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

Exercise 10.1

1. You start from Floor +2 and press –3 in the lift. Where will you reach? Write an expression for this movement.

Ans: You start from Floor +2 and press -3 in the lift. Where will you reach? Write an expression for this movement.

Starting floor: +2

Movement: -3

Expression: +2 + (-3)

Answer: +2 + (-3) = -1

You will reach Floor -1.

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). 

a. (+1)+(+4) = _______ 

b. (+4)+(+1) = _______ 

c. (+4)+(– 3) = _______ 

d. (–1)+(+2) = _______ 

e. (–1)+(+1) = _______ 

f. 0+(+2) = _________ 

g. 0+(–2) = _________

Ans: Evaluate these expressions:

a. +1 + (+4) = +5

b. +4 + (+1) = +5

c. +4 + (-3) = +1

d. -1 + (+2) = +1

e. -1 + (+1) = 0

f. 0 + (+2) = +2

g. 0 + (-2) = -2

3. Starting from different floors, find the movements required to reach Floor –5. For example, if I start at Floor +2, I must press –7 to reach Floor –5. The expression is (+2) + (–7) = –5. 

Find more such starting positions and the movements needed to reach Floor –5 and write the expressions.

Ans: Starting from different floors, find the movements required to reach Floor -5.

Example 1: If you start at Floor +2, press -7 to reach Floor -5.

Expression: +2 + (-7) = -5

Example 2: Start at Floor +4, press -9 to reach Floor -5.

Expression: +4 + (-9) = -5

Example 3: Start at Floor 0, press -5 to reach Floor -5.

Expression: 0 + (-5) = -5

Example 4: Start at Floor -3, press -2 to reach Floor -5.

Expression: -3 + (-2) = -5

4. Evaluate these expressions by thinking of them as the resulting movement of combining button presses: 

a. (+1)+(+4) = _____________ 

b. (+4)+(+1) = _____________ 

c. (+4)+(– 3)+(–2) = _______ 

d. (–1)+(+2)+(–3) = _______

Ans: 

a.  (+1)+(+4) = +5

b. (+4)+(+1) = +5

c. (+4)+(– 3)+(–2) = (+4)-3-2 =(+4)-(3+2) = (+4)-(5) = -1

d. (-1)+2-3 = -1-1 = -2 

5. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. 

a. –2 __ +5 

b. –5 __ +4 

c. –5 __ –3

d. +6 __ –6

e. 0 __ –4 

f. 0 __ +4 

Notice that all negative number floors are below Floor 0. So, all negative numbers are less than 0. All the positive number floors are above Floor 0. So, all positive numbers are greater than 0.

Ans: 

a. - 2 < + 5

b. - 5 < + 4

c. - 5 < - 3

d. + 6 > - 6

e. 0 > - 4

f. 0 < + 4

6. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: 

a. –10 __ –12 

b. +17 __  –10 

c. 0 __  –20 

d. +9 __  –9 

e. –25 __ –7 

f. +15 __  –17

Ans:

a. –10 > – 12 

b. +17 > –10 

c. 0 > –20 

d. +9 > –9 

e. –25 < –7 

7. If Floor A = –12, Floor D = –1 and Floor E = +1 in the building shown on the right as a line, find the numbers of Floors B, C, F, G, and H.

Ans: 

8. Mark the following floors of the building shown on the right. 

a. –7 b. – 4 c. +3 d. – 10

Ans:

9. Complete these expressions. You may think of them as finding the movement needed to reach the Target Floor from the Starting Floor. 

a. (+1) –(+4) = _______ 

b. (0) –(+2) = _________ 

c. (+4) –(+1) = _______ 

d. (0)– (–2) = _________ 

e. (+4) – (–3) = _______ 

f. (–4)–(–3) = ________ 

g. (–1) –(+2) = _______ 

h. (–2) – (–2) = ________ 

i. (–1) – (+1) = _______ 

j. (+3)– (–3) = ________

Ans:

a. (+1) –(+4) = -3

b. (0) –(+2) = -2 

c. (+4) –(+1) = 3 

d. (0)– (–2) = 2 

e. (+4) – (–3) =7 

f. (–4)–(–3) = -1 

g. (–1) –(+2) = -3 

h. (–2) – (–2) = 0

i. (–1) – (+1) = -2 

j. (+3)– (–3) = 6

10. Complete these expressions. 

a. (+40)+ ______ = +200 

b. (+40)+_______=–200 

c. (–50)+ ______ = +200 

d. (–50)+_______=–200 

e. (–200) – (–40) = _______ 

f. (+200)– (+40) = _______ 

g. (–200) –(+40) = _______ 

Ans:

a. (+40)+(+160) = +200 

b. (+40)+ (-240) =–200 

c. (–50)+ (+250) = +200 

d. (–50)+ (-150) =–200 

e. (–200) – (–40) = -160 

f. (+200)– (+40) = +160 

g. (–200) –(+40) = -240 

11. Mark 3 positive numbers and 3 negative numbers on the number line above. 

Ans: 

12. Write down the above 3 marked negative numbers in the following boxes: ___ ___ ___.

Ans: -7, -3. -2

13. Is 2 > – 3? Why? Is –2 < 3? Why?

Ans: Yes, 2 > -3. This is because on the number line, 2 is to the right of -3, meaning it is greater than -3.

Yes, -2 < 3. This is also because on the number line, -2 is to the left of 3, meaning it is less than 3. 

Positive numbers are always greater than negative numbers, and among negative numbers, the one closer to zero is greater.

14. What are a. –5+0    b. 7+(–7)    c. –10+20    d. 10 –20    e. 7–(–7)    f. –8–(–10)?

Ans:

a. - 5 + 0 = -5

b. 7 + (+7) = 14

c. - 10 + 20 = 10

d. 10 - 20 = - 10

e. 7 - ( -7) = 14

f. -8 - (-10) = 2

Exercise 10.2

1. Complete the additions using tokens.

a. (+6)+(+4) 

Ans:

For (+6), we use 6 positive tokens 

And for (+4), we use 4 positive tokens

Therefore, by combining we will get 

(+6) + ( +4) = (+10)

Hence, Counting all the positive tokens we get (+10)

b. (−3)+(−2)

Ans: For (-3) we use 3 negative tokens

For (-2) we use 2 negative tokens

Token

Now combining them we will get 

(-3) + (-2) = -5

Hence, counting all the negative tokens we get (-5)

c. (+5)+(−7)

Ans: For (+5) we use 5 positive tokens

For (-7) we use 7 negative tokens

Now, combining them we will get 

Therefore, remaining tokens -= -2

d. (−2)+(+6)

Ans: For (-2) we use 2 negative tokens

For (+6) we use 6 positive tokens

Now, combining we will get 

(-2) +(+6) = +4

Hence, Remaining tokens are (+4)

2. Cancel the zero pairs in the following two sets of tokens. On what floor is the lift attendant in each case? What is the corresponding addition statement in each case?

Ans: 

a. From the given image we can remove 3 pairs 

Therefore the remaining are 2 negative tokens 

The lift attendant is on the second floor below the ground floor.

The corresponding addition statement = (+3)+(-5) = (-2)

b. From the given image we can remove 3 pairs.

The remaining are 3 positive tokens.

The lift attendant is on the third floor above the ground floor.

The corresponding addition statement = (+6) + (-3) = (+3)

3. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: 

a. (+10) –(+7) b. (–8)– (–4) c. (–9) –(–4) 

d. (+9) –(+12) e. (–5)– (–7) f. (–2) –(–6)

Ans:

a. Here there are 10 positive tokens take away 7 positive tokens

Therefore, (+10)-(+7) = (+3)

b. Here there are -8 negative tokens take away -4 negative tokens

Therefore, (-8) -( - 4) = (-4)

c. Here there are -9 negative tokens take away -4 negative tokens

Therefore, (-9) -( - 4) = (-5)

d. There are not enough tokens to take out 12 positive tokens from 9 positive tokens.

So we put an extra 3 zero pairs ( 1 pair = a positive and a negative token)

Now we can take out 12 positive tokens 

Therefore, (+9) - (+12) = (-3)

e. There are ot enough tokens to take out -7 negative tokens from -5 negative tokens.

So, we put extra 2 zero pairs( 1 pair = a positive and a negative token)

Now, we can take out -7 negative tokens.

Hence, (-5)-(-7) = +2

f. There are not enough tokens to take out -6 negative tokens from -2 negative tokens

So, we put extra 4 zero pairs (1 pair = a negative and a positive token)

Now we can take out -6 negative tokens 

Hence, (-2)-(-6) = (+4)

4. Complete the subtractions: 

a. (–5) –(–7) b. (+10) –(+13) c. (–7) –(–9) 

d. (+3) – (+8) e. (–2) –(–7) f. (+3)–(+15) 

Ans:

a. There are not enough tokens to take out -7 negative tokens from -5 negative tokens.

So, we take an extra 2 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out -7 negative tokens.

Hence, (-5)-(-7) = +2

b. There are not enough tokens to take out +10 positive tokens from +13 positive tokens.

So, we take an extra 3 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out +13 positive tokens.

Hence, (+10)-(+13) = -3

c. There are not enough tokens to take out -9 negative tokens from -7 negative tokens.

So, we take an extra 2 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out -9 negative tokens.

Hence, (-7)-(-9) = +2

d. There are not enough tokens to take out +8 positive tokens from +3 positive tokens.

So, we take an extra 5 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out +8 positive tokens.

Hence, (+3) -(+8) = -5

e.  There are not enough tokens to take out -7 negative tokens from -2 negative tokens.

So, we take an extra 2 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out -7 negative tokens.

Hence, (-2) - (-7) = +5

f. There are not enough tokens to take out +15 positive tokens from +3 positive tokens.

So, we take an extra 12 Zero pairs (1 pair = a positive and a negative token) 

Now, we can take out +15 positive tokens.

Hence, (+3) - (+15) = -12

5. Try to subtract: –3– (+5). 

How many zero pairs will you have to put in? What is the result?

Ans: Here we have to take out 5 positives from -3 negatives. But there are not enough positive tokens.

So, we put down an extra 5 zero pairs (1 pair = a negative and positive token)

Now, we can take out 5 positive tokens

Hence, -3-(+5) = - 8

6. Evaluate the following using tokens. 

a. (–3)–(+10) b. (+8)– (–7) c. (–5)–(+9) 

d. (–9) – (+10) e. (+6) –(–4) f. (–2)–(+7)

Ans:

a. Here we have to take 10 positive tokens from -3 negative tokens. But there are not enough positive tokens.

So, we put down an extra 10 zero pairs (1 pair = a positive and a negative token)

Now, we can take out 10 positive tokens 

Hence (-3) - (+10) = -13

b. Here we have to take out -7 negative tokens from 8 positives. But there are not enough negative tokens.

So, we put down an extra 10 zero pairs.(1 pair = a negative and a positive pair)

Now, we can take out 10 positive tokens.

Hence, (-3)-(+10) = -13

b. Here we have to take out -7 negative tokens from 8 positive tokens. Butu there are not enough negative tokens.

So, we put down an extra 7 zero pairs (1 pair = a negative and a positive pair)

Now we can take out -7 negatives tokens

Hence, -5-(+9) = - 14

c. Here we have to take out 9 positives from -5 negative tokens. But there are not enough positive tokens.

So, we put extra 9 zero pairs (1 pair = a positive and a negative token)

Now we can take out 9 positive tokens.

Hence -5 - (-9) = =-14

d. Here we have to take out 10 positive tokens from -9 negative tokens. Butu there are not enough positive tokens.

So, we put down an extra 10 zero pairs (1 pair = a negative and a positive pair)

Now we can take out 10 positive tokens

Hence, -9-(+10) = -19

e. Here we have to take out -4 negative tokens from 6 positive tokens. But there are not enough negative tokens.

So, we put down an extra 4 zero pairs (1 pair = a negative and a positive pair)

Now we can take out -4 negative tokens

Hence, (+6)-(-4) =10

f. Here we have to take out 7 positive tokens from -2 negative tokens. Butu there are not enough positive tokens.

So, we put down an extra 7 zero pairs (1 pair = a negative and a positive pair)

Now we can take out 7 positive tokens

Hence, (-2) - (+7) = -9

Exercise 10.3

1. Suppose you start with ₹ 0 in your bank account, and then you have credits of ₹ 30, ₹ 40, and ₹ 50, and debits of ₹ 40, ₹ 50, and ₹ 60. What is your bank account balance now? 

Ans: Given 

Credits = ₹ 30 + ₹ 40 + ₹ 50 = ₹ 120

Debits = ₹ 40 + ₹ 50 + ₹ 60 = ₹ 150

Therefore balance = Credits - Debits 

= ₹ 120 - ₹ 150

= - ₹ 30

Hence, your bank account balance is - ₹30.

2. Suppose you start with `0 in your bank account, and then you have debits of ₹ 1, 2, 4, 8, 16, 32, 64, and 128, and then a single credit of ₹ 256. What is your bank account balance now? 

Ans: Given

Debits = ₹ 1 + ₹ 2 + ₹ 4 + ₹ 8 + ₹ 16 + ₹ 32 + ₹ 64 + ₹ 128 = ₹ 255

Credit = ₹ 256

Therefore, Balance = Credits - Debits 

= ₹ 256 - ₹ 255 

= ₹ 1

Hence, your bank account balance is ₹ 1.

3. Why is it generally better to try and maintain a positive balance in your bank account? What are circumstances under which it may be worthwhile to temporarily have a negative balance?

Ans: Having a positive balance in your bank account is generally preferable because:

• Avoids Overdraft Fees: Many banks charge fees if your account goes into a negative balance.

• Provides Financial Security: A positive balance ensures you have money available for unexpected expenses or emergencies.

• Helps Build Good Credit History: Maintaining a positive balance can improve your credit score, making it easier to obtain loans or credit cards in the future.

However, there are some situations where having a temporary negative balance could be acceptable:

• Overdraft Protection: Some banks offer overdraft protection, preventing bounced checks or failed transactions.

• Planned Large Expenses: If you anticipate receiving a large income soon and need to make necessary purchases, a brief negative balance may be manageable.

4. Looking at the geographical cross section, fill in the respective heights:

a. ______ b. _____ c. _____ d. _____

e. ______ f. ______ g. _____

Ans: You will need to estimate the height of points A to G from the graph.  

Check the vertical axis to find the height value for each point.  

Here are the heights:

A = +1500 m

B = -500m

C = +300m

D = -1200m

E = +1200m

F = -200m

G = +100m

5. Which is the highest point in this geographical cross section? Which is the lowest point? 

Ans: Find the point on the graph that reaches the highest point above sea level. Point A represents the highest point. Likewise, identify the lowest point, including those below sea level. Point D represents the lowest point.

6. Can you write the points A, B, …, G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights? 

Ans: Using the heights determined in question 1, arrange the points from highest to lowest (descending order) as A > E > C > G > F > B > D, and from lowest to highest (ascending order) as D < B < F < G < C < E < A.

7. What is the highest point above sea level on Earth? What is its height? 

Ans: Mount Everest is the highest point on Earth, standing at 8,848 metres above sea level.

8. What is the lowest point with respect to sea level on land or on the ocean floor? What is its height? (This height should be negative)

Ans: The deepest known point on Earth is the Mariana Trench, located in the Pacific Ocean, with a depth of 11,034 metres below sea level.

9. Do you know that there are some places in India where temperatures can go below 0°C? Find out the places in India where temperatures sometimes go below 0°C. What is common among these places? Why does it become colder there and not in other places?

Ans: 

• Ladakh: This region is famous for its extremely cold winters, with temperatures frequently falling below 0°C.

• Himachal Pradesh: High-altitude areas, particularly in northern Himachal Pradesh, often experience sub-zero temperatures.

• Jammu & Kashmir: Similar to Ladakh, parts of Jammu and Kashmir, especially the mountainous regions, endure freezing temperatures.

• Sikkim: As a mountainous state, Sikkim experiences sub-zero temperatures in some areas.

• Arunachal Pradesh: The higher elevations in Arunachal Pradesh can face cold conditions, with temperatures dropping below 0°C.

Common Factor: All these regions are located in the Himalayan range, known for its high altitudes.

Reason for Colder Temperatures:  

• High Altitude: As altitude increases, temperatures decrease due to the thinner atmosphere at higher elevations, which leads to less heat retention.  

• Distance from the Equator: These regions are farther from the equator, receiving less direct sunlight, contributing to colder temperatures.

10. Leh in Ladakh gets very cold during the winter. The following is a table of temperature readings taken during different times of the day and night in Leh on a day in November. Match the temperature with the appropriate time of the day and night.

Ans:

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.

Exercise 10.5

1. Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or terms of a number line? 

Ans: Brahmagupta’s Rules

1. Addition of Positive Numbers:

Rule: When adding two positive numbers, the result is positive.
Bela’s Building: Imagine Bela starts on the 3rd floor. If she moves up 2 floors, she reaches the 5th floor.
Number Line: This is like moving from 3 to 5 by adding 2.
Example: 3 + 2 = 5

2. Addition of Negative Numbers:

Rule: Adding two negative numbers results in a negative number.
Bela’s Building: If Bela begins 3 floors below ground level (-3) and goes down another 2 floors, she ends up 5 floors below ground (-5).
Number Line: This is represented by moving from -3 to -5 by adding -2.
Example: −3 + (−2) = −5

3. Addition of a Positive and a Negative Number:

Rule: Subtract the smaller absolute value from the larger absolute value and keep the sign of the larger one.
Bela’s Building: If Bela starts on the 3rd floor and moves down 5 floors, she ends up 2 floors below ground (-2).
Number Line: This is like moving from 3 to -2 by adding -5.
Example: 3 + (−5) = −2

4. Subtraction of a Positive Number from a Negative Number:

Rule: Subtracting a positive number from a negative one is the same as adding both numbers and keeping the negative sign.
Bela’s Building: If Bela is 3 floors below ground (-3) and goes down 2 more floors, she ends up 5 floors below ground (-5).
Number Line: This is shown as moving from -3 to -5 by subtracting 2.
Example: −3 − 2 = −5

5. Subtraction of a Negative Number from a Positive Number:

Rule: Subtracting a negative number from a positive number is equivalent to adding the two numbers.
Bela’s Building: If Bela starts on the 3rd floor and goes up 2 floors, she reaches the 5th floor.
Number Line: This is like moving from 3 to 5 by subtracting -2.
Example: 3 − (−2) = 5

6. Subtraction of a Negative Number from a Negative Number:

Rule: Subtracting a negative number from another negative number is the same as adding their absolute values while keeping the negative sign.
Bela’s Building: If Bela is 3 floors below ground (-3) and moves up 2 floors, she ends up 1 floor below ground (-1).
Number Line: This is like moving from -3 to -1 by subtracting -2.
Example: −3 − (−2) = −1

2. Give your own examples of each rule.

Ans:

1. Addition of Positive Numbers:

Example: 6 + 3 = 9
Bela’s Building: Starting on the 6th floor and moving up 3 floors to reach the 9th floor.

2. Addition of Negative Numbers:

Example: −7 + (−4) = −11
Bela’s Building: Starting 7 floors below ground and moving down 4 more floors, reaching 11 floors below ground.

3. Addition of a Positive and a Negative Number:

Example: 9 + (−6) = 3
Bela’s Building: Starting on the 9th floor and moving down 6 floors, arriving at the 3rd floor.

4. Subtraction of a Positive Number from a Negative Number:

Example: −5 − 4 = −9
Bela’s Building: Starting 5 floors below ground and moving down 4 more floors to 9 floors below ground.

5. Subtraction of a Negative Number from a Positive Number:

Example: 4 − (−3) = 7
Bela’s Building: Starting on the 4th floor and moving up 3 floors to reach the 7th floor.

6. Subtraction of a Negative Number from a Negative Number:

Example: −6 − (−2) = −4
Bela’s Building: Starting 6 floors below ground and moving up 2 floors to end up 4 floors below ground.

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

• The solutions help clarify the concept of positive and negative numbers, making it easier for students to understand their meanings and applications.

• Each solution provides detailed steps, allowing students to follow the logical process of solving problems related to integers.

• The solutions provide a variety of practice problems that reinforce learning and improve problem-solving skills.

• Students learn how to apply the concept of integers to real-life situations, enhancing their understanding of its relevance.

• By solving different types of problems, students gain confidence in their ability to handle mathematical challenges involving integers.

Important Study Material Links for Maths Chapter 2 Class 6

Conclusion

Vedantu's Class 6 Maths NCERT Solutions for Chapter 10, "The Other Side of Zero," provide essential support for students learning about positive and negative numbers. These solutions break down complex concepts into easy steps, making it simpler for students to understand and apply their knowledge. By practising with a variety of problems, students build confidence in their skills and prepare effectively for exams. The clear explanations and structured format help in quick revision and error correction, ensuring that students grasp the material fully. Overall, Vedantu’s solutions not only enhance comprehension but also encourage a practical approach to mathematics, making learning engaging and effective for young learners.

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.