By using Ganita Prakash Book Class 6 Solutions and Chapter 10 The Other Side of Zero Class 6 NCERT Solutions Question Answer, students can improve their problem-solving skills.
Class 6 Maths Chapter 10 The Other Side of Zero Solutions
The Other Side of Zero Class 6 Solutions Questions and Answers
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:
Here, the starting floor is +2 (Art center) and the number of button presses is -3.
Therefore, Target floor = Starting floor + Movement
= (+2) + (-3)
= -1 (Toys Store).
Question 2.
Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the guiding 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) +5
(b) +5
(c) +1
(d) +1
(e) 0
(f) +2
(g) -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 such starting positions and the movements needed to reach Floor -5 and write the expressions.
Solution:
Do it yourself.
10.1 Bela’s Building of Fun Figure it Out (Page No. 246)
Question 1.
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) = __________
Solution:
(a)(+1) + (+4) = + 5
Moving up 1 floor, then moving up 4 more floors results in being on floor + 5.
(b) (+ 4) + (+1) = + 5
Moving up 4 floors, then moving up 1 more floor also results in being on floor + 5.
(c) (+ 4) + (- 3) + (- 2) = -1
Moving up 4 floors, then moving down 3 floors and then moving down 2 more floors results in being on floor – 1.
(d) (-1) + (+ 2) + (- 3) = – 2
Moving down 1 floor, then moving up 2 floors and then moving down 3 floors results in-being on floor-2.
10.1 Bela’s Building of Fun Figure it Out (Page No. 247)
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) Here, floor -2 is lower than floor +5 then -2 < +5.
(b) Here, floor -5 is lower than floor +4 then -5 < +4.
(c) Here, floor -5 is lower than floor -3 then -5 < -3.
(d) Here, floor +6 is greater than floor -6 then +6 > -6.
(e) Here, floor 0 is greater than floor -4 then 0 > -4.
(f) Here, floor 0 is lower than floor +4 then 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) o -20
(d) +9 -9
(e) -25 -7
(f) +15 -17
Solution:
(a) Here, floor -10 is greater than floor -12 then -10 > -12
(b) Here, floor +17 is greater than floor -10 then +17 > -10
(c) Here, floor 0 is greater than floor -20 then 0 > -20
(d) Here, floor +9 is greater than floor -9 then + 9 > -9
(e) Here, floor -25 is lower than floor -7 then -25 < -7
(f) Here, floor +15 is greater than floor -17 then +15 > -17
Question 3.
If Floor A = -12, Floor D = -1, and Floor E = +1 in the building shown above as a line, find the number of floors B, C, F, G, and H.
Solution:
B = -9
C = -6
F = 2
G = 6
H = 11
Question 4.
Mark the following floors of the building shown above.
(a) -7
(b) -4
(c) +3
(d) -10
Solution:
10.1 Bela’s Building of Fun Figure it Out (Page No. 249)
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) = -3
(b) (0) – (-2) = -200
(c) (+4) – (+1) = +3
(d) (o) – (-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
10.1 Bela’s Building of Fun Figure it Out (Page No. 251)
Question 1.
Complete these expressions.
(a) (+40) + ____ = +200
(b) (+40) + ____ = -200
(c) (-50) +____ = +200
(d) (-50) + ____ = -200
(e) (-200) – (-40) = ____
(f) (+200) – (+40) = ____
(g) (-200) – (+40) = ____
Check your answers by thinking about the movement in the mineshaft.
Solution:
(a) (+ 40) + (+ 160) = + 200
Starting at Floor + 40, moving up 160 floors lands you at Floor + 200.
(b) (+ 40) + (- 240) = – 200
Starting at Floor + 40, moving down 240 floors lands you at Floor – 200.
(c) (- 50) + (250) = + 200
Starting at Floor – 50, moving up 250 floors lands you at Floor + 200.
(d) (-50)+ (-150) = -200
Starting at Floor – 50, moving down 150 floors lands you at Floor – 200.
(e) (- 200) – (- 40) = -160
Starting at Floor – 200, moving up 40 floors (since ‘ subtracting a negative is like adding a positive) lands you at Floor – 160.
(f) (+ 200) – (+ 40) = +160
Starting at Floor + 200, moving down 40 floors lands you at Floor + 160.
(g) (-200) – (+40) = -240
Starting at Floor – 200, moving down 40 floors lands you at Floor – 240.
10.1 Bela’s Building of Fun Figure it Out (Page No. 253 – 254)
Question 1.
Mark 3 positive numbers and 3 negative numbers on the number line above.
Solution:
On the Number Line:
Positive numbers: We can mark any three positive numbers, e.g., 3, 6 and 9.
Negative numbers: We can mark any three negative numbers e.g., -2, -5 and -8.
Question 2.
Write down the above 3 marked negative numbers in the following boxes:
Solution:
-2, -5, and -8 are three negative numbers.
Question 3.
Is 2 > -3? Why? Is -2 < 3? Why?
Solution:
Yes, 2 > -3.
Numbers on the right side of a number line are greater than the numbers on the left side of the number line. So, 2 > -3.
Similarly, -2 < 3, as 3 lies right side on the number line w.r.t -2.
Question 4.
What are
(i) -5 + 0
(ii) 7 + (-7)
(iii) -10 + 20
(iv) 10 – 20
(v) 7 – (-7)
(vi) -8 – (-10)
Solution:
(i) -5 + 0 = -5
(ii) 7 + (-7) = 0
(iii) -10 + 20 = 10
(iv) 10 – 20 = -10
(v) 7 – (-7) = 7 + 7 = 14
(vi) -8 – (-10) = -8 + 10 = 2
10.2 The Token Model Figure it Out (Page No. 257)
Question 1.
Complete the additions using tokens.
(a) (+6) + (+4)
(b) (- 3) + (- 2)
(c) (+ 5) + (- 7)
(d) (-2) + (+6)
Solution:
(a) (+ 6) + (+ 4)
Tokens 6 positive tokens and 4 positive tokens. Result 10 positive tokens.
Expression (+ 6) + (+ 4) = +10
(b) (-3) +(-2)
Tokens 3 negative tokens and 2 negative tokens. Result 5 negative tokens.
Expression (- 3) + (- 2) = – 5
(c) (+ 5) + (— 7)
Tokens 5 positive tokens and 7 negative tokens.
Result The 5 positive tokens cancel out 5 of the 7 negative tokens, leaving 2 negative tokens.
Expression (+ 5) + (- 7) = – 2
(d) (-2) +(+6)
Tokens 2 negative tokens and 6 positive tokens.
Result The 2 negative tokens cancel out 2 of the 6 positive tokens, leaving 4 positive tokens.
Expression (- 2) + (+ 6) = + 4
Question 2.
Cancel the zero pairs in the following two sets of token. On what floor is the lift attendant in each case? What is the corresponding addition statement in each case?
Solution:
(a)-2 floor, (+3) + (-5) = (-2)
(b) + 3 floor, (+ 6) + (- 3) = (+ 3)
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)
(b) (-8) – (-4)
(c) (-9) – (-4)
(d) (+9) – (+12)
(e) (-5) – (-7)
(f) (-2) – (-6)
Solution:
Question 2.
Complete the subtractions.
(a) (-5) – (-7)
(b) (+10) – (+13)
(c) (-7) – (-9)
(d) (+3) – (+8)
(e) (-2) – (-7)
(f) (+3) – (+15)
Answer:
(a) (-5) – (-7) = +2
(b) (+10) – (+13) = -3
(c) (-7) – (-9) = +2
(d) (+3) – (+8) = -5
(e) (-2) – (-7) = +5
(f) (+3) – (+15) = -12
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:
We have to take out 5 positives from -3 negatives. But there are not enough positives.
So, we put down an extra 5 zero pairs (1 pair = a positive and a negative).
Now we can take out 5 positives.
Hence, -3 – (+5) = -8
Question 2.
Evaluate the following using tokens.
(a) (-3) – (+10)
Solution:
Start with 3 negatives.
Since, we need to take out 10 positives. So, we will add 10 extra zero pairs and then take out 10 positives.
∴ (-3) – (+10) = -13
(b) (+8) – (-7)
Solution:
Start with 8 positives.
Since we need to take out 7 negatives. So, we will add 7 extra zero pairs and then take out 7 negatives.
∴ (+8) – (-7) = + 15
(c) (-5) – (+9)
Solution:
Start with 5 negatives.
Add 9 extra zero pairs and then take out 9 positives.
∴ (-5) – (+9) = -14
(d) (-9) – (+10)
Solution:
Start with 9 negatives.
Add 10 extra zero pairs and then take out 10 positives.
∴ (-9) – (+10) = -19
(e) (+6) – (-4)
Solution:
Start with 6 positives.
Add 4 extra zero pairs and then take out 4 negatives.
∴ (+6) – (-4) = +10
(f) (-2) – (+7)
Solution:
Start with 2 negatives.
Add 7 extra zero pairs and then take out 7 positives.
∴ (-2) – (+7) = – 9
10.3 Integers in Other Places Figure it Out (Page No. 260)
Question 1.
Suppose yo ‘art with O 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:
(1) Total Credits =30 + 40 + 50 = 120
Solution:Total Debits = 40 + 50 + 60 = 150
Final balance = 120 (credit) – 150 (debit) = – 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:
Total amount of debits = – ₹ (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128) = – ₹ 255
Single credit = ₹ 256
Account balance = ₹ 256 – ₹ 255 = ₹ 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 may be worthwhile to temporarily have a negative balance?
Solution:
Having a positive balance in your bank account is generally better because:
- Avoids Overdraft Fees: Many banks charge fees if your account balance goes negative.
- Provide Financial Security: A positive balance ensures you have funds available for unexpected expenses or emergencies.
- Builds Good Credit History: Maintaining a positive balance can help to improve your credit score, making it easier to get loans or credit cards in the future.
There might be a few specific situations where temporarily having a negative balance could be considered:
- Overdraft Protection: Some banks offer overdraft protection which can help avoid bounced checks or declared transactions.
- Planned Large Expenses: If you know you will have a large income soon and need to make essential purchases a temporary negative balance might be acceptable.
10.3 Integers in Other Places Figure it Out (Page No. 261)
Question 1.
Looking at the geographical cross-section fill in the respective heights:
Solution:
A = 1500 m
B = -500 m
C = 300 m
D = -1200 m
E = 1200 m
F = -200m
G = 100 m
Question 2.
Which ¡s the highest point In this geographical cross-section? Which Is the lowest points?
Solution:
Highest point A
Lowest point = D
Question 3.
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?
Solution:
Yes, we can write the points in a sequence of decreasing and increasing order of heights as
A > E > C > G > F > B > D and
D < B < F < G < C < E < A
Question 4.
What is the highest point above sea level on Earth? What is its height?
Solution:
The highest point is A
Height = +1500 m
Question 5.
What is the lowest point concerning sea level on land or the ocean floor? What is its height? (This height should be negative)
Solution:
Lowest Point: D
Height = -1200 m
10.3 Integers in Other Places Figure it Out (Page No. 262)
Question 1.
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?
Solution:
Places:
- Ladakh: This region is well-known for it’s extremely cold winters, with temperatures often dropping below 0°C.
- Himachal Pradesh: Some high-altitude areas in Himachal Pradesh, especially the northern parts, can experience sub-zero temperatures.
- Jammu & Kashmir: Similar to Ladakh, parts of Jammu and Kashmir, particularly the mountainous regions, face freezing temperatures.
- Sikkim: Being a mountainous state, Sikkim also witness sub-zero temperatures in certain areas.
- Arunachal Pradesh: The higher reaches of Arunachal Pradesh can experience cold conditions with temperatures below 0°C.
Common Factor:
All these places are located in the Himalayan region, which is characterized by high altitudes.
Reason for Colder Temperatures:
- High Altitude: As altitude increases, the temperature decreases. This is primarily due to the thinner atmosphere at higher altitudes, which results in less heat retention.
- Distance from the Equator: These regions are farther from the equator receiving less direct sunlight, leading to colder temperatures.
Question 2.
Left in Ladakh gets very cold during winter. The following is a table of temperature readings taken during different times of the day/night in Leh on a day in November. Match the temperature with the appropriate time of the day/night.
Solution:
10.4 Explorations with Integers Figure it Out (Page No. 263 – 264)
Question 1.
Do the calculations for the given grid and find the border sum.
Solution:
Top row: 5 + (-3) + (-5) = -3
Bottom row: (-8) + (-2) + 7 = -3
Left column: 5 + 0 + (-8) = -3
Right column: (-5) + (-5) + 7 = -3
Therefore, The border sum of the given grid is ‘-3’.
Question 2.
Complete the grids to make the required border sum:
Solution:
Question 3.
For the last grid above, find more than one way of filling the numbers to get border sum -4.
Solution:
Question 4.
Which other grids can be filled in multiple ways? What could be the reason?
Solution:
Each grid could be filled in multiple numbers of ways. As there could be different combinations to get a particular integer value.
10.4 Explorations with Integers Figure it Out (Page No. 265)
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 such numbers are: 0,1, and -9 as 0 + 1 + (-9) = -8 (Note: Answer my vary.)
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:
Let’s find the sums that are not possible when rolling these two dice.
The faces of the dice are: -1, 2, -3, 4, -5, and 6.
First, let’s list 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 sums 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 are:
-9, -7, -5, 0, 2, 7, 9, 11
Question 4.
Solve these:
Solution:
8 – 13 = -5
(-8) – (13) = -21
(-13) – (-8) = -5
(-13) + (-8) = -21
8 + (-13) = -5
(-8) – (-13) = 5
13 – 8 = 5
13 – (-8) = 21
Question 5.
Find the years below.
(a) From the present year, which year was it 150 years ago? _______________
Solution:
150 years ago from the present year (2024):
[2024 – 150 = 1874]
So, 150 years ago, it was the year 1874.
(b) From the present year, which year was it 2200 years ago? _______________
(Hint: Recall that there was no year 0.)
Solution:
2200 years ago from the present year (2024):
Since there was no year 0, we need to account for this in our calculation:
[2024 – 2200 = -176].
The year -176 corresponds to 177 BCE (Before the Common Era).
So, 2200 years ago, it was the year 177 BCE.
(c) What will be the year 320 years after 680 BCE? _______________
Solution:
As BCE is before Christ, hence Let’s write 680 BCE = -680
Hence, 320 years after 680 BCE = -680 + 320 = -360 = 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) (-40) + 6 = (-34); (-34) + 6 = (-28);
(-28) + 6 = (-22); (-22) + 6 = (-16);
(-16) + 6 = (-10), (-10) + 6 = (-4),
Therefore, (-40), (-34), (-28), (-22), (-16), (-10), (-4)
(b) 3 + 1 = 4; 4 – 2 = 2; 2 + 3 = 5; 5 – 4 = 1; 1 + 5 = 6; 6 – 6 = 0; 0 + 7 = 7; 7 – 8 = -1; -1 + 9 = 8; 8 – 10 = -2
3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2
(c) Since, 27 – 8 = 19; 19 – 7 = 12; 12 – 6 = 6; 6 – 5 = 1; 1 – 4 =-3; -3 – 3 = -6; -6-2 = -8; -8 – 1 = -9
27, 19, 12, 6, 1, (-3), (-6), -8, -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:
(-5) +(-9) – (+18)-(-2)
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) (b) (positive) + (negative)
(c) (negative) + (negative) (d) (negative) – (negative)
(e) (negative) – (positive) (f) (negative) + (positive)
Solution:
(a) Can be positive or negative.
(b) Can be positive or negative.
(c) Negative
(d) Can be positive or negative.
(e) Negative
(f) Can be positive or negative.
Question 9.
This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?
Solution:
For every 10 tokens, there are +6 tokens and -4 tokens.
So, the sum for 10 tokens is +2.
Then for 100 such tokens are +2 × 10 = +20
10.5 A Pinch of History Figure it Out (Page No. 268)
Question 1.
Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or in terms of a number line?
Solution:
Let’s break down Brahmagupta’s rules using the concept of Bela’s Building of Fun and a number line. Brahmagupta’s rules primarily deal with operations involving positive and negative numbers. Here’s how we can understand them:
Brahmagupta’s Rules
1. Addition of Positive Numbers:
- Rule: Adding two positive numbers results in a positive number.
- Bela’s Building: If Bela starts on the 3rd floor and moves up 2 floors, she ends up on the 5th floor.
- Number Line: On a number line, 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 starts 3 floors below ground level (-3) and moves down 2 more floors, she ends up 5 floors below ground (-5).
- Number Line: 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 absolute value.
- 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: 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 number is like adding the two numbers and keeping the negative sign.
- Bela’s Building: If Bela is 3 floors below ground (-3) and moves down 2 more floors, she ends up 5 floors below ground (-5).
- Number Line: 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 like adding two numbers.
- Bela’s Building: If Bela starts on the 3rd floor and moves up 2 floors, she ends up on the 5th floor.
- Number Line: 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 like adding the absolute values and 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: Moving from -3 to -1 by subtracting -2.
- Example: (-3 – (-2) = -1)
Question 2.
Give your own examples of each rule.
Solution:
Do it yourself