Number PlayClass 6 Mathematics NCERT Solutions
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Q1Chapter 3 Introductory Question
Think about various situations where we use numbers. List five different situations in which numbers are used.
Solution
Solution:
Numbers are used in many aspects of our daily lives. Here are five different situations:
- Telling Time: We use numbers to read the time on clocks and watches (e.g., 8:30 AM, 14:00 hours).
- Using a Calendar: We use numbers for dates, months, and years to keep track of important events and schedules (e.g., 15th August 2023).
- Handling Money: Numbers are essential for counting money, making payments, calculating change, and managing budgets (e.g., buying an item for ₹50 and paying with a ₹100 note).
- Measurement: We use numbers to measure various quantities like height (e.g., 150 cm), weight (e.g., 45 kg), distance (e.g., 5 km), and temperature (e.g., 32°C).
- Scores in Games and Academics: Numbers are used to represent scores in sports (e.g., a cricket score of 180/2) and marks in school examinations (e.g., 95 out of 100).
Q1Section 3.1 Numbers can Tell us Things
Can the children rearrange themselves so that the children standing at the ends say '2'?
Solution
Answer: No.
Reasoning:
A child says '2' if both the children standing next to them are taller. A child standing at either end of the line has only one neighbor. Since they do not have two neighbors, it is impossible for them to have two taller neighbors. Therefore, a child at the end can say '0' or '1', but never '2'.
Q2Section 3.1 Numbers can Tell us Things
Can we arrange the children in a line so that all would say only 0s?
Solution
Answer: Yes.
Reasoning:
A child says '0' if neither of the children standing next to them are taller. This can happen in two ways:
- All children are of the same height: If all children have the same height, no child has a taller neighbor. So, every child would say '0'.
- Arrange by decreasing height: If the children are arranged in a line from tallest to shortest, each child will have a shorter neighbor (or no neighbor on one side). No child will have a taller neighbor. Therefore, all of them would say '0'.
Q3Section 3.1 Numbers can Tell us Things
Can two children standing next to each other say the same number?
Solution
Answer: Yes.
Reasoning:
It is possible for two adjacent children to say the same number. For example, consider four children with heights 1, 2, 3, 4 arranged in increasing order of height.
Arrangement: 1, 2, 3, 4
Let's look at the two middle children:
- The child with height 2 has one shorter neighbor (1) and one taller neighbor (3). So, this child says '1'.
- The child with height 3 has one shorter neighbor (2) and one taller neighbor (4). So, this child also says '1'.
Thus, it is possible for two children standing next to each other to say the same number.
Q4Section 3.1 Numbers can Tell us Things
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?
Solution
Answer: Yes, it is possible.
Reasoning:
A child says '1' if they have exactly one taller neighbor. A child says '0' if they have no taller neighbors. The tallest child in any arrangement will always say '0' because no one is taller than them.
To make four children say '1', each of them must have exactly one taller neighbor. This can be achieved by placing the tallest child in the middle and arranging the other children around them.
Let the heights of the 5 children be H1 < H2 < H3 < H4 < H5.
Consider the following arrangement:
H2, H4, H5, H3, H1
Let's check what each child would say:
- Child H2 (at one end): Has one neighbor, H4, who is taller. So, H2 says '1'.
- Child H4: Has two neighbors, H2 (shorter) and H5 (taller). So, H4 says '1'.
- Child H5 (tallest): Has two neighbors, H4 and H3, who are both shorter. So, H5 says '0'.
- Child H3: Has two neighbors, H5 (taller) and H1 (shorter). So, H3 says '1'.
- Child H1 (at the other end): Has one neighbor, H3, who is taller. So, H1 says '1'.
The sequence of numbers said is 1, 1, 0, 1, 1. This arrangement has four children saying '1' and one child saying '0'. Therefore, it is possible.
Q5Section 3.1 Numbers can Tell us Things
For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
Solution
Answer: No, it is not possible.
Reasoning:
In any group of children with different heights, there will always be one child who is the tallest. The tallest child, by definition, has no one taller than them. Therefore, the tallest child will always have zero taller neighbors, regardless of their position in the line. This means the tallest child will always say '0'.
Since at least one child must say '0', it is impossible for all five children to say '1'.
Q6Section 3.1 Numbers can Tell us Things
Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
Solution
Answer: Yes, it is possible.
Reasoning:
Let the heights of the 5 children be represented by numbers 1 (shortest) to 5 (tallest). We need to find an arrangement of these five heights that produces the sequence 0, 1, 2, 1, 0.
Let the arrangement be A, B, C, D, E.
- The children at the ends (A and E) say '0'. This means they must be taller than their only neighbor. So, A > B and E > D.
- The middle child (C) says '2'. This means both of their neighbors are taller. So, B > C and D > C.
- The other two children (B and D) say '1'. This means they each have exactly one taller neighbor. For B, the neighbors are A and C. Since A > B, C must be shorter than B. For D, the neighbors are C and E. Since E > D, C must be shorter than D.
Let's summarize the conditions:
- A > B
- E > D
- B > C
- D > C
Combining these, we get: A > B > C and E > D > C. This shows that C must be the shortest child. Let C = 1.
Now we have A > B > 1 and E > D > 1. The remaining heights are 2, 3, 4, 5.
Let's try to assign values. Let B=2 and D=3. Then A > 2 and E > 3. We can set A=4 and E=5.
Let's check the arrangement: A=4, B=2, C=1, D=3, E=5.
- Child A (height 4): Neighbor is B (height 2). 4 > 2. No taller neighbors. Says '0'. Correct.
- Child B (height 2): Neighbors are A (4, taller) and C (1, shorter). One taller neighbor. Says '1'. Correct.
- Child C (height 1): Neighbors are B (2, taller) and D (3, taller). Two taller neighbors. Says '2'. Correct.
- Child D (height 3): Neighbors are C (1, shorter) and E (5, taller). One taller neighbor. Says '1'. Correct.
- Child E (height 5): Neighbor is D (height 3). 5 > 3. No taller neighbors. Says '0'. Correct.
The arrangement of heights (from left to right) 4, 2, 1, 3, 5 results in the sequence 0, 1, 2, 1, 0. Therefore, it is possible.
Q7Section 3.1 Numbers can Tell us Things
How would you rearrange the five children so that the maximum number of children say '2'?
Solution
Answer: The maximum number of children who can say '2' is two.
Reasoning:
A child says '2' if both their neighbors are taller. Children at the ends of the line cannot say '2' because they only have one neighbor. This means at most, the three children in the middle can say '2'.
It is impossible for three adjacent children to all say '2'. If we have three children B, C, and D in a line, for C to say '2', B > C and D > C. But for B to say '2', its other neighbor must also be taller, and for D to say '2', its other neighbor must be taller. This creates contradictions. Specifically, for B and C to both say '2', we would need C > B and B > C, which is impossible.
Therefore, the maximum number of children saying '2' is less than three.
To get two children to say '2', we can arrange them such that the two shortest children are surrounded by taller children. Let the heights be 1, 2, 3, 4, 5.
Consider the arrangement: 4, 1, 5, 2, 3
- Child with height 4 (at the end): Has one neighbor (1) who is shorter. Says '0'.
- Child with height 1: Has two neighbors (4 and 5) who are both taller. Says '2'.
- Child with height 5: Has two neighbors (1 and 2) who are both shorter. Says '0'.
- Child with height 2: Has two neighbors (5 and 3) who are both taller. Says '2'.
- Child with height 3 (at the end): Has one neighbor (2) who is shorter. Says '0'.
The sequence of numbers said is 0, 2, 0, 2, 0. This arrangement successfully makes two children say '2'. Therefore, the maximum is two.