375

Check with prime factors: 8=238 = 2^3. A fraction (in lowest terms) with denominator of the form 2m5n2^m 5^n has a terminating decimal.

Conclusion: 38=0.375\frac{3}{8} = 0.375, which is a terminating decimal.

Common mistake:
Students stop the division too early or misplace the decimal point, writing 0.3750.375 as 0.350.35 or 3.753.75.
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Exercise 1.2 Q2 • 3 marks

Express the recurring decimal 0.6‾0.\overline{6} (i.e. 0.6666…0.6666\dots) in the form pq\frac{p}{q}.
Hint (Socratic — try this first)▾
If you let xx equal the decimal, what does multiplying by 10 do to the repeating tail?
Step-by-step solution▾

Let x=0.6666…x = 0.6666\dots

Since one digit repeats, multiply by 1010: 10x=6.6666…10x = 6.6666\dots

Subtract the first equation from the second: 10x−x=6.6666⋯−0.6666…10x - x = 6.6666\dots - 0.6666\dots 9x=69x = 6 x=69=23x = \frac{6}{9} = \frac{2}{3}

Conclusion: 0.6‾=230.\overline{6} = \frac{2}{3}.

Common mistake:
Students forget to multiply by the correct power of 10 (one 10 for one repeating digit) or write 0.6‾=6100.\overline{6} = \frac{6}{10}, ignoring the repetition.
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How to approach Exercise 1.2

  1. Re-read the chapter summary first. Open Number Systems and refresh the key concepts: Rational numbers, Irrational numbers, Decimal expansions, Laws of exponents.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Number Systems

  1. Exercise 1.1 — Rational vs irrational numbers — closure, density on the line, and locating simple surds.
  2. Exercise 1.2 — Decimal expansions of rationals — terminating and non-terminating recurring forms.
  3. Exercise 1.3 — Real numbers — operations and representability; approximating irrational numbers.
  4. Exercise 1.4 — Laws of exponents extended to rational and real bases and exponents.
  5. Exercise 1.5 — Rationalising denominators with surds and revisiting identities with roots.
  6. Exercise 1.6 — Representing irrational numbers like √x on the number line using constructions.

← Exercise 1.1

Rational vs irrational numbers — closure, density on the line, and locating simple surds.

Exercise 1.3 →

Real numbers — operations and representability; approximating irrational numbers.

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