NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10 PDF Download

NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10 are prepared according to the latest CBSE syllabus to help students clearly understand the concept of numbers on both sides of zero. 

Class 6 Maths Chapter 10  The Other Side of Zero introduces students to the idea of numbers less than zero and how they can be placed and compared on a number line. 

These easy-to-follow solutions make it simple for students to grasp new concepts and apply them in practical situations. 

Class 6 Maths Chapter 10 The Other Side of Zero Solutions

Class 6 Maths Chapter 10 The Other Side of Zero introduces students to the concept of negative numbers and how they are used in daily life.

This chapter helps learners understand how numbers exist on both sides of zero using a number line. It explains the meaning and use of integers in a fun and easy way, laying a strong foundation for higher classes.

Class 6 Maths Ganita Prakash Chapter 10 Question Answers

Below are the NCERT Solutions for Class 6 Maths Chapter 10  The Other Side of Zero.

These step-by-step answers will help students understand the concepts easily and prepare well for their exams

10.1 Bela’s Building of Fun Figure it Out (Page No. 245)

Question 1.
You start from Floor + 2 and press – 3 in the lift. Where will you reach? Write an expression for this movement.
Solution:
The starting floor is (+ 2)
and the number on the button pressed is (- 3).
∴ The target floor (+2) + (- 3) = + 2 – 3 = -1

Question 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) = _______________
Solution:
(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

Question 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 starting positions and the movements needed to reach Floor -5 and write the expressions.
Solution:
Other such expressions are:
(+3) + (-8) = -5
(+4) + (-9) = -5
(+5) + (-10) = -5
(+6) + (-11) = -5
And there could be infinite such expressions.

10.1 Bela’s Building of Fun Figure it Out (Page No. 246)

Evaluate these expressions by thinking of them as the resulting movement of combining button presses:
(a) (+1) + (+4) = _______________
Solution:
Target floor = (+1) + (+4) = +5

(b) (+4) + (+1) = _______________
Solution:
Target floor = (+4) + (+1) = +5

(c) (+4) + (-3) + (-2) = _______________
Solution:
Target floor = (+4) + (-3) + (-2)
= 4 + (-5)
= -1

(d) (-1) + (+2) + (-3) = _______________
Solution:
Target floor = (-1) + (+2) + (-3)
= (-4) + (2)
= -2

10.1 Bela’s Building of Fun Figure it Out (Page No. 247)

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.

Question 1.
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
Solution:
(a) Floor – 2 is lower than the floor + 5.
So, – 2 < + 5
(b) Floor – 5 is lower than the floor + 4.
So, – 5 < + 4
(c) Floor – 5 is lower than the floor – 3.
So, – 5 < – 3
(d) Floor + 6 is higher than the floor – 6.
So, + 6 > – 6
(e) Floor 0 is higher than the floor – 4.
So, 0 > – 4
(f) Floor 0 is lower than the floor + 4.
So, 0 < + 4

Question 2.
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
Solution:
(a) -10 > -12
(b) +17 > -10
(c) 0 > – 20
(d) + 9 > – 9
(e) – 25 < – 7
(f) +15 > -17

Question 3.
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.

Solution:
Floor B is 9 floors lower than Floor 0.
So, the number of Floor B is -9.
Floor C is 6 floors lower than Floor 0.
So, the number of Floor C is -6.
Floor F is 2 floors higher than Floor 0.
So, the number of Floor F is +2.
Floor G is 6 floors higher than Floor 0.
So, the number of Floor G is +6.
Floor H is 11 floors higher than Floor 0.
So, the number of Floor H is +11.

Question 4.
Mark the following floors of the building shown on the right.
(a) -7
(b) -4
(c) +3
(d) -10
Solution:
Floors -7, -4, +3, and -10 of the building are marked on the line given on previous page.

10.1 Bela’s Building of Fun Figure it Out (Page No. 249)

Question 1.
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) = _______________
Solution:
(a) (+1) – (+4) = 1 – 4 = -3
(b) (0) – (+2) = 0 – 2 = -2
(c) (+4) – (+1) = 4 – 1 = 3
(d) (0) – (-2) = 0 + 2 = 2
(e) (+4) – (-3) = 4 + 3 = 7
(f) (-4) – (-3) = -4 + 3 = -1
(g) (-1) – (+2) = -1 – 2 = -3
(h) (-2) – (-2) = -2 + 2 = 0
(i) (-1) – (+1) = -1 – 1 = -2
(j) (+3) – (-3) = 3 + 3 = 6

10.1 Bela’s Building of Fun Figure it Out (Page No. 251)

Complete these expressions. Check your answers by thinking about the movement in the mineshaft.
(a) (+40) + _______________ = +200
Solution:
Given (+40) + _______________ = +200
Let (+40) + x = +200
⇒ +x = 200 – 40 = 160
∴ (+40) + (+160) = +200

(b) (+40) + _______________ = -200
Solution:
Given (+40) + _______________ = -200
Let (+40) + x = -200
⇒ x = -200 – 40 = -240
∴ (+40) + (-240) = -200

(c) (-50)+ _______________ = +200
Solution:
Given (-50) + _______________ = +200
Let (-50) + x = +200
⇒ x = +200 – (-50) = +250
∴ (-50) + (+250) = +200

(d) (-50) + _______________ = -200
Solution:
Given (-50) + _______________ = -200
Let (-50) + x = -200
⇒ x = -200 – (-50) = -150
∴ (-50) + (-150) = -200

(e) (-200) – (-40) = _______________
Solution:
Given (-200) – (-40) = _______________
Let (-200) – (-40) = x
⇒ (-200) – (-40) = -160 = x
∴ (-200) – (-40) = -160

(f) (+200) – (+40) = _______________
Solution:
Given (+200) – (+40) = _______________
Let (+200) – (+40) = x
⇒ +160 = x
∴ (+200) – (+40) = +160

(g) (-200) – (+40) = _______________
Solution:
Given (-200) – (+40) = _______________
Let (-200) – (+40) = x
⇒ (-200) + (-40) = -240 = x
∴ (-200) + (-40) = -240

10.1 Bela’s Building of Fun Figure it Out (Page No. 253 – 254)

Question 1.
Write down the above 3 marked negative numbers in the following boxes:

Solution:

Question 2.
Is 2 > -3? Why? Is -2 < 3? Why?
Solution:
Represent the numbers 2, -3, -2 and 3 on a number line.

2 is to the right of-3 on the number line.
So, 2 > -3.
And, -2 is to the left of 3 on the number line. So, -2 < 3.

Question 3.
(i) -5 + 0
Solution:

(ii) 7 +(-7)
Solution:

(iii) -10 + 20
Solution:

(iv) 10 – 20
Solution:

(v) 7 – (-7)
Solution:

(vi) -8 – (-10)?
Solution:

10.2 The Token Model Figure it Out (Page No. 258)

Question 1.
Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know:
(a) (+10) – (+7)
Solution:

(+10) – (+7) = 3

(b) (-8) – (-4)
Solution:

(-8) – (-4) = -4

(c) (-9) – (-4)
Solution:

(-9) – (-4) = -5

(d) (+9) – (+12)
Solution:
Start with 9 positives.

But there are not enough tokens to take out 12 positives. So, we will add 3 extra zero pairs and then take out 12 positives.

(+9) – (+12) = -3

(e) (-5) – (-7)
Solution:
Start with 5 negatives.

But there are not enough tokens to take out 7 negatives. So, we will add 2 extra zero pairs and then take out 7 negatives.

(-5) – (-7) = +2

(f) (-2) – (-6)
Solution:
Start with 2 negatives.

But there are not enough tokens to take out 6 negatives. So, we will add 4 extra zero pairs and then take out 6 negatives.

(-2) – (-6) = +4

10.2 The Token Model Figure it Out (Page No. 259)

Question 1.
Try to subtract – 3 – (+ 5).
How many zero pairs will you have to put in? What is the result?
Solution:
You have – 3 (3 negative tokens). Subtracting + 5 means you need to remove 5 positive tokens.
Add 5 zero pairs to introduce 5 positive tokens that can be removed.
Remove the 5 positive tokens.
After removing these, you are left with 3 original negative tokens and 5 additional negative tokens.
3 original negative tokens + 5 additional negative tokens
= 8 negative tokens
So, the result is – 8 and you needed to add 5 zero pairs to perform the subtraction.
– 3 – (+ 5) = – 8

10.3 Integers in Other Places Figure it Out (Page No. 260)

Question 1.
Suppose you start with 0 rupees 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?
Solution:
Here, Credits = ₹ 30 + ₹ 40 + ₹ 50 = ₹ 120
and Debits = ₹ 40 + ₹ 50 + ₹ 60 = ₹ 150
∴ Balance = Credits – Debits
= ₹ 120 – ₹ 150
= – ₹30
Therefore, your bank account balance is – ₹ 30.

Question 2.
Suppose you start with 0 rupees 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?
Solution:
Consider ‘credits’ as positive numbers and ‘debits’ as negative numbers.
Total credits = +256
Total debits = (-1) + (-2) + (-4) + (-8) + (-16) + (-32) + (-64)+ (-128) =-255
Account balance = Total credits + Total debits = (+256) + (-255) = +1
Hence, the account balance is ₹ 1.

Question 3.
Why is it generally better to try and maintain a positive balance in your bank account? What are circumstances under which it many be worthwhile to temporarily have a negative balance?
Solution:
Maintaining a positive balance ensures you avoid fees or interest charges.
In situations like making an investment that will quickly pay off a profit, a temporary negative balance might be worthwhile.

Question 4.
Which other grids can be filled in multiple ways? What could be the reason?
Solution:
We can also fill up the grid 1 in multiple ways. Any grid with 3 or fewer prefilled numbers can be filled in multiple ways.

10.4 Explorations with Integers Figure it Out-2 (Page No. 265 – 266)

Question 1.
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
Solution:
(a) -6, -5, -4, -3, -2, -1
(b) -3, -2, -1, 0, 1, 2, 3
(c) -14, -13, -12, -11, -10, -9
(d) -29, -28, -27, -26, -25, -24

Question 2.
Give three numbers such that their sum is – 8.
Solution:
Three numbers whose sum is – 8, are – 2, – 5, -1.

Question 3.
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.
Solution:
The required numbers are – 9, 0, 7, 9 and 11.

Question 4.
Solve these:

Solution:

Question 5.
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?
Solution:
(a) 1874
(b) 176 BCE
(c) 360 BCE

Question 6.
Complete the following sequences:
(a) (-40), (-34), (-28), (-22) ______, ______, ______
(b) 3, 4, 2, 5, 1, 6, 0, 7, ______, ______, ______
(c) ______, ______, 12, 6, 1, (-3), (-6), ______, ______, ______
Solution:
(a) In this sequence, each consecutive number is obtained by adding 6 to the previous number.
The required sequence is -40, -34, -28, -22, -16, -10, -4,…..

(b) Identify the Differences Between Consecutive Terms:
3 to 4: +1
4 to 2: -2
2 to 5: +3
5 to 1: -4
1 to 6: +5
6 to 0: -6
0 to 7: +7
So, now 7 – 8 = -1
-1 + 9 = 8
8 – 10 = -2
Therefore, the required numbers are -1, 8 and -2
3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2

(c) The numbers to the right of 12 are decreasing as per the following rule:
12 to 6 = -6
6 to 1 = -5
1 to -3 = -4
-3 to -6 = -3
So, further, they should decrease as follows:
-6 – 2 = -8
-8 – 1 = -9
Now, the numbers to the left of 12 should each increase as follows:
12 + 7 = 19
19 + 8 = 27
27, 19, 12, 6, 1, (-3), (-6), -8, -9, -9

Question 7.
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).
Solution:
Let’s try to create an expression that gets 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)
Let’s calculate the value step by step:

1. 1. (-9) + (-5) = -14

2. 2. -14 + (-2) = -16

3. 3. -16 + (-18) = -34

4. 4. -34 + (+1) = -33

Hence, the value of this expression is (-33), which is quite close to (-30).

Question 8.
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)
Solution:
(Positive) – (Negative):
Subtracting a negative number is the same as adding its positive counterpart. So, this will always be positive.
For example, 5 – (-3) = 5 + 3 = 8.

(b) (Positive) + (Negative)
Solution:
(Positive) + (Negative):
This depends on the magnitudes of the numbers. If the positive number is larger, the result is positive; if the negative number is larger, the result is negative.
For example,
7 + (-4) = 3 (positive)
4 + (-7) = -3 (negative)

(c) (Negative) + (Negative)
Solution:
(Negative) + (Negative):
Adding two negative numbers always results in a negative number.
For example, -2 + (-3) = -5.

(d) (Negative) – (Negative)
Solution:
(Negative) – (Negative):
This is like adding the positive counterpart of the second number to the first negative number.
If the first negative number is larger in magnitude, the result is negative.
However, if the first negative number is smaller than the second negative number, then it is positive.
For example,
7 + (4) = 3 (positive)
4 + (-7) = -3 (negative)

(e) (Negative) – (Positive)
Solution:
(Negative) – (Positive):
This will always be negative because you’re subtracting a positive number from a negative number.
For example, -4 – 2 = -6.

(f) (Negative) + (Positive)
Solution:
(Negative) + (Positive):
Similar to (Positive) + (Negative), it depends on the magnitudes. If the positive number is larger, the result is positive; if the negative number is larger, the result is negative.
For example,
-3 + 5 = 2 (positive)
-5 + 3 = -2 (negative)

Question 9.
This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?

Solution:
The group of 5 tokens hasthe value, (+3) + (-2) = +1
There will be 20 such groups in a string of 100 tokens.
So, the total value will be +20.

10.5 A Pinch of History Figure it Out (Page No. 268)

Question 1.
Give your examples of each rule.
Solution:
Rules for Addition:
1. The sum of two positives is positive.
3 + 4 = 7

2. The sum of two negatives is negative.
(-4) + (-6) = -10

3. To add a positive number and a negative number, subtract the smaller number (without the sign) from the greater number (without the sign), and place the sign of the greater number to obtain the result.
(-3) + 4 = 1

4. The sum of a number and its inverse is zero.
(-4) + 4 = 0

5. The sum of any number and zero is the same number.
(-7) + 0 = -7

Rules for Subtraction:
1. If a smaller positive is subtracted from a larger positive, the result is positive.
9 – 8 = 1

2. If a larger positive is subtracted from a smaller positive, the result is negative.
7 – 8 = -1

3. Subtracting a negative number is the same as adding the corresponding positive number.
3 – (-5) = 3 + 5 = 8

4. Subtracting a number from itself gives zero.
9 – 9 = 0

5. Subtracting zero from a number gives the same number.
8 – 0 = 8

Adding, Subtracting, and Comparing any Numbers

Try evaluating the following expressions by similarly drawing or imagining a suitable lift: (Page No. 251)
(a) -125 + (-30)
(b) +105 – (-55)
(c) +105 + (+55)
(d) +80 – (-150)
(e) +80 + (+150)
(f) -99 – (-200)
(g) -99 + (+200)
(h) +1500 – (-1500)
Solution:
(a) -125 + (-30) = -125 – 30 = -155
(b) +105 – (-55) = 105 + 55 = +160
(c) +105 + (+55) = 105 + 55 = +160
(d) +80 – (-150) = 80 + 150 = +230
(e) +80 + (+150) = 80 + 150 = +230
(f) -99 – (-200) = -99 + 200 = +101
(g) -99 + (+200) = -99 + 200 = +101
(h) +1500 – (-1500) = +1500 + 1500 = 3,000

In the other exercises that you did above, did you notice that subtracting a negative number was the same as adding the corresponding positive number? (Page No. 252)
Solution:
Subtracting a number is the same as adding it’s opposite. So subtracting a positive number is like adding a negative number – you move to the left on the number line. Subtracting a negative number is like adding a positive number – you move to the right on the number line.
For example: Subtract -2 – (-3)
Start at -2 and move 3 units to the right.

So, -2 – (-3) = +1

Take a look at the ‘infinite lift’ above. Does it remind you of a number line? In what ways? (Page No. 252)
Solution:
A number line is a way of representing numbers visually on a straight line. For instance, the number line has arrows at the end to represent this, idea of having no bounds. The symbol used to represent infinity is ∞.

NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10

Students can now download the NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10 The Other Side of Zero Class 6 in PDF format from below. 

These solutions provide step-by-step explanations to help learners grasp the concept of numbers beyond zero, including both positive and negative integers. 

Benefits of Using NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10

Here are the benefits of using NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 10  The Other Side of Zero:

1. Helps students learn negative numbers and integers in a simple and effective way.

2. The solutions are based on the CBSE syllabus and exam pattern.

3. Each question is solved with clear steps to help students understand the logic.

4. Regular practice strengthens mathematical thinking and accuracy.

5. Prepares students for more advanced concepts in higher classes.

6. Helps in revising key topics quickly before tests and exams.