NCERT Solutions for Class 6 Maths Chapter 10 - The Other Side of Zero Exercise 10.4

Exercise 10.4

1. Do the calculations for the second grid above and find the border sum.

Ans:  Let’s examine the provided grid to determine the border sum.

Understanding the Grid:

The grid consists of a 3 × 3 arrangement of numbers, where the sum of the numbers in each row and column should be equal.

Calculating the Border Sum:

Top Row: 5 + (-3) + (-5) = -3

Bottom Row: (-8) + (-2) + 7 = -3

Left Column: 5 + 0 + (-8) = -3

Right Column: (-5) + (-5) + 7 = -3

Thus, the border sum of the grid is -3.

2. Complete the grids to make the required border sum:

Ans:

The border sum is +4.

Here, we added the missing numbers to make sure the sum of each row and column matches the specified border sum. 

For instance, in the first grid, to achieve a border sum of +4, the missing numbers in the 

Top row must be 6 and 8 (because -10 + 6 + 8 = +4).

Bottom row must be 2 and -7 (because 9 + 2 - 7 = +4).

Left column must be 5 (because -10 + 5 + 9 = +4).

Do it by yourself for other grids.

3. For the last grid above, find more than one way of filling the numbers to get border sum –4

Ans: There are multiple ways to fill the last grid with border sum -4, for example

4. Which other grids can be filled in multiple ways? What could be the reason?

Ans: A grid with a larger size (more rows and columns) is likely to have multiple solutions. This occurs because there are more options for distributing numbers while still achieving the desired border sum.

5. Make a border integer square puzzle and challenge your classmates.

Ans: Below is the puzzle

Border sum = +2

6. Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!

Ans: Let’s circle the number -5. According to the game rules, we will now cross out the row and column that contain the number -5.

Let’s circle the number 3 once more. Now, let’s cross out the row and column that contain the number 3.

Once more, let’s circle the number -1. Now, let’s cross out the row and column that contain the number -1.

Once again, let’s highlight the balance number 2. Now, let’s cross out the row and column that contain the number 2.

Now, let’s sum the circled numbers: (-5) + 3 + (-1) + 2 = -6 + 5 = -1. 

Thus, we arrive at the same value, -1. 

7. Play the same game with the grids below. What answer did you get?

Ans: For the first grid:

Lets circle the number 4

According to the game, let’s strike out the row and column with number 4

Now, circle the number 10.

Again strike the crow and column with number 10.

Now, circle the number -20

Now strike the row and column with number -20

Again, circle the number -2

Now strike the row and column with number -2

Let’s sum all the circled numbers = 4 + (10) + (-20) + (-2) = -8  

For second Grid: Lets circle the number -6

According to the game, let’s strike out the row and column with number -6

Now, circle the number 4.

Again strike the crow and column with number 4.

Now, circle the number -9

Now strike the row and column with number -9

Again, circle the number -3

Now strike the row and column with number -3

Let’s sum all the circled numbers = -6 + 4 + (-9) + (-3) = -2 -12 = -14 

8. What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

Ans: Grids can be intriguing due to both the numbers involved and their arrangement. Here’s why:

Numbers: The numbers in a grid can exhibit specific patterns or sequences, like in magic squares, where the sums of numbers in each row, column, and diagonal are equal.

Arrangement: The layout of elements in a grid can establish visual balance and harmony.

9. Write all the integers between the given pairs, in increasing order.

a. 0 and –7 b. –4 and 4

c. –8 and –15 d. –30 and –23

Ans: 

a. The integers between 0 and -7, in increasing order: -6, -5, -4, -3, -2, -1

b. The integers between -4 and 4, in increasing order: -3, -2, -1, 0, 1, 2, 3

c. The integers between -8 and -15, in increasing order: -14, -13, -12, -11, -10, -9

d. The integers between -30 and -23, in increasing order: -29, -28, -27, -26, -25, -24

10. Give three numbers such that their sum is –8.

Ans: 3 numbers that add up to -8 are -10, 4 and -2

Let’s add them together, we will get -10 + 4 + (-2) = -8

11. There are two dice whose faces have these numbers: –1, 2, –3, 4, –5, 6. The smallest possible sum upon rolling these dice is –10 = (–5) + (–5) and the largest possible sum is 12 = (6)+(6). Some numbers between (–10) and (+12) are not possible to get by adding numbers on these two dice. Find those numbers.

Ans: Let’s identify the sums that cannot be achieved when rolling these two dice. The faces of the dice show: -1, 2, -3, 4, -5, and 6. 

First, let’s outline all possible sums:  

The sum of two negative numbers:

• (-1) + (-1) = -2

• (-1) + (-3) = -4

• (-1) + (-5) = -6

• (-3) + (-3) = -6

• (-3) + (-5) = -8

• (-5) + (-5) = -10

The sum of one negative and one positive number:

• (-1) + 2 = 1

• (-1) + 4 = 3

• (-1) + 6 = 5

• (-3) + 2 = -1

• (-3) + 4 = 1

• (-3) + 6 = 3

• (-5) + 2 = -3

• (-5) + 4 = -1

• (-5) + 6 =1

The sum of two positive numbers:

• 2 + 2 = 4

• 2 + 4 = 6

• 2 + 6 = 8

• 4 + 4 = 8

• 4 + 6 = 10

• 6 + 6 = 12

Now, let’s list all the possible sum in ascending order:

-10, -8, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 10, 12

The sum of numbers between -10 and 12 that are not the sum of possible to get ate -9, -7, -5, 0, 2, 7, 9, 11

12. Solve these:

Ans: 

(a) 8 - 13 = -5

(b) -8 - (13) = -8 -13 = -21

(c) (-13) - (-8) = -13 + 8 = -5

(d) (-13) + (-8) = -13 - 8 = -21

(e) 8 + (-13) = 8 - 13 = -5

(f) (-8) - (-13) = -8 +13 = 5

(g) (13) - 8 = 13 - 8 = 5

(h) 13 - (-8) = 13 + 8 = 21

13. Find the years below.

a. From the present year, which year was it 150 years ago? ________

b. From the present year, which year was it 2200 years ago? _______

Hint: Recall that there was no year 0.

c. What will be the year 320 years after 680 BCE? ________ 

Ans:

a. 150 years ago from the current year (2024) is calculated as follows:  

2024 - 150 = 1874

Therefore, 150 years ago, it was the year 1874.

b. 2200 years ago from the present year (2024):

Since there is no year 0, we need to consider this in our calculation:  

2024 - 2200 = -176.  

The year -176 corresponds to 177 BCE (Before the Common Era).  

Therefore, 2200 years ago, it was the year 177 BCE.

c. Since BCE refers to "before Christ," we can express 680 BCE as -680. Therefore, 320 years after 680 BCE is calculated as -680 + 320 = -360, which corresponds to 360 BCE.

14. Complete the following sequences: 

a. (–40), (–34), (–28), (–22), _____, ______, ______ 

b. 3, 4, 2, 5, 1, 6, 0, 7, _____, _____, _____ 

c. _____, ______, 12, 6, 1, (–3), (–6), _____, ______, ______ 

Ans: 

a. (-40), (-34), (-28), (-22), _____, ______, ______

Let’s find the next numbers in the sequence by subtracting each number from the one before it:  

-22 - (-28) = -22 + 28 = 6

-28 - (-34) = -28 + 34 = 6  

-34 - (-40) = -34 + 40 = 6  

This indicates that each term in the sequence increases by 6.  

Therefore, the next term is:  

-22 + 6 = -16  

The following term is:  

-16 + 6 = -10  

And the final term is:  

-10 + 6 = -4  

Thus, the complete sequence is (-40), (-34), (-28), (-22), (-16), (-10), (-4).

b. Let’s calculate the differences between the last number in the sequence and the one before it:  

7 - 0 = 7  

0 - 6 = -6  

6 - 1 = 5  

1 - 5 = -4  

5 - 2 = 3  

2 - 4 = -2  

4 - 3 = 1  

In this sequence, the numbers decrease by 1, alternating between positive and negative integers. 

Thus, the next numbers are:  

7 + (-8) = -1  

-1 + 9 = 8  

8 - 10 = -2  

-2 + 11 = 9  

and so on.

The complete sequence is 3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2, 9, $\ldots$

Let’s verify:  

-1 - 7 = -8  

8 + 1 = 9  

-2 - 8 = -10

c. _____, 12, 6, 1, (-3), (-6), ____, _____, _____

Let’s find the differences by subtracting each number from the one before it:  

(-6) - (-3) = -6 + 3 = -3  

(-3) - 1 = -4  

1 - 6 = -5  

6 - 12 = -6  

This shows that in this sequence, a negative integer is added to each number. 

Let’s denote the first number as x and the second number as y.  

For the second number:  

12 - y = -7  

Thus, y = 12 + 7 = 19.  

Now for the first number, let it be x:  

From 19 - x = -8  

So, x = 19 + 8 = 27.  

Next, let’s find the 8th number, which we’ll call a:  

From a - (-6) = -2  

Thus, a = -2 - 6 = -8.  

Now, let’s determine the 9th number, referred to as b:  

From b - (-8) = -1  

So, b = -1 - 8 = -9.  

Therefore, the complete sequence is: 27, 19, 12, 6, 1, (-3), (-6), (-8), (-9).

15. Here are six integer cards: (+1), (+7), (+18), (–5), (–2), (–9). You can pick any of these and make an expression using addition(s) and subtraction(s).

Here is an expression: (+18)+(+1)–(+7) – (–2) which gives a value (+14). Now, pick cards and make an expression such that its value is closer to (– 30).

Ans:  Let’s attempt to create an expression that comes as close to -30 as possible using the given cards: (+1, +7, +18, -5, -2, -9). 

One possible expression is: (-9) + (-5) + (-2) + (-18) + (+1). 

Now, let’s calculate the value step by step:

1. (-9) + (-5) = -14  

2. -14 + (-2) = -16  

3. -16 + (-18) = -34  

4. -34 + (+1) = -33  

Therefore, the value of this expression is -33, which is quite close to -30.

16. The sum of two positive integers is always positive but a (positive integer) – (positive integer) can be positive or negative. What about

a. (positive) – (negative) b. (positive) + (negative)

c. (negative) + (negative) d. (negative) – (negative)

e. (negative) – (positive) f. (negative) + (positive)

Ans: 

a. (Positive) - (Negative)  

Subtracting a negative number is equivalent to adding its positive counterpart, which will always yield a positive result. For instance, 5 - (-3) = 5 + 3 = 8.

b. (Positive) + (Negative):  

The outcome depends on the magnitudes of the numbers involved. If the positive number is greater, the result will be positive; if the negative number is greater, the result will be negative.  

For example:  

7 + (-4) = 3 (positive)  

4 + (-7) = -3 (negative)

c. (Negative) + (Negative):  

Adding two negative numbers always produces a negative result.  

For instance, -2 + (-3) = -5.

d. (Negative) – (Negative):  

This is similar to adding the positive version of the second negative number to the first negative number. If the first negative number has a larger magnitude, the result will be negative. Conversely, if the first negative number has a smaller magnitude than the second, the result will be positive. 

For example:  

7 + (-4) = 3 (positive)  

-4 + (-7) = -11 (negative).

e. (Negative) – (Positive):  

This will always result in a negative value because you are subtracting a positive number from a negative one. For instance, -4 - 2 = -6.

f. (Negative) + (Positive):  

Just like with (Positive) + (Negative), the result depends on the magnitudes of the numbers involved. If the positive number is larger, the outcome will be positive; if the negative number has a greater magnitude, the result will be negative. 

For example:  

-3 + 5 = 2 (positive)  

-5 + 3 = -2 (negative)

17. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?

Ans:  Let’s examine the sequence of the string:  

3, -2, 3, -2, 3, -2  

Since the pattern repeats, we can group the tokens into sets of 5, where the total for each set is 3 - 2 = 1.  

With 100 tokens in the string, we have:  

Total sets = $\frac{100}{ 5} = 20 sets.  

The total value of one set is 1.  

Therefore, the overall value of the string is 1 $\times$ 20 = 20.

Benefits of NCERT Solutions for Class 6 Maths Chapter 10 Exercise 10.4 The Other Side of Zero

• Exercise 10.3 solutions help students understand the concept of positive and negative numbers, making it easier to differentiate between them.

• These solutions use the number lines so that students can visualise how integers interact, making abstract concepts more concrete.

• Students get to practise a variety of problems, from basic integer operations to more complex applications, strengthening their understanding of the topic.

• Students get a clear understanding of how zero functions as a middle point between positive and negative numbers in integer operations..

• The solutions are based on the NCERT curriculum, ensuring that all the important topics are covered thoroughly.

Class 6 Maths Chapter 10: Exercises Breakdown

Important Study Material Links for Maths Chapter 2 Class 10

Conclusion

Vedantu’s NCERT Solutions for Class 6 Maths Chapter 10, The Other Side of Zero, Exercise 10.4, provides a clear understanding of integers and their operations. These solutions simplify complex concepts like the addition and subtraction of positive and negative numbers, making it easier for students to learn through step-by-step explanations. By using number lines and real-life examples, students can visualise how integers work, enhancing their problem-solving skills. The detailed solutions help build confidence in solving integer-related questions and provide a strong base for future mathematical topics. With these solutions, students can practise effectively and strengthen their understanding of integer operations, helping them succeed in their exams.

Chapter-wise NCERT Solutions Class 6 Maths 

The chapter-wise NCERT Solutions for Class 6 Maths are given below. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

Related Important Links for Maths Class 6

Along with this, students can also download additional study materials provided by Vedantu for Maths Class 6.