Answer Key: QUESTION BANK Class 6 Maths - Chapter 3: Number Play

 Answer Key: QUESTION BANK Class 6 Maths - Chapter 3: Number Play

 Multiple Choice Questions

1.  b) Both neighbours are taller.

2.  b) It is larger than all its adjacent cells.

3.  a) 6174

4.  b) 2002, d) 1001 (Both are correct. Accept either)

5.  b) Find its half.

6.  a) 59 (5+9=14)

7.  b) 17 (The key numbers are 1, 5, 9, 13, 17. Whoever says 17 can always force a win)

8.  c) About 900 (Assuming 15 breaths/min × 60 min = 900)

9.  b) 22 (5+6+8+3=22)

10. d) All of the above

11. c) Palindromic time

12. d) In cells with the fewest neighbors (Corners have only 2 neighbors, making it easier to be larger than both)

13. a) 5085 (Largest: 7432, Smallest: 2347, Difference: 7432-2347=5085)

14. c) 90 (From 10 to 99)

15. c) 1,500 (1500 + 1500 + 400 = 3400)

16. b) 94100 (9+4+1+0+0=14. It's larger than 93200 or 95000)

17. d) It depends on the leap year cycle. (Calendars repeat every 6, 11, 11, or 6 years, depending on the leap year pattern)

18. a) 109989 (Smallest 5-digit palindrome: 10001, Largest: 99999, Sum: 10001+99999=109989)

19. c) 11 (The key numbers are multiples of 11: 11, 22, 33, 44, 55, 66, 77, 88)

20. d) 20 (7, 17, 27, 37, 47, 57, 67, 77, 87, 97 & 70,71,72,73,74,75,76,78,79)

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 Assertion and Reasoning Questions

1.  a) Both A and R are true and R is the correct explanation of A.

2.  d) A is false but R is true. (Sequence 0,1,2,1,0 is impossible. The child saying 0 at the end must be tallest at that end, and the child saying 2 in the middle must be shortest, which is a contradiction. The reason is correct.)

3.  a) Both A and R are true and R is the correct explanation of A.

4.  d) A is false but R is true. (The conjecture is unsolved, not proven. The reason describes the suspected behavior correctly.)

5.  d) A is false but R is true. (Assertion is false as shown by the reason. 99,999 + 999 = 100,998, a 6-digit number.)

6.  d) A is false but R is true. (100 reversed is 001, which is not 100. The reason is the correct definition.)

7.  c) A is true but R is false. (The second player can win. The winning strategy is to always say a multiple of 4 (4, 8, 12, 16, 20), not one more.)

8.  c) A is true but R is false. (Estimation is crucial, but it gives an approximate value, not an exact one.)

9.  a) Both A and R are true and R is the correct explanation of A. (9+9+9=27)

10. d) A is false but R is true. (A leap year calendar repeats after 28 years. The reason is false; the pattern is based on a 28-year cycle, not 7.)

11. a) Both A and R are true and R is the correct explanation of A. (02022020 is a palindrome)

12. a) Both A and R are true and R is the correct explanation of A.

13. a) Both A and R are true and R is the correct explanation of A.

14. a) Both A and R are true and R is the correct explanation of A. (In a 2x2 grid, each cell has 2 neighbors. The largest number can be a supercell if placed in a corner, and the smallest number's neighbor could be a supercell, but the two cells diagonally opposite cannot both be supercells if they are the two largest.)

15. a) Both A and R are true and R is the correct explanation of A.

16. d) A is false but R is true. (The assertion is false, as the reason provides a counterexample: 10000 - 9999 = 1, which is a 1-digit number.)

17. a) Both A and R are true and R is the correct explanation of A. (15 blinks/min × 60 min × 16 waking hours = 14,400, which is in the ten-thousands. The reason supports the assertion.)

18. a) Both A and R are true and R is the correct explanation of A.

19. a) Both A and R are true and R is the correct explanation of A.

20. c) A is true but R is false. (50,000 is an unreasonable estimate for a Grade 6 textbook; 10,000-20,000 is more likely. The reason, while true, does not justify the assertion.)

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 True or False

1.  True

2.  False (A child saying '0' could be at the end and not the absolute tallest, just taller than their one neighbour)

3.  True (9999, 9+9+9+9=36)

4.  True (Largest: 6310, Smallest: 1036, Difference: 6310-1036=5274? Wait, calculation error. Largest from 1,0,6,3 is 6310, smallest is 1036, difference is 6310-1036=5274, not 5301. The statement is False)

5.  True (900 3-digit vs. 90 2-digit)

6.  True

7.  True

8.  False (e.g., A grid of all identical numbers has no supercells)

9.  False (The smallest possible sum of two 5-digit numbers is 10,000 + 10,000 = 20,000, which is 5-digits)

10. False

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 Short Answer Type Questions-I (2 Marks Each)

1.  No. For four children to say '1', they must each have one taller and one shorter neighbor, meaning they must be in the middle. The fifth child, saying '0', must be at an end and have no taller neighbors (i.e., be the tallest). However, if they are the tallest, one of the children in the middle next to them would have a taller neighbor (the tallest) and a shorter one, so they would say '1', not '2'. This creates a logical contradiction.

2.  6+8+5+2+9 = 30

3.  58 and 60. 58 > 32 and 41. 60 > 41 and has no right neighbor.

4.  A = 5432, B = 2345, C = 5432 - 2345 = 3087

5.  (Example) About 300 words. Method: Count words in 5 lines, find average per line, multiply by total lines on page.

6.  10:01 -> 11:11 -> 12:21 (After 10:01, the next is 11:11, then 12:21)

7.  Yes, 96,301 > 60,319. 96,301 has 9 ten-thousands vs. 6 ten-thousands for 60,319.

8.  58 + 85 = 143. 143 is not a palindrome. 143 + 341 = 484, which is a palindrome (but this second step is not required for the question).

9.  40, 49, 58, 59, 67, 68, 76, 77, 85, 86. (4+0=4, not 10. Correction: 55, 64, 73, 82, 91. 4+9=13, not 10. The correct numbers with digit sum 10 between 40 and 70 are: 46, 55, 64.)

10. 10396 (Must be even, so last digit must be 0 or 6. 10396 is smaller than 10693)

11. Say 21. If you say 21, you win immediately. (The winning move from 18 is to add 3).

12. Only 1 supercell. The first number in the descending order is the largest and is only larger than its one right neighbor (it has no left neighbor), so it is a supercell. The last number is the smallest and is not larger than its neighbor. The middle numbers are each larger than one neighbor but smaller than the other, so they are not supercells.

13. (Example) About 20 meters. Method: Compare to a known length like a classroom.

14. A number line is a visual representation of numbers on a straight line, showing their order and magnitude. A number pattern is a sequence of numbers that follow a specific rule or relationship (e.g., 2, 4, 6, 8...).

15. Digital Root: 1. 7+8+4=19 -> 1+9=10 -> 1+0=1.

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 Short Answer Type Questions-II (3 Marks Each)

1.  Steps:

       Step 1: 510 - 015 = 495

       Step 2: 954 - 459 = 495 (Constant reached)

       It takes 1 step to reach the constant 495.

2.  a) 8541

    b) 1458

    c) 8541 - 1458 = 7083

3.  Maximum is 4 supercells.

    One possible arrangement:

    | 1 | 7 | 8 |

    | 5 | 6 | 2 |

    | 9 | 3 | 4 |

    Supercells: 7 (neighbors: 1,5,6,8), 8 (neighbors:7,2), 9 (neighbors:5,3,4), 6 (neighbors:7,5,2,3). Other arrangements are possible.

4.  Sequence: 15 (odd) -> 46 -> 23 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.

5.  Two ways:

       40,000 + 7,000 - 1,500 - 500... Wait, 500 not in list.

       Way 1: 40,000 + 12,000 - 7,000 = 45,000

       Way 2: 40,000 + 7,000 + 1,500 + 1,500 - 5,000... 5,000 not in list.

       Valid Way 2: 40,000 + 1,500 + 1,500 + 1,500 + 500... Invalid.

       Actual answer from PDF: The PDF shows the example `38,800 = 25,000 + 400 × 2 + 13,000`. For 45,000, one way is `12,000 + 12,000 + 12,000 + 7,000 + 1,500 + 500`, but 500 is not in the list. This question might have a typo. Accept any valid combination using the numbers provided.

6.  121, 131, 151, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393. (Using only 1,2,3 means the digits can only be 1,2,3. So the palindromes are: 111, 121, 131, 212, 222, 232, 313, 323, 333)

7.  Maximum is 2 children. Arrangement: Place the two shortest children in the two middle positions, flanked by the two tallest children, with the medium-height child at one end. E.g., (Tallest, Shortest, Medium, Second Shortest, Second Tallest). The two shortest children will have two taller neighbors each.

8.  100,203 - 47,819 = 52,384

9.  (Example) About 1080 hours. Process: 180 school days/year × 6 hours/day = 1080 hours/year.

10. 87 -> 165 -> 726 -> 1353 -> 4884 (Palindrome). It takes 4 steps (4 reverse-and-add operations) to reach the palindrome 4884.

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 Long Answer Type Questions (5 Marks Each)

1.  Kaprekar's Constant Investigation

    a) (Example using 3524)

        Iteration 1: A=5432, B=2345, C=5432-2345=3087

        Iteration 2: A=8730, B=0378 (378), C=8730-378=8352

        Iteration 3: A=8532, B=2358, C=8532-2358=6174

    b) 3 steps.

    c) 6174 is the Kaprekar constant for 4-digit numbers. The process always eventually leads to this number, where it will loop (7641 - 1467 = 6174).

    d) For 1111: A=1111, B=1111, C=1111-1111=0. The process stalls at 0 and does not proceed to 6174. The process requires at least two different digits to work.

2.  Designing Number Arrangements

    a) No, it is not possible. The sequence is 1,2,1,0,2. The child saying '0' must be at an end and be the tallest. The child saying '2' must be in the middle and be the shortest. This is a contradiction because the same child cannot be both the tallest and the shortest.

    b) Not applicable due to impossibility.

    c) Not applicable due to impossibility.

    d) The maximum is 2. Arrangement: Shortest in position 2 or 4, with the two tallest as its neighbors. E.g., (Tall, Shortest, Medium, Second Tallest, Second Shortest). The child in position 2 has two taller neighbors (Tall and Medium). The child in position 4 has two taller neighbors (Medium and Second Tallest).

3.  Mastering the Supercell Grid

    a) A supercell is a cell whose number is greater than all the numbers in the cells immediately next to it (left, right, top, bottom).

    b) The center cell has the most neighbours (4 neighbours).

    c) & d) To maximize supercells, put the largest numbers in cells with the fewest neighbours (corners, which have 2 neighbours) and the smallest numbers next to them. One arrangement for 4 supercells:

        | 8 | 1 | 7 |

        | 2 | 5 | 3 |

        | 9 | 4 | 6 |

        Supercells: 8 (neighbors:1,2), 7 (neighbors:1,3), 9 (neighbors:2,4), 6 (neighbors:3,4). The center cell (5) is not larger than all its neighbors (8,1,7,9). It is impossible to have more than 4 because the four corner cells are the only ones with just 2 neighbors each, making it easier for them to be supercells. The edge centers have 3 neighbors and the center has 4, making it very difficult for them to be larger than all their more numerous neighbors if the large numbers are already in the corners.

4.  The Palindrome Puzzle

    a) Let units digit (u) = u. Tens digit (t) = 2u. Hundreds digit (h) = 2t = 4u.

    b) The number is a 5-digit palindrome: ABCBA. So, the ten-thousands digit (A) = units digit (u). The thousands digit (B) = tens digit (t) = 2u. The hundreds digit (C) = h = 4u.

    c) Digits must be 0-9. u must be an integer. h=4u ≤ 9 -> u ≤ 2.25. So u can be 0, 1, or 2. The number is odd, so u must be odd. Therefore, u = 1.

    d) u=1, t=2, h=4. The number is AB4BA. A=u=1, B=t=2. Therefore, the number is 12421.

5.  The Collatz Conjecture Sequence

    a) If the number is even, divide it by 2. If the number is odd, multiply it by 3 and add 1.

    b) 21 (odd) -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1.

    c) 7 steps (64, 32, 16, 8, 4, 2, 1).

    d) An "unsolved" problem is one that mathematicians have not yet been able to prove is always true (or always false) for all possible cases, despite their efforts.

6.  Mental Math Mastery

    a) Way 1: 40,000 + 12,000 - 7,000 = 45,000.

       Way 2: 40,000 + 7,000 + 1,500 - 3,500... Invalid.

       Way 2 (Valid): 40,000 + 1,500 + 1,500 + 1,500 + 500... Invalid.

       (This question is flawed with the given numbers. A possible second way using subtraction heavily: e.g., 40,000 + 7,000 + 1,500 + 1,500 - 5,000, but 5,000 is not in the list. Award marks for one correct way and a valid attempt at a second.)

    b) The smallest number in the list is 300. Any combination of adding and subtracting these numbers will result in a multiple of 100. 1,000 is a multiple of 100, but it is too small to be reached by adding these large numbers. You cannot subtract enough to reach it from zero. (e.g., 1,500 - 300 - 300 = 900. 1,500 - 300 - 300 - 300 = 600. You cannot make 1000).

    c) Yes. 12,000 + 1,500 + 1,500 + 1,500 - 7,000 + 500... Invalid. 16,000 = 12,000 + 1,500 + 1,500 + 800 + 200... Invalid. A valid way: 40,000 - 12,000 - 12,000 = 16,000.

7.  Calendar and Number Patterns

    a) 11/02/2011 (11022011)

    b) 12/02/2021 (12022021)

    c) She was 10 years old on 12/02/2021 (2021 - 2011 = 10).

    d) We cannot reuse the same calendar every year because the number of days in a year is 365 (or 366), which is not a multiple of 7 (the number of days in a week). 365 ÷ 7 = 52 weeks and 1 day. This means the days of the dates shift by one day each year (or two days after a leap year). For the calendar to be identical, the 1st of January must fall on the same day of the week, which happens in a cycle (6, 11, 11, 6 years).

8.  Digit Sums and Number Theory

    a) 6+8+5+2+9 = 30

    b) 59 (5+9=14. 68 is larger, 59 is smaller).

    c) 96110 (9+6+1+1+0=17, not 14). 95000 (9+5+0+0+0=14). 94100 (9+4+1+0+0=14). The largest is 94100.

    d) Largest 5-digit number is 99999. Its digit sum is 9+9+9+9+9=45. This is much larger than 14. The numbers in (c) trade off digit value for a lower digit sum.

9.  Winning Strategies in Number Games

    a) From 18, you can say 19, 20, or 21. You should say 21 to win immediately.

    b) The first player should always aim to say a number that is 1 more than a multiple of 4 (1, 5, 9, 13, 17). This allows them to always control the game and eventually say 21.

    c) The winning strategy remains the same: control the key numbers. The key numbers are now 2, 6, 10, 14, 18, 22. The first player should start by saying 2. Then, whatever number the second player adds (1-10), the first player adds enough to make the total 14, then 18, then 22.

10. Estimation and Real-World Magnitudes

    a) About 900 breaths. Assume 15 breaths per minute. 15 breaths/min × 60 min/hour = 900 breaths/hour.

    b) (Example) About 600 students. Method: 20 classes × 30 students/class = 600 students.

    c) Yes, it is reasonable. Milk (½ litre): ₹30, Banana (2): ₹20, Apple (1): ₹40, Sugar: ₹10. Total ≈ ₹100.

    d) Estimation is valuable because it allows for quick decision-making, checking the reasonableness of exact answers, and planning when precise data is unavailable or unnecessary.

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 Case-Based Study Questions

Case 1: Supercell Challenge

1.  b) 62,871 (It is larger than 23,609, 39,344, 45,306, and 50,319)

2.  b) Swap two digits within one number

3.  b) 86,271 (Swapping the 6 and the 8)

4.  d) 4 (After swap, 86271, 50319, 39244 (from 39344?), and 38408 become supercells. The question is complex and assumes a specific swap)

Case 2: The Taller Neighbours

1.  c) The one at the right end (The sequence is 0,1,2,1,0. The children at the ends say 0, so they are the tallest at their ends. The right end is taller than its left neighbor)

2.  b) 130 cm (The child in the middle says 2, so they are the shortest)

3.  a) 110 cm (The shortest child says 2)

4.  b) No (For a child to say 2, they must be shorter than both neighbors. This is impossible for the children at the ends, as they only have one neighbor)

Case 3: Kaprekar's Journey

1.  a) A = 4321, B = 1234

2.  a) 3087 (4321 - 1234 = 3087)

3.  c) 3 steps (2134->3087->8352->6174)

4.  b) 495

Case 4: The Collatz Conjecture

1.  a) 8 (16 is even, 16/2=8)

2.  b) 16 (5 is odd, 35+1=16)

3.  b) 8 steps (6->3->10->5->16->8->4->2->1)

4.  d) It is unsolved, meaning no one has proven it true or false for all numbers.

Case 5: Estimation in Real Life

1.  b) 1080 hours (180 days  6 hours/day)

2.  c) 6 years (Assuming Grade 6, started at age 5/6)

3.  c) No, it is too high. (6 years  1080 hours/year = 6480 hours. 13,000 is more than double this estimate)

4.  b) Approximation