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:
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) = 1 + 4 = 5
(b) (+4) + (+1) = 4 + 1 = 5
(c) (+ 4) + (- 3) = 4 – 3 = 1
(d) (-1) + (+ 2) = -1 + 2 = 1
(e) (-1) + (+1) = – 1 + 1 = 0
(f) 0 + (+ 2) = 0 + 2 = 2
(g) 0 + (-2) = 0 – 2 = -2
Question 3.
Starting from different floors, find the movements required to reach floor – 5.
e.g. if I start at floor + 2,1 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:
Starting at Floor + 3
Movement needed – 8
Expression (+3) + (- 8) = – 5
Starting at Floor 0
Movement needed – 5
Expression (0) + (- 5) = – 5
Starting at Floor – 2
Movement needed – 3
Expression (- 2) + (- 3) = – 5
Starting at Floor – 1
Movement needed – 4
Expression (- 1) + (- 4) = – 5
Starting at Floor + 7
Movement needed – 12
Expression (+ 7) + (- 12) = – 5
Starting at Floor – 3
Movement needed – 2
Expression (- 3) + (- 2) = – 5
Starting at Floor +10
Movement needed -15
Expression (+ 10) + (- 15) = – 5
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) 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, then find the numbers of floors B, C, F, G and H.
Solution:
(a) Floor B =- 9
Floor C = – 6
Floor F = + 2
Floor G = + 6
Floor H = +11
Question 4.
Mark the following floors of the building shown on the right.
(a) – 7
(b) – 4
(c) +3
(d) – 10
Solution:
(a)
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) = -3
Starting at Floor + 1, moving down 4 floors lands you at Floor-3.
(b) (0) – (+ 2) = — 2
Starting at Floor 0, moving down 2 floors lands you at Floor – 2.
(c) (+4) – (+ 1) = + 3
Starting at Floor + 4, moving down 1 floor lands you at Floor + 3.
(d) (0) — (-2) = + 2
Starting at Floor 0, moving up 2 floors (since, subtracting a negative is like adding a positive) lands you at Floor + 2.
(e) (+ 4) — (— 3) = + 7
Starting at Floor + 4, moving up 3 floors (subtracting a negative) lands you at Floor + 7.
(f) (— 4) – (—3) = — 1
Starting at Floor – 4, moving up 3 floors (subtracting a negative) lands you at Floor – 1.
(g) (-1) – (+ 2)=- 3
Starting at Floor – 1, moving down 2 floors lands you at Floor-3.
(h) (- 2) – (- 2) = 0
Starting at Floor – 2, moving up 2 floors (subtracting a negative) lands you at Floor 0.
(i) (- 1) – (+1) = -2
Starting at Floor – 1, moving down 1 floor lands you at Floor – 2.
(j) (+3) – (-3) = + 6
Starting at Floor + 3, moving up 3 floors (subtracting a negative) lands you at Floor + 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:
Question 2.
Write down the above 3 marked negative numbers in the following boxes:
Solution:
Question 3.
If 2 > – 3 ? Why? ls-2<3? Why?
Solution:
Yes,2 > -3
Because 2 is to the right side of – 3.
Yes, – 2 < 3
Because – 2 is to the left side of 3.
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 (5 steps backward from 0)
(ii) 7 + (- 7) =0
(iii) -10 + 20 = + 10
(iv) 10 – 20 = -10
(v) 7 – (-7) = 14
(vi) – 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)
Solution:
(+10)-(+7)
Tokens
Start with 10 positive tokens.
Remove 7 positive tokens.
Remaining tokens = 3 positive tokens.
Result + 3
Verification (+10) – (+ 7) = + 3
(b) (- 8) – (- 4)
Solution:
(-8)-(-4)
Tokens
Start with 8 negative tokens.
Remove 4 negative tokens.
Remaining tokens = 4 negative tokens.
Result – 4
Verification (- 8) – (- 4) = – 4
(c) (- 9) – (- 4)
Solution:
(-9)-(-4)
Tokens
Start with 9 negative tokens.
Remove 4 negative tokens.
Remaining Tokens = 5 negative tokens.
Result – 5
Verification (- 9) – (- 4) = – 5
(d) (+ 9) – (+12)
Solution:
(+9)-(+12)
Tokens
Start with 9 positive tokens.
Attempt to remove 12 positive tokens but you only have 9.
Additional Tokens
You need 3 more negative tokens to subtract from. Remaining tokens = 3 negative tokens.
Result – 3
Verification (+ 9) – (+12) = – 3
(e) (- 5) – (- 7)
Solution:
(-5)-(-7)
Tokens
Start with 5 negative tokens.
Attempt to remove 7 negative tokens, but you only have 5.
Additional Tokens You need 2 more positive tokens to subtract from.
Remaining tokens = 2 positive tokens. ,
Result + 2
Verification (- 5) – (- 7) = + 2
(f) (- 2) – (- 6)
Solution:
(-2)-(-6)
Tokens
Start with 2 negative tokens.
Attempt to remove 6 negative tokens, but you only have 2.
Additional Tokens
You need 4 more positive tokens to subtract from.
Remaining tokens = 4 positive tokens.
Result + 4
Verification (- 2) – (- 6) = + 4
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:
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
Question 2.
Evaluate the following using tokens.
(a) (-3)-(+10)
(b) (+ 8) – (- 7)
(c) (- 5) – (+ 9)
(d) (-9)- (+10)
(e)(+6)-(-4)
(f) (- 2) – (+ 7)
Solution:
(a) You need to subtract +10 (remove 10 positive tokens), but you do not have any positive tokens available. Adding 10 zero pairs gives you 10 positive tokens and 10 negative tokens. Now, remove the + 10 (positive tokens).You are left with the original – 3 tokens plus the additional – 10 tokens from the zero pairs.
(-3)- (+10) = -13
(result of the other problems can be solved in the same way)
Do yourself
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 Debits =1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255
Credit of 256, so add 256.
Final balance = 256 (credit) – 255 (debit) = 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.
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
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:
8,848 m ( Mt. Everest in Nepal)
Question 5.
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).
Solution:
400 m (the dead sea is the lowest place on earths surface)
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:
Thangu Valley (North Sikkim), Leh, Ladakh, Spiti Valley. Dras Valley, Sinchen, etc. All these are high altitude places.
Therefore, it becomes colder there and not in other places.
Question 2.
Leh 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.
Answer:
14°C → 02: 00 PM
8°C → 11:00 AM
10.4 Explorations with Integers Figure it Out (Page No. 263 – 264)
Question 1.
Do the calculations for the second grid above 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
The board sum of the second 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:
First and last gird with border sums + 4 and – 4, respectively.
Can be filled in many ways because there are less number of filled boxes.
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 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) (40), (- 34), (- 28), (- 22), (- 16), (- 10), (- 4)
(b) 3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2
(c) 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:
(-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:
+ 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 buildings of fun, or in terms of a number line?
Solution:
Do it yourself
Question 2.
Give your own examples of each rule.
Solution:
Rule 1 If a smaller positive number is subtracted from a larger positive number, then the result is positive.
- Building of Fun If you are on Floor 3 and go down 2 floors, you end up on Floor 1 (which is above the ground) positive.
- Number Line If you are at Floor + 3 on the number line and move left 2 units, you reach +1, which is positive.
Rule 2 If a larger positive number is subtracted from a smaller positive number, the result is negative.
- Building of Fun If you are on the floor 3 and go down 5 floors, you end up on Floor (- 2), which is negative.
- Number Line If you are at + 3 on the number line and move left 5 units, you end up at – 2, which is negative.
- Similarly,’other rules can be explained in terms of Bela’s Building of fun or a number line.