Chapter 02 Lines and Angles Solutions
November 5, 2024Chapter 04 Data Handling and Presentation Solutions
November 5, 2024Chapter 03 Number Play Text Book Solutions
Page 56
Q1: Can the children rearrange themselves so that the children standing at the ends say ‘2’?
Ans: No, the children at the ends of the line cannot say ‘2’ because they have only one neighbour. To say ‘2’, a child must have two taller neighbours, but children at the ends can only have one neighbour, so they can say ‘0’ or ‘1’ depending on the height of that one neighbour.
Q2: Can we arrange the children in a line so that all would say only 0s?
Ans: Yes, if all the children are of the same height, then every child will say ‘0’ because no one will have a taller neighbor.
Q3: Can two children standing next to each other say the same number?
Ans: Yes, two children standing next to each other can say the same number if they have the same number of taller neighbors. For example, if both have one taller neighbor, they will both say ‘1’.
Q4: There are 5 children in a group, all of different heights. Can they stand such that four of them say ‘1’ and the last one says ‘0’? Why or why not?
Ans: No, this is not possible. If one child says ‘0’, that means there is no one taller than them, so this child must be the tallest. The others cannot all say ‘1’ because they cannot each have only one taller neighbor if there is a tallest child who has none.
Q5: For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
Ans: No, this sequence is not possible because if everyone says ‘1’, it would mean that each child has exactly one taller neighbor. This can’t happen when the heights are different because someone has to be the tallest and someone has to be the shortest.
Q6: Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
Ans: Yes, this sequence is possible. The children can be arranged in a way that the first and last are the shortest (0), the second and fourth have one taller neighbor (1), and the middle child has two taller neighbors (2).
Q7: How would you rearrange the five children so that the maximum number of children say ‘2’?
Ans: To maximize the number of children saying ‘2’, place the shortest child in the middle, with two children of increasing height on either side. This way, the three middle children will have two taller neighbors each, and the children at the ends will have one taller neighbor each, so they will say ‘1’.
Page 57
Q1: Colour or mark the supercells in the table below.
Ans: Supercells are cells where the number is greater than all the numbers in its neighboring cells. The supercells in the provided table are:
9435
8000
Q2: Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.
Ans: Here is one possible arrangement:
5346 | 1258 | 9635 |
|---|---|---|
2543 | 4123 | 5123 |
6789 | 2345 | 5678 |
Q3: Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.
Ans: Here is one possible arrangement:
938 | 876 | 945 |
|---|---|---|
342 | 523 | 745 |
453 | 298 | 634 |
Q4: Out of the 9 numbers, how many supercells are there in the table above?
Ans: There are 4 supercells in the table.
Q5: Find out how many supercells are possible for different numbers of cells.
Ans: The number of supercells depends on the arrangement of the numbers. The maximum number of supercells can be found if the highest numbers are placed in positions surrounded by lower numbers.
Q6: Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?
Ans: No, it is not possible because at least one number will always be greater than its neighbouring cells unless all the numbers are identical, but that would involve repetition.
Q7: Will the cell having the largest number in a table always be a supercell? Can the cell having the smallest number in a table be a supercell? Why or why not?
Ans: The cell with the largest number will always be a supercell because it is greater than all neighboring cells. The cell with the smallest number cannot be a supercell because it is smaller than its neighbors.
Q8: Fill a table such that the cell having the second largest number is not a supercell.
Ans: Here is one possible arrangement:
9365 | 4000 | 1254 |
|---|---|---|
6782 | 9342 | 2345 |
4533 | 1987 | 9635 |
Q9: Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?
Ans: Yes, it is possible. Here is one possible arrangement:
9623 | 1254 | 9345 |
|---|---|---|
6782 | 2943 | 8234 |
4533 | 1234 | 6543 |
Q10: Make other variations of this puzzle and challenge your classmates.
Ans: Students should try to solve this question on their own, but here’s a hint to help them along the way: Create tables with different numbers and try to arrange them in various patterns to create interesting supercell challenges.
Page 59
Q1: Identify the numbers marked on the number lines below, and label the remaining positions.
Ans:
(a) The numbers are: 2010, 2015, 2020, 2025
(b) The numbers are: 9996, 9997, 9998, 9999
(c) The numbers are: 86705, 87205, 87705, 88205
Q2: Put a circle around the smallest number and a box around the largest number in each of the sequences above.
Ans:
(a) Circle: 2010, Box: 2025
(b) Circle: 9996, Box: 9999
(c) Circle: 86705, Box: 88205
Page 60
Q1: Digit sum 14: Write other numbers whose digits add up to 14.
Ans: Some numbers are 59, 95, 68, 176, 545, 4761.
Q2: What is the smallest number whose digit sum is 14?
Ans: The smallest number is 59.
Q3: What is the largest 5-digit number whose digit sum is 14?
Ans: The largest 5-digit number is 99500.
Q4: How big a number can you form having the digit sum 14? Can you make an even bigger number?
Ans: The largest number you can form is 9950000000. You cannot make a bigger number because adding more digits will not maintain the digit sum of 14.
Q5: Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.
Ans: The digit sums vary. For example:
40: 4
41: 5
42: 6
43: 7
44: 8
Q6: Calculate the digit sums of 3-digit numbers whose digits are consecutive (e.g., 345). Do you see a pattern? Will this pattern continue?
Ans: Yes, the digit sums for consecutive numbers (like 345, 456, 567) increase by a constant value. For example:
345: 12
456: 15
567: 18 This pattern continues with each consecutive sequence.
Page 61
Q1: Among the numbers 1–100, how many times will the digit ‘7’ occur?
Ans: The digit ‘7’ appears 20 times in the numbers from 1 to 100.
Q2: Among the numbers 1–1000, how many times will the digit ‘7’ occur?
Ans: The digit ‘7’ appears 300 times in the numbers from 1 to 1000.
Page 62
Q1: Write all possible 3-digit palindromes using the digits 1, 2, and 3.
Ans: The possible palindromes are:
121, 131, 232, 323, 212, 313
Q2: Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome?
Ans: Yes, for 2-digit numbers, the process of reversing and adding will eventually lead to a palindrome.
Page 63
Q1: Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got.
Ans: Every 4-digit number eventually reaches the number 6174, known as the Kaprekar constant.
Q2: Carry out these same steps with a few 3-digit numbers. What number will start repeating?
Ans: For 3-digit numbers, the process will repeat at the number 495.
Page 64
Q1: Choose 4-digits to make the difference between the largest and smallest numbers greater than 5085.
Ans: Choosing digits 9, 1, 3, 7, the largest number is 9731 and the smallest is 1379. The difference is 9731 – 1379 = 8352.
Q2: Choose 4-digits to make the difference between the largest and smallest numbers less than 5085.
Ans: Choosing digits 6, 5, 3, 2, the largest number is 6532 and the smallest is 2356. The difference is 6532 – 2356 = 4176.
Q3: What is the sum of the smallest and largest 5-digit palindrome? What is their difference?
Ans: The smallest 5-digit palindrome is 10001, and the largest is 99999. Their sum is 109000, and their difference is 89998.
Q4: The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?
Ans: The next palindromic time after 10:01 is 11:11, which is in 70 minutes. The one after that is 12:21, which is in 130 minutes.
Page 66
Q1: Write an example for each of the below scenarios whenever possible.
Ans:
5-digit + 5-digit to give a 5-digit sum more than 90,250: 50000 + 40350 = 90350
5-digit – 5-digit to give a difference less than 56,503: 76543 – 21212 = 55331
4-digit + 4-digit to give a 6-digit sum: 1234 + 8765 = 9999
5-digit − 4-digit to give a 4-digit difference: 50000 – 1234 = 48766
5-digit + 3-digit to give a 6-digit sum: 50000 + 987 = 50987
5-digit – 3-digit to give a 4-digit difference: 40000 – 987 = 39013
5-digit + 5-digit to give a 6-digit sum: 50000 + 50000 = 100000
5-digit − 5-digit to give a 3-digit difference: 10000 – 9000 = 1000
5-digit + 5-digit to give 18,500: 10000 + 8500 = 18500
5-digit − 5-digit to give 91,500: 100000 – 8500 = 91500
Q2: Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates.
Ans: Yes, examples are found for all cases.
Page 67
Q1: Always, Sometimes, Never?
Ans:
5-digit number + 5-digit number gives a 5-digit number: Sometimes true
4-digit number + 2-digit number gives a 4-digit number: Sometimes true
4-digit number + 2-digit number gives a 6-digit number: Never true
5-digit number – 5-digit number gives a 5-digit number: Sometimes true
5-digit number – 2-digit number gives a 3-digit number: Sometimes true
Page 69
Q1: Estimate the number of liters a mug, a bucket, and an overhead tank can hold.
Ans:
Mug: 0.25 liters
Bucket: 15 liters
Overhead tank: 1000 liters
Q2: How many rounds does your year of birth take to reach the Kaprekar constant?
Ans: If your year of birth is 2000, after the steps, the Kaprekar constant is reached in 3 rounds.
Page 70
Q1: Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.
Ans: Choose 250. The pattern could be:
40 + 40 + 40 + 40 + 40 + 25 = 250
Page 72
Q1: Estimate the number of holidays you get in a year including weekends, festivals, and vacation. Then try to get an exact number and see how close your estimate is.
Ans: Estimate: 100 days. Exact number: 105 days.