{"id":3340,"date":"2026-06-04T11:37:51","date_gmt":"2026-06-04T06:07:51","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=3340"},"modified":"2026-06-04T11:46:53","modified_gmt":"2026-06-04T06:16:53","slug":"ncert-class-9-maths-exercise-3-5-solutions-decimal-expansion-of-numbers","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/ncert-class-9-maths-exercise-3-5-solutions-decimal-expansion-of-numbers\/","title":{"rendered":"NCERT Class 9 Maths Exercise 3.5 Solutions | Decimal Expansion of Numbers"},"content":{"rendered":"
\"image_print\"Print<\/span><\/a> <\/span><\/div>\n

Short Intro<\/h1>\n\n\n\n

In this article, we provide complete step-by-step solutions of Exercise 3.5 from Chapter 3 \u201cThe World of Numbers\u201d. This exercise focuses on decimal expansion of rational numbers, terminating and repeating decimals, rational and irrational numbers, and cyclic number patterns.<\/p>\n\n\n\n

Quick Information Box<\/h1>\n\n\n\n

Chapter: The World of Numbers
Exercise: 3.5
Class: 9 Mathematics
Main Topic: Decimal Expansion of Rational and Irrational Numbers
Sub-topics: Terminating decimals, repeating decimals, rational numbers, irrational numbers, cyclic numbers, 0.999\u2026 = 1<\/p>\n\n\n\n

Concepts Used<\/h1>\n\n\n\n
    \n
  1. Rational numbers can be written in the form p\/q, where p and q are integers and q \u2260 0.<\/li>\n\n\n\n
  2. A rational number has either a terminating decimal or a repeating decimal.<\/li>\n\n\n\n
  3. If the denominator of a rational number in lowest form has only prime factors 2 and\/or 5, then its decimal expansion terminates.<\/li>\n\n\n\n
  4. If the denominator has any prime factor other than 2 or 5, then its decimal expansion is non-terminating repeating.<\/li>\n\n\n\n
  5. Irrational numbers have non-terminating and non-repeating decimal expansions.<\/li>\n\n\n\n
  6. Some repeating blocks show cyclic properties.<\/li>\n<\/ol>\n\n\n\n

    Important Formulas<\/h1>\n\n\n\n
      \n
    1. Terminating Decimal Test:
      If q = 2\u1d50 \u00d7 5\u207f, then p\/q has a terminating decimal.<\/li>\n\n\n\n
    2. Repeating Decimal Test:
      If q has prime factors other than 2 and 5, then p\/q has a repeating decimal.<\/li>\n\n\n\n
    3. Pure Repeating Decimal:
      0.(abc) = abc\/999<\/li>\n\n\n\n
    4. Algebraic Method for 0.999\u2026:
      Let x = 0.999\u2026
      10x = 9.999\u2026
      10x \u2212 x = 9
      9x = 9
      x = 1<\/li>\n<\/ol>\n\n\n\n

      Questions and Step-by-Step Solutions<\/h1>\n\n\n\n

      Question 1<\/h2>\n\n\n\n

      Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7\/20, 4\/15 and 13\/250. Then check your answers by explicitly performing long divisions and expressing these rational numbers as decimals.<\/p>\n\n\n\n

      Solution<\/h2>\n\n\n\n

      (i) 7\/20<\/h3>\n\n\n\n

      Step 1: Check the denominator.
      20 = 2\u00b2 \u00d7 5<\/p>\n\n\n\n

      Step 2: Since the denominator has only the prime factors 2 and 5, the decimal expansion will terminate.<\/p>\n\n\n\n

      Step 3: Convert into decimal.
      7\/20 = 35\/100 = 0.35<\/p>\n\n\n\n

      Answer:
      7\/20 = 0.35, terminating decimal.<\/p>\n\n\n\n

      (ii) 4\/15<\/h3>\n\n\n\n

      Step 1: Check the denominator.
      15 = 3 \u00d7 5<\/p>\n\n\n\n

      Step 2: Since the denominator contains 3, which is not 2 or 5, the decimal expansion will be non-terminating repeating.<\/p>\n\n\n\n

      Step 3: Convert into decimal.
      4 \u00f7 15 = 0.2666\u2026<\/p>\n\n\n\n

      Answer:
      4\/15 = 0.2666\u2026 = 0.2(6), repeating decimal.<\/p>\n\n\n\n

      (iii) 13\/250<\/h3>\n\n\n\n

      Step 1: Check the denominator.
      250 = 2 \u00d7 5\u00b3<\/p>\n\n\n\n

      Step 2: Since the denominator has only 2 and 5 as prime factors, the decimal expansion will terminate.<\/p>\n\n\n\n

      Step 3: Convert into decimal.
      13\/250 = 52\/1000 = 0.052<\/p>\n\n\n\n

      Answer:
      13\/250 = 0.052, terminating decimal.<\/p>\n\n\n\n

      Final Answer:
      7\/20 = terminating
      4\/15 = repeating
      13\/250 = terminating<\/p>\n\n\n\n

      \n\n\n\n

      Question 2<\/h2>\n\n\n\n

      Perform the long division for 1\/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2\/13? Now compute 3\/13, 4\/13, etc. What do you notice?<\/p>\n\n\n\n

      Solution<\/h2>\n\n\n\n

      First, divide 1 by 13.<\/p>\n\n\n\n

      1\/13 = 0.076923076923\u2026<\/p>\n\n\n\n

      Repeating block = 076923<\/p>\n\n\n\n

      Now calculate other fractions:<\/p>\n\n\n\n

      2\/13 = 0.153846153846\u2026
      3\/13 = 0.230769230769\u2026
      4\/13 = 0.307692307692\u2026
      5\/13 = 0.384615384615\u2026
      6\/13 = 0.461538461538\u2026
      7\/13 = 0.538461538461\u2026
      8\/13 = 0.615384615384\u2026
      9\/13 = 0.692307692307\u2026
      10\/13 = 0.769230769230\u2026
      11\/13 = 0.846153846153\u2026
      12\/13 = 0.923076923076\u2026<\/p>\n\n\n\n

      Observation:
      The repeating blocks are related to each other. They are not all simple rotations of 076923, but they form two cyclic groups.<\/p>\n\n\n\n

      Group 1:
      076923, 230769, 307692, 692307, 769230, 923076<\/p>\n\n\n\n

      Group 2:
      153846, 384615, 461538, 538461, 615384, 846153<\/p>\n\n\n\n

      Conclusion:
      The decimal expansion of 1\/13 has repeating block 076923. The multiples of 1\/13 show cyclic patterns in two groups.<\/p>\n\n\n\n

      \n\n\n\n

      Question 3<\/h2>\n\n\n\n

      Classify the following numbers as rational or irrational. Find explicit fractions in case they are rational.<\/p>\n\n\n\n

      (i) \u221a81
      (ii) \u221a12
      (iii) 0.33333\u2026
      (iv) 0.123451234512345\u2026
      (v) 1.01001000100001\u2026
      (vi) 23.560185612239874790120<\/p>\n\n\n\n

      Solution<\/h2>\n\n\n\n

      (i) \u221a81<\/h3>\n\n\n\n

      \u221a81 = 9<\/p>\n\n\n\n

      9 can be written as 9\/1.<\/p>\n\n\n\n

      Answer:
      \u221a81 is rational.
      Explicit fraction = 9\/1<\/p>\n\n\n\n

      (ii) \u221a12<\/h3>\n\n\n\n

      \u221a12 = \u221a(4 \u00d7 3) = 2\u221a3<\/p>\n\n\n\n

      Since \u221a3 is irrational, 2\u221a3 is also irrational.<\/p>\n\n\n\n

      Answer:
      \u221a12 is irrational.<\/p>\n\n\n\n

      (iii) 0.33333\u2026<\/h3>\n\n\n\n

      Let x = 0.33333\u2026
      10x = 3.33333\u2026<\/p>\n\n\n\n

      Subtract:
      10x \u2212 x = 3.33333\u2026 \u2212 0.33333\u2026
      9x = 3
      x = 3\/9 = 1\/3<\/p>\n\n\n\n

      Answer:
      0.33333\u2026 is rational.
      Explicit fraction = 1\/3<\/p>\n\n\n\n

      (iv) 0.123451234512345\u2026<\/h3>\n\n\n\n

      The block 12345 repeats continuously.<\/p>\n\n\n\n

      So,
      0.1234512345\u2026 = 0.(12345)<\/p>\n\n\n\n

      For a five-digit repeating block:
      0.(12345) = 12345\/99999<\/p>\n\n\n\n

      Simplify:
      12345\/99999 = 4115\/33333<\/p>\n\n\n\n

      Answer:
      0.123451234512345\u2026 is rational.
      Explicit fraction = 4115\/33333<\/p>\n\n\n\n

      (v) 1.01001000100001\u2026<\/h3>\n\n\n\n

      Here the number of zeros keeps increasing.
      The pattern is not repeating as one fixed block.<\/p>\n\n\n\n

      So, it is non-terminating and non-repeating.<\/p>\n\n\n\n

      Answer:
      1.01001000100001\u2026 is irrational.<\/p>\n\n\n\n

      (vi) 23.560185612239874790120<\/h3>\n\n\n\n

      This decimal is terminating because it has a finite number of decimal digits.<\/p>\n\n\n\n

      Any terminating decimal is rational.<\/p>\n\n\n\n

      There are 21 digits after the decimal point.<\/p>\n\n\n\n

      So,
      23.560185612239874790120
      = 23560185612239874790120 \/ 1000000000000000000000<\/p>\n\n\n\n

      On simplification:
      = 589004640305996869753 \/ 25000000000000000000<\/p>\n\n\n\n

      Answer:
      23.560185612239874790120 is rational.
      Explicit fraction = 589004640305996869753 \/ 25000000000000000000<\/p>\n\n\n\n

      Final Classification:
      (i) Rational
      (ii) Irrational
      (iii) Rational
      (iv) Rational
      (v) Irrational
      (vi) Rational<\/p>\n\n\n\n

      \n\n\n\n

      Question 4<\/h2>\n\n\n\n

      The number 0.9, which means 0.99999\u2026, is a rational number. Using algebra, explain why 0.9 is exactly equal to 1.<\/p>\n\n\n\n

      Solution<\/h2>\n\n\n\n

      Let
      x = 0.99999\u2026<\/p>\n\n\n\n

      Multiply both sides by 10:
      10x = 9.99999\u2026<\/p>\n\n\n\n

      Now subtract the first equation from the second equation:<\/p>\n\n\n\n

      10x \u2212 x = 9.99999\u2026 \u2212 0.99999\u2026<\/p>\n\n\n\n

      9x = 9<\/p>\n\n\n\n

      Divide both sides by 9:
      x = 1<\/p>\n\n\n\n

      But x = 0.99999\u2026<\/p>\n\n\n\n

      Therefore,
      0.99999\u2026 = 1<\/p>\n\n\n\n

      Answer:
      0.9 repeating is exactly equal to 1.<\/p>\n\n\n\n

      Explanation:
      0.999\u2026 is not slightly less than 1. It is exactly equal to 1 because the infinite repeating decimal has no gap left between itself and 1.<\/p>\n\n\n\n

      \n\n\n\n

      Question 5<\/h2>\n\n\n\n

      We have seen that the repeating block of 1\/7 is a cyclic number. Try to find more numbers n whose reciprocals 1\/n produce decimals with repeating blocks that are cyclic.<\/p>\n\n\n\n

      Solution<\/h2>\n\n\n\n

      We know:
      1\/7 = 0.142857142857\u2026<\/p>\n\n\n\n

      Repeating block = 142857<\/p>\n\n\n\n

      Now check its multiples:<\/p>\n\n\n\n

      142857 \u00d7 1 = 142857
      142857 \u00d7 2 = 285714
      142857 \u00d7 3 = 428571
      142857 \u00d7 4 = 571428
      142857 \u00d7 5 = 714285
      142857 \u00d7 6 = 857142<\/p>\n\n\n\n

      The same digits rotate cyclically. So 7 gives a cyclic number.<\/p>\n\n\n\n

      Now let us find more such numbers.<\/p>\n\n\n\n

      Example 1: n = 17<\/h3>\n\n\n\n

      1\/17 = 0.0588235294117647\u2026<\/p>\n\n\n\n

      Repeating block = 0588235294117647<\/p>\n\n\n\n

      This block also shows cyclic behaviour when multiplied by 1, 2, 3, \u2026, 16.<\/p>\n\n\n\n

      So, n = 17 is another example.<\/p>\n\n\n\n

      Example 2: n = 19<\/h3>\n\n\n\n

      1\/19 = 0.052631578947368421\u2026<\/p>\n\n\n\n

      Repeating block = 052631578947368421<\/p>\n\n\n\n

      This also produces a cyclic pattern.<\/p>\n\n\n\n

      So, n = 19 is another example.<\/p>\n\n\n\n

      Example 3: n = 23<\/h3>\n\n\n\n

      1\/23 = 0.0434782608695652173913\u2026<\/p>\n\n\n\n

      This also has a long repeating block with cyclic behaviour.<\/p>\n\n\n\n

      Answer:
      Some values of n whose reciprocals produce cyclic repeating blocks are:<\/p>\n\n\n\n

      7, 17, 19, 23<\/p>\n\n\n\n

      Note:
      Such cyclic behaviour is usually seen when the decimal expansion of 1\/n has a repeating block of length n \u2212 1.<\/p>\n\n\n\n

      Common Mistakes<\/h1>\n\n\n\n
        \n
      1. Students often think every non-terminating decimal is irrational. This is wrong. A non-terminating repeating decimal is rational.<\/li>\n\n\n\n
      2. Students forget to reduce the fraction to its lowest form before applying the terminating decimal test.<\/li>\n\n\n\n
      3. Many students classify 0.999\u2026 as less than 1, but it is exactly equal to 1.<\/li>\n\n\n\n
      4. Students confuse repeating pattern with increasing pattern. For example, 1.01001000100001\u2026 is not repeating, so it is irrational.<\/li>\n\n\n\n
      5. In recurring decimals, students sometimes write only a few digits and forget to use \u201c\u2026\u201d or a repeating notation.<\/li>\n<\/ol>\n\n\n\n

        Exam Tips<\/h1>\n\n\n\n
          \n
        1. Always reduce p\/q to lowest form before checking the denominator.<\/li>\n\n\n\n
        2. If the denominator has only 2 and\/or 5, the decimal terminates.<\/li>\n\n\n\n
        3. If the denominator has any prime factor other than 2 or 5, the decimal repeats.<\/li>\n\n\n\n
        4. A terminating decimal is always rational.<\/li>\n\n\n\n
        5. A repeating decimal is always rational.<\/li>\n\n\n\n
        6. A non-terminating and non-repeating decimal is irrational.<\/li>\n\n\n\n
        7. For 0.999\u2026 = 1 type questions, use the algebraic method.<\/li>\n<\/ol>\n\n\n\n

          Practice MCQs<\/h1>\n\n\n\n

          MCQ 1<\/h2>\n\n\n\n

          Which of the following has a terminating decimal expansion?
          A. 4\/15
          B. 7\/20
          C. 1\/13
          D. 2\/3<\/p>\n\n\n\n

          Answer: B. 7\/20<\/p>\n\n\n\n

          MCQ 2<\/h2>\n\n\n\n

          The decimal expansion of 1\/13 is:
          A. Terminating
          B. Non-terminating non-repeating
          C. Non-terminating repeating
          D. Irrational<\/p>\n\n\n\n

          Answer: C. Non-terminating repeating<\/p>\n\n\n\n

          MCQ 3<\/h2>\n\n\n\n

          0.33333\u2026 is equal to:
          A. 3\/10
          B. 1\/3
          C. 33\/100
          D. 3\/100<\/p>\n\n\n\n

          Answer: B. 1\/3<\/p>\n\n\n\n

          MCQ 4<\/h2>\n\n\n\n

          Which number is irrational?
          A. \u221a81
          B. 0.333\u2026
          C. \u221a12
          D. 23.56<\/p>\n\n\n\n

          Answer: C. \u221a12<\/p>\n\n\n\n

          MCQ 5<\/h2>\n\n\n\n

          0.999\u2026 is equal to:
          A. 0.9
          B. 0.99
          C. Slightly less than 1
          D. 1<\/p>\n\n\n\n

          Answer: D. 1<\/p>\n\n\n\n

          FAQ Section<\/h1>\n\n\n\n

          Q1. What is a terminating decimal?<\/h2>\n\n\n\n

          A decimal that ends after a finite number of digits is called a terminating decimal. Example: 0.35.<\/p>\n\n\n\n

          Q2. What is a repeating decimal?<\/h2>\n\n\n\n

          A decimal in which one digit or a group of digits repeats forever is called a repeating decimal. Example: 0.333\u2026<\/p>\n\n\n\n

          Q3. How do we know whether p\/q is terminating?<\/h2>\n\n\n\n

          First reduce p\/q to lowest form. If the denominator has only 2 and\/or 5 as prime factors, the decimal is terminating.<\/p>\n\n\n\n

          Q4. Is every repeating decimal rational?<\/h2>\n\n\n\n

          Yes, every repeating decimal can be converted into a fraction, so it is rational.<\/p>\n\n\n\n

          Q5. Is 0.999\u2026 really equal to 1?<\/h2>\n\n\n\n

          Yes. Algebraically, if x = 0.999\u2026, then 10x = 9.999\u2026, so 9x = 9 and x = 1.<\/p>\n\n\n\n

          Q6. Why is 1.01001000100001\u2026 irrational?<\/h2>\n\n\n\n

          Because the number of zeros keeps increasing and no fixed block repeats forever. Therefore, it is non-terminating and non-repeating.<\/p>\n\n\n\n

          For more Class 9 Maths chapter-wise solutions, mock tests, practice questions, and exam preparation resources, visit www.mymockmate.com<\/a>. Practice regularly with MyMockMate and improve your speed, accuracy, and confidence.<\/p>\n\n\n\n

          <\/p>\n

          \"image_print\"Print<\/span><\/a> <\/span><\/div>\n
          \n\n\t\t\n
            \n\t\t\t <\/ul>\n <\/div> \n","protected":false},"excerpt":{"rendered":"

            Short Intro In this article, we provide complete step-by-step solutions of Exercise 3.5 from Chapter 3 \u201cThe<\/p>\n","protected":false},"author":1,"featured_media":3341,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[11],"tags":[136,2602,2599,945,49,946,2601,2600,893],"class_list":["post-3340","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths-class-9","tag-class-9-maths","tag-cyclic-numbers","tag-decimal-expansion","tag-exercise-3-5-solutions","tag-irrational-numbers","tag-rational-numbers","tag-repeating-decimals","tag-terminating-decimals","tag-the-world-of-numbers"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/3340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=3340"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/3340\/revisions"}],"predecessor-version":[{"id":3342,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/3340\/revisions\/3342"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/3341"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=3340"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=3340"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=3340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}