NCERT Solutions Class 6 Maths Chapter 10 the Other Side of Zero

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 6 · M AT H S

NCERT Solutions

Chapter 10: The Other Side of
Zero

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

242 – 271 27 76 English

Solutions, notes, sample papers & more at 61 pages

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 10: The Other Side of Zero
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 10 The Other Side of Zero from the
NCERT textbook Ganita Prakash. Every Figure it Out question and every in-text question from pages 243 to
268 is solved with diagrams — Bela's Building of Fun and its lift, positive and negative numbers, integers on
the number line, the token model, credits and debits, sea level, temperature, border-sum grids and
Brahmagupta's rules.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 242 – 271

SECTIONS QUESTIONS

27 76

MEDIUM

English

In-text Questions — Page 243
Section 10.1 Bela’s Building of Fun

Q1 Can there be a number less than 0? Can you think of any ways to have less than 0 of
something?

Yes. A number less than 0 makes perfect sense the moment we start counting in two opposite
directions from a starting mark.

Floors of a building — the ground floor is 0, and the basement floors below it are – 1, – 2, –
3, …
Temperature — in Leh or Drass the thermometer often shows – 4 °C, that is 4 degrees below
the freezing point of water.
Money — if you have ₹100 and you owe a shopkeeper ₹150, you are ₹50 short. Your balance
is – ₹50.
Sea level — the shore of the Dead Sea is about 430 m below sea level, so its height is – 430
m.
A lift in a mine — a level 90 m below the ground is marked – 90 m.

Why it works: zero here does not mean ‘nothing’. It is a reference point — the
ground floor, the freezing point, the sea level. Once we fix a zero, we can go both
above it (positive) and below it (negative).

Page 1 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q2 Observe that some of the floors in the ‘Building of Fun’ are below the ground. What
are the shops that you find on these floors? What is there on the ground floor?

Look at the picture of Bela's Building of Fun. Below the ground there are five floors:

Toy Store — the first floor below the ground
Video Games — the second floor below
Cinema — the third floor below
Ghost House — the fourth floor below
Dinosaur show — the fifth (lowest) floor

On the ground floor there is the Welcome Hall — the entrance where every visitor comes in.

Tip: Above the ground there are six floors — Food Court, Art Centre, Book Store, Ice
Cream, Sports Centre and Space. So the building has 6 floors up and 5 floors down.

Q3 A lift is used to go up and down between the floors. It has two buttons: ‘+’ to go up
and ‘–’ to go down. Can you spot the lift?

Yes — the lift is the tall narrow shaft drawn on the right-hand side of the building. It runs all
the way from the topmost floor (Space) down to the lowest floor (Dinosaur), passing through
the ground floor.
Its panel has just two buttons:

‘+’ → go up one floor for each press

‘–’ → go down one floor for each press

So pressing + + + (that is + 3) takes you up three floors, and pressing – – – – (that is – 4) takes you
down four floors.

Page 2 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q4 What do you press to go four floors up? What do you press to go three floors down?

Four floors up → press ‘+’ four times → + + + + , written as + 4

Three floors down → press ‘–’ three times → – – – , written as – 3

For example, from the Welcome Hall (Floor 0) pressing + 4 takes you to the Ice Cream floor, and
pressing – 3 takes you to the Cinema.

Tip: The number after the sign is simply how many times the button is pressed; the
sign tells you the direction.

In-text Questions — Page 244
Numbering the floors in the Building of Fun

Q1 The ground floor is called Floor 0. Can you see why?

Because you have to press the lift button zero times to stay on it.

Starting Floor + Movement = Target Floor

Ground floor + 0 = Ground floor

The ground floor is the starting point from which every other floor is counted. Floors above it
need some presses of ‘+’, floors below need some presses of ‘–’, and the ground floor itself
needs none.

Why this matters: the ground floor is the reference floor. That is exactly the job of 0
among the numbers — it separates the positive numbers from the negative
numbers and belongs to neither group.

Page 3 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Number all the floors in the Building of Fun.
e
Q2

m l as
.co a g
a s em
aglat the Welcome Hall as Floor 0. Count upwards with + and downwards with –.
Start

co m
m . ag
l a se Space
ag
+6

+5 Sports Centre
co m
em.
m +4 l as
.co g
Ice Cream
m a
l a se
a g +3 Book Store above

m a s
agl
+2 ArtoCentre
.c
a s em
+1 agl Food Court
0 Welcome Hall (ground)
co m
m .
m as e
.co –1 Toy Store a g l
a sem
agl –2 Video Games
se m
com g l a
–3
m . Cinema below a
a s e
–4 agl Ghost House

co m
–5 Dinosaur
m .
m as e
.co a g l
se m
g l a
a c
.
Every floor of Bela's Building of Fun with its number. The brown line is the ground; above it the
numbers are positive, below it they are negative.
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 4 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

FLOOR WHAT IS THERE BUTTON PRESS FROM THE GROUND
NUMBER FLOOR

+6 Space ++++++

+5 Sports Centre +++++

+4 Ice Cream ++++

+3 Book Store +++

+2 Art Centre ++

+1 Food Court +

0 Welcome Hall (ground no press at all
floor)

–1 Toy Store –

–2 Video Games ––

–3 Cinema –––

–4 Ghost House ––––

–5 Dinosaur –––––

The neat part: the floor number and the button press are the same number. + 3 is
the Book Store and the movement of three floors up; – 3 is the Cinema and the
movement of three floors down.

In-text Question — Page 245
Addition to keep track of movement

Q1 Start from the Food Court and press + 2 in the lift. Where will you reach?

The Food Court is Floor + 1. Pressing + 2 lifts you two floors higher.

Page 5 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Starting Floor + Movement = Target Floor

(+ 1) + (+ 2) = + 3

Answer: Floor + 3 — the Book Store.

Check it yourself: Food Court → Art Centre (one press) → Book Store (second
press). Two presses, two floors up.

Figure it Out — Page 245
Addition to keep track of movement

Q1 You start from Floor + 2 and press – 3 in the lift. Where will you reach? Write an
expression for this movement.

Floor + 2 is the Art Centre. Pressing – 3 takes you three floors down: + 2 → + 1 → 0 → – 1.

Starting Floor + Movement = Target Floor
(+ 2) + (– 3) = – 1

Answer: Floor – 1 — the Toy Store.

Why the answer is negative: you go down 3 floors but you were only 2 floors above
the ground, so you end up 1 floor below the ground.

Page 6 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q2 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)

EXPRESSION JOURNEY IN THE LIFT ANSWER

a. (+ 1) + (+ 4) From Food Court go 4 floors up +5

b. (+ 4) + (+ 1) From Ice Cream go 1 floor up +5

c. (+ 4) + (– 3) From Ice Cream go 3 floors down +1

d. (– 1) + (+ 2) From Toy Store go 2 floors up +1

e. (– 1) + (+ 1) From Toy Store go 1 floor up 0

f. 0 + (+ 2) From Welcome Hall go 2 floors up +2

g. 0 + (– 2) From Welcome Hall go 2 floors down –2

Two things worth noticing: parts (a) and (b) give the same answer — the order of
adding does not matter. In part (e) the two numbers are inverses of each other, so
you land back on the ground floor, 0.

Q3 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.

Use Movement = Target Floor – Starting Floor with Target Floor = – 5 every time.

Page 7 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

STARTING FLOOR BUTTON PRESS NEEDED EXPRESSION

+ 6 (Space) – 11 (+ 6) + (– 11) = – 5

+ 5 (Sports Centre) – 10 (+ 5) + (– 10) = – 5

+ 3 (Book Store) –8 (+ 3) + (– 8) = – 5

+ 1 (Food Court) –6 (+ 1) + (– 6) = – 5

0 (Welcome Hall) –5 0 + (– 5) = – 5

– 2 (Video Games) –3 (– 2) + (– 3) = – 5

– 4 (Ghost House) –1 (– 4) + (– 1) = – 5

– 5 (Dinosaur) 0 (– 5) + 0 = – 5

Why every press is ‘–’ (or 0): Floor – 5 is the lowest floor of the building, so from
anywhere else you can only come down to it. If you are already there, you press
nothing — a movement of 0.

Figure it Out — Page 246
Combining button presses is also addition

Q1 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)

Here the numbers are movements, not floors. Add them one after another to get the total
movement.

a. (+ 1) + (+ 4) = + 5 → 1 floor up, then 4 more up = 5 floors up

b. (+ 4) + (+ 1) = + 5 → 4 floors up, then 1 more up = 5 floors up
c. (+ 4) + (– 3) + (– 2) = (+ 1) + (– 2) = – 1 → 4 up, 3 down, 2 down = 1 floor down

d. (– 1) + (+ 2) + (– 3) = (+ 1) + (– 3) = – 2 → 1 down, 2 up, 3 down = 2 floors down

Page 8 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Why this is the same as before: whether the first number is a floor or a movement,

as e
comthe same effect as one press of – 1. l
the rule for combining is the same addition. Three separate presses of + 4, – 3 and –
. a g
em
2 have exactly
a s
agl
Tip: To add a long string quickly, total the ‘+’ numbers and the ‘–’ numbers

co m
ag
separately. In (c): + 4 against – 3 – 2 = – 5, and 4 – 5 = – 1.
m .
as e
a g l
In-text Questions — Pages 246 & 247
co m
m.
Back to zero! · Comparing numbers using floors

o m l a se
Q1 m
c g will he reach?
If .Basant now presses + 4 and then presses – 4 in the lift, where
a
l a se
ag
s

m a
.co agl
He reaches the place he started from — the ground floor, Floor 0 (Welcome Hall).

se m
g l a
0 + (+ 4) = + 4 (Ice Cream) a
(+ 4) + (– 4) = 0 (back to the Welcome Hall)
co m
m .
m as e
.co a g l
Four presses of ‘+’ take him four floors up; four presses of ‘–’ bring him four floors back down.

se m
a
The two movements cancel out completely.

a g l
Why: – 4 is the inverse of + 4. A number added to its inverse always gives 0.
se m
com g l a
m . a
ase
Q2 agl
Write the inverses of these numbers: + 4, – 4, – 3, 0, + 2, – 1.

co m
m .
e

om
csign. g l as
.
The inverse of a number is the number that brings you back to Floor 0. Keep the digits, change
a
sem
the
a
agl c
NUMBER +4 –4 –3 0 +2 –1
m .
m a s e
. co agl
Inverse –4 +4 +3 0 –2 +1
m
se– 4 + 4 = 0
l a
ag
Check 4 + (– 4) = 0 –3+3=0 0+0=0 2 + (– 2) = 0 –1+1=0

co m
m .
m as e
.co


a g l Page 9 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Did you know? 0 is the only number that is its own inverse. Pressing the button zero
times, and then zero times again, still leaves you on the ground floor.

Q3 Connect the inverses by drawing lines. + 5, – 7, – 8, + 9 / – 9, + 8, – 5, + 7

+5 –7 –8 +9

–9 +8 –5 +7

Page 246 — the eight numbers as printed. One join, – 7 to + 7, is already drawn in the
book.

Join each number in the top row to the number in the bottom row that has the same digits but
the opposite sign.

+5 –7 –8 +9

–9 +8 –5 +7

Each pair joined by a line adds up to 0: (+ 5) and (– 5), (– 7) and (+ 7), (– 8) and (+ 8), (+ 9) and (– 9).

Page 10 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

(+ 5) + (– 5) = 0 (– 7) + (+ 7) = 0 (– 8) + (+ 8) = 0 (+ 9) + (– 9) = 0

Q4 Who is on the lowest floor? 1. Jay is in the Art Centre. So, he is on Floor + 2. 2. Asin is
in the Sports Centre. So, she is on Floor ___. 3. Binnu is in the Cinema Centre. So, she
is on Floor ___. 4. Aman is in the Toy Store. So, he is on Floor ___.

CHILD WHERE FLOOR

Jay Art Centre +2

Asin Sports Centre +5

Binnu Cinema –3

Aman Toy Store –1

Arrange the four floors the way they sit in the building:

+ 5 (Asin) > + 2 (Jay) > – 1 (Aman) > – 3 (Binnu)

Answer: Binnu is on the lowest floor — the Cinema, Floor – 3. Asin is on the highest.

Q5 Should we write – 3 < – 4 or – 4 < – 3?

We should write – 4 < – 3 (and the same fact can be written as – 3 > – 4).
Look at the building: the Ghost House (Floor – 4) is below the Cinema (Floor – 3). Anything lower
down is the smaller number.

Floor – 4 is lower than Floor – 3 → – 4 < – 3

Careful — a common mistake: with negative numbers, the one with the bigger
digits is the smaller number. 4 > 3, but – 4 < – 3, because going down 4 floors takes
you lower than going down 3 floors.

Page 11 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Figure it Out — Page 247
Comparing numbers using floors

Q1 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

COMPARISON REASON FROM THE BUILDING

a. –2<+5 Video Games is below the Sports Centre

b. –5<+4 The Dinosaur floor is below Ice Cream

c. –5<–3 Floor – 5 is below Floor – 3

d. +6>–6 Six floors up is above six floors down

e. 0>–4 The ground floor is above every basement floor

f. 0<+4 The ground floor is below every floor above it

Two rules you can now say aloud: every negative number is less than 0, and
every positive number is greater than 0. So any negative number is smaller than
any positive number, no matter how big its digits look.

Q2 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

a. – 10 > – 12 b. + 17 > – 10 c. 0 > – 20

d. + 9 > – 9 e. – 25 < – 7 f. + 15 > – 17

a. Floor – 10 is only 10 floors down, Floor – 12 is 12 floors down — so – 12 is lower, and – 10 is
greater.
b, d, f. A positive floor is always above a negative floor.
c. The ground floor is above the 20th basement.
e. Floor – 25 is much deeper than Floor – 7, so – 25 is the smaller number.

Page 12 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Tip: For two negative numbers, the one with the larger digits is smaller. Compare the
digits, then flip your answer.

Q3 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.

The ticks on the line are one floor apart, so simply count. Between D (– 1) and E (+ 1) there is one
unmarked tick — that is Floor 0, which fixes the scale.

Page 13 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
e m.
m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
ag

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

H + 11

G +6

F +2
E +1
0
D –1

C –6

B –9

A – 12

Page 15 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

The building drawn as a line. The three given floors A = – 12, D = – 1 and E = + 1 fix the scale; the
floors found are shown in red.

B is 3 ticks above A: – 12 + 3 = – 9

C is 3 ticks above B: – 9 + 3 = – 6

F is 1 tick above E: + 1 + 1 = + 2

G is 4 ticks above F: + 2 + 4 = + 6

H is 5 ticks above G: + 6 + 5 = + 11

Answer: B = – 9, C = – 6, F = + 2, G = + 6, H = + 11.

Check it yourself: count the ticks from D (– 1) straight up to H — there are 12 of
them, and – 1 + 12 = + 11. ✔

Q4 Mark the following floors of the building shown on the right. a. – 7 b. – 4 c. + 3 d. – 10

Count ticks from a floor you already know. Each tick is one floor.

– 7 — two ticks above B (– 9), that is one tick below C (– 6).
– 4 — two ticks above C (– 6), or three ticks below D (– 1).
+ 3 — one tick above F (+ 2).
– 10 — two ticks above A (– 12), or one tick below B (– 9).

Page 16 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

c. + 3

0
D=–1

b. – 4

a. – 7

d. – 10

A = – 12

Page 17 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

The four floors marked on the building line: + 3 above the ground, and – 4, – 7 and – 10 below it.

Order from lowest to highest: – 10 < – 7 < – 4 < + 3

In-text Question — Page 248
Subtraction to find which button to press

Q1 Evaluate 15 – 5, 100 – 10 and 74 – 34 from this perspective.

‘This perspective’ means reading a subtraction as finding the missing number to be added —
how much must the smaller number grow to become the bigger one?

15 – 5 → 5 + ? = 15 → ? = 10

100 – 10 → 10 + ? = 100 → ? = 90

74 – 34 → 34 + ? = 74 → ? = 40

Why this way of looking helps: ‘take away’ makes no sense when you want (– 1) – (+
3) — you cannot take 3 positives away from – 1 easily. But ‘what must be added to +
3 to reach – 1?’ always makes sense: the answer is the movement – 4. This is the
meaning of subtraction used through the whole chapter.

Tip: In lift language, 74 – 34 = 40 says: to travel from Floor 34 to Floor 74 you must
press + 40.

Figure it Out — Page 249

Page 18 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

Subtraction to find which button to press
co m
em.
m l as
.co
Complete these expressions. You may think of them as finding the movement
a g
em to reach the Target Floor from the Starting Floor. a. (+ 1) – (+ 4) b. (0) – (+ 2) c.
Q1

a s
needed

a gl (+ 4) – (+ 1) d. (0) – (– 2) e. (+ 4) – (– 3) f. (– 4) – (– 3) g. (– 1) – (+ 2) h. (– 2) – (– 2) i. (– 1) – (+
1) j. (+ 3) – (– 3)

com
e m . ag
as

a g l
Remember: Target Floor – Starting Floor = Movement needed. The first number is where you
want to go, the second is where you are.

. om
cMOVEMENT
EXPRESSION FROM … TO …
e m
m
(+.c1)o– (+ 4) g l as
a.
em
from Floor + 4 to Floor + 1 → 3 floors down
a –3

a s
aglb. (0) – (+ 2) from + 2 to 0 → 2 floors down –2

a s
com agl
c. (+ 4) – (+ 1) from + 1 to + 4 → 3 floors up +3

m .
ase
d. from – 2 to 0 → 2 floors up

agl
(0) – (– 2) +2

e. (+ 4) – (– 3) from – 3 to + 4 → 3 floors up to 0, then 4 more +7

co m– 1
.
f. (– 4) – (– 3) from – 3 to – 4 → 1 floor down

e m
c
(–m
o g l as
. a
g. 1) – (+ 2) from + 2 to – 1 → 2 floors down to 0, then 1 more –3

a s eh.m (– 2) – (– 2)
gl
from – 2 to – 2 → you are already there 0
a
i. (– 1) – (+ 1) from + 1 to – 1 → 2 floors down –2
se m
c→om g l a
m . a
se
j. (+ 3) – (– 3) from – 3 to + 3 3 floors up to 0, then 3 more +6

l a
ag
Look at e, d and j: every time a negative number is subtracted the answer grows. (+
4) – (– 3) = + 7 is the same as (+ 4) + (+ 3). Subtracting a negative number is the same
co m
m .
e
as adding the matching positive number — you will meet this rule again on page
m l as
.co g
252.
m a
l a se
ag
.c
s e m
m a
Figure it Out — Page 251

em . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 19 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Adding and subtracting larger numbers (the mine)

Q1 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.

In the mine, Starting Level + Movement = Target Level, so the missing movement is always
Target Level – Starting Level.

QUESTION JOURNEY IN THE MINESHAFT ANSWER

a. (+ 40) + ___ = + 200 from + 40 m up to + 200 m + 160

b. (+ 40) + ___ = – 200 40 m down to 0, then 200 m more down – 240

c. (– 50) + ___ = + 200 50 m up to 0, then 200 m more up + 250

d. (– 50) + ___ = – 200 from – 50 m down to – 200 m – 150

e. (– 200) – (– 40) from – 40 m down to – 200 m – 160

f. (+ 200) – (+ 40) from + 40 m up to + 200 m + 160

g. (– 200) – (+ 40) 40 m down to 0, then 200 m more down – 240

Check (b): (+ 40) + (– 240) = – 200 ✔ Check (c): (– 50) + (+ 250) = + 200 ✔

Notice the pairs: (a) and (f) give the same answer, and (b) and (g) give the same
answer. That is because ‘find the missing movement’ and ‘Target – Starting’ are two
ways of asking exactly the same question.

In-text Question — Page 251

Page 20 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Adding, subtracting, and comparing any numbers

Q1 Try evaluating the following expressions by similarly drawing or imagining a
suitable lift: a. – 125 + (– 30) b. + 105 – (– 55) c. + 105 + (+ 55) d. + 80 – (– 150) e. + 80 + (+
150) f. – 99 – (– 200) g. – 99 + (+ 200) h. + 1500 – (– 1500)

Imagine an ‘infinite lift’ with Level 0 in the middle. Read each expression as a journey.

EXPRESSION JOURNEY IN THE IMAGINARY LIFT ANSWER

a. – 125 + (– 30) Start at – 125 and go 30 more levels down – 155

b. + 105 – (– 55) From – 55 to + 105: 55 up to 0, then 105 more up + 160

c. + 105 + (+ 55) Start at + 105 and go 55 levels up + 160

d. + 80 – (– 150) From – 150 to + 80: 150 up to 0, then 80 more up + 230

e. + 80 + (+ 150) Start at + 80 and go 150 levels up + 230

f. – 99 – (– 200) From – 200 to – 99: 101 levels up + 101

g. – 99 + (+ 200) Start at – 99, go 200 up: 99 to reach 0, 101 above it + 101

h. + 1500 – (– 1500) From – 1500 to + 1500: 1500 up to 0, then 1500 more + 3000

The pattern to spot: b and c give the same answer, d and e give the same answer, f
and g give the same answer. In each pair the first one subtracts a negative and the
second one adds the matching positive. So – (– n) = + n.

In-text Questions — Pages 252 & 253
Back to the number line

MATH TALK

Q1 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?

Yes — it happened in every single pair.

Page 21 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

+ 105 – (– 55) = + 160 and + 105 + (+ 55) = + 160

+ 80 – (– 150) = + 230 and + 80 + (+ 150) = + 230

– 99 – (– 200) = + 101 and – 99 + (+ 200) = + 101

Why it happens: ‘– (– 55)’ asks how far above – 55 is + 105. To climb out of a level that
is 55 below the ground you must first come up those 55 levels — so the 55 gets
added to the journey, not removed from it. In short, a – (– b) = a + b.

Q2 Take a look at the ‘infinite lift’ above. Does it remind you of a number line? In what
ways?

Yes — the infinite lift is a number line, only standing up instead of lying down.

Level 0 of the lift is the 0 of the number line.
Levels above 0 are the positive numbers; levels below 0 are the negative numbers.
The levels are equally spaced, just like the marks on a number line.
Pressing ‘+’ is moving forward; pressing ‘–’ is moving backward.
A higher level means a greater number, exactly as a mark further right means a greater
number.

Turn the infinite lift through 90° and it becomes

… – 3, – 2, – 1, 0, 1, 2, 3 … — a complete number line.

Math Talk: This is exactly the answer to the question asked at the very start of the
chapter — the number ray starting at 0 becomes a full number line once the
negative numbers are put to the left of 0.

Page 22 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q3 If, from 5 you wish to go over to 9, how far must you travel along the number line?

0 1 2 3 4 5 6 7 8 9 10

The number line from 0 to 10, page 252.

You must travel 4 steps forward (to the right).

+4

0 1 2 3 4 5 6 7 8 9 10

Four steps forward take you from 5 to 9.

Starting Number + Movement = Target Number → 5 + 4 = 9

Target Number – Starting Number = Movement needed → 9 – 5 = 4

Q4 Now, from 9, if you wish to go to 3, how much must you travel along the number
line?

0 1 2 3 4 5 6 7 8 9 10

The number line from 0 to 10, page 253.

You must move 6 steps backward, that is a movement of – 6.

Page 23 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
–6
e m.
m l as
m .co a g
l a s0e
g
1 2 3 4 5 6 7 8 9 10
a

co m
ag
Six steps backward take you from 9 to 3, so the movement is – 6.

m .
as e
a g l
Starting Number + Movement = Target Number → 9 + (– 6) = 3

Target Number – Starting Number = Movement needed → 3 – 9 = – 6
co m
se m.
o m l a
g (right) is +,
m .c answer is negative: the sign shows the direction. Forward
a
se
Why the

g l a
a
backward (left) is –. The 6 tells you how far, the minus tells you which way.

m a s
m .co agl
Q5
l a se
Now, from 3, if you wish to go to – 2, how far must you travel?

ag
m

. co
m
You must travel – 5, that is 5 steps backward. Three steps bring you to 0, and two more take you

m as e
.co l
to – 2.
a g
se m
g l a
a –5

se m
–c2o
m g l a
m . a
ase
– 10 –8 –6 –4 0 2 4 6 8 10

agl
From 3 to – 2 is five steps backward — three to reach 0 and two more beyond it.

co m
m .
m as e
.co
3 + (– 5) = – 2
a g l
se m The matching subtraction statement: – 2 – 3 = – 5
g l a
a c
m .
a s e
. c om agl
Figure it Out — Pages 253em
s
& 254
a
agl

co m
m .
m ase
.co


a g l Page 24 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Back to the number line

Q1 Mark 3 positive numbers and 3 negative numbers on the number line above.

– 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 9 10

The number line printed above this exercise, page 253.

Any three numbers on each side will do. Here is one choice — positives 2, 5, 8 and negatives – 1,
– 3, – 7.

R Q P A B C

– 10 –7 –3 –1 0 2 5 8 10

Three positive numbers A = 2, B = 5, C = 8 (right of 0, in green) and three negative numbers P = – 1, Q
= – 3, R = – 7 (left of 0, in red).

Tip: Every positive number must be marked to the right of 0 and every negative
number to the left of 0. Your friend's three numbers may be different — that is
perfectly fine.

Q2 Write down the above 3 marked negative numbers in the following boxes:

Write them in the order in which they appear on the number line, that is smallest first (left to
right):

–7 < –3 < –1

Page 25 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Why this is the increasing order: – 7 lies furthest to the left, so it is the smallest; – 1
is nearest to 0, so of the three it is the greatest. All three are still less than 0.

Q3 Is 2 > – 3? Why? Is – 2 < 3? Why?

Yes to both.

2 > – 3 — on the number line 2 lies to the right of – 3, and whatever is further right is greater.
In the building, Floor 2 is above Floor – 3.
– 2 < 3 — here – 2 lies to the left of 3, and whatever is further left is smaller. Floor – 2 is below
Floor 3.

–3<0<2 → 2>–3

–2<0<3 → –2<3

Tip: A positive number is always greater than a negative number — you never even
need to look at the digits.

Q4 What are a. – 5 + 0 b. 7 + (– 7) c. – 10 + 20 d. 10 – 20 e. 7 – (– 7) f. – 8 – (– 10)?

EXPRESSION WORKING ANSWER

a. –5+0 Adding 0 changes nothing –5

b. 7 + (– 7) A number plus its inverse 0

c. – 10 + 20 From – 10 move 20 forward: 10 to reach 0, 10 beyond + 10

d. 10 – 20 10 + (– 20): from 10 move 20 backward – 10

e. 7 – (– 7) = 7 + 7; from – 7 to 7 is 14 steps + 14

f. – 8 – (– 10) = – 8 + 10; from – 10 up to – 8 is 2 steps +2

Page 26 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Compare (b) and (e): 7 + (– 7) = 0 but 7 – (– 7) = 14. Adding the inverse cancels;
subtracting the inverse doubles. The sign in front of the bracket makes all the
difference.

In-text Question — Page 255
Using the unmarked number line to add and subtract

Q1 Use unmarked number lines to evaluate these expressions: a. – 125 + (– 30) b. + 105 –
(– 55) c. + 80 – (– 150) d. – 99 – (– 200)

On an unmarked number line (UNL) only 0 is shown. Mark the starting number roughly in the
right place, draw the jump, and read the answer.

– 30 a. – 125 + (– 30) = – 155

– 155 – 125 0

Start at – 125 and jump 30 further backwards — you land on – 155.

+ 160

b. + 105 – (– 55)
= + 160

– 55 0 + 105

Subtraction asks for the movement from – 55 to + 105: 55 steps to reach 0 and 105 more, that is + 160
in all.

Page 27 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

a. – 125 + (– 30) = – 155

b. + 105 – (– 55) = 105 + 55 = + 160

c. + 80 – (– 150) = 80 + 150 = + 230 (from – 150 up to 0 is 150, then 80 more)
d. – 99 – (– 200) = – 99 + 200 = + 101 (from – 200 up to – 99 is 101 steps)

Tip: The unmarked number line does not need a scale. All it has to show is which side
of 0 each number is on and which is further from 0.

Figure it Out — Page 257
Section 10.2 The Token Model — using tokens for addition

Q1 Complete the additions using tokens. a. (+ 6) + (+ 4) b. (– 3) + (– 2) c. (+ 5) + (– 7) d. (– 2)
+ (+ 6)

Put down the tokens, throw away every zero pair (one green with one red), and read what is left.

TOKENS PUT DOWN ZERO PAIRS REMOVED WHAT IS LEFT ANSWER

a. 6 green and 4 more green none — all are green 10 green + 10

b. 3 red and 2 more red none — all are red 5 red –5

c. 5 green and 7 red 5 pairs 2 red –2

d. 2 red and 6 green 2 pairs 4 green +4

+ + + + +
left: 2 red = – 2
– – – – – – –

Part (c): 5 green and 7 red tokens. Five zero pairs are struck out, and 2 red tokens are left — so (+ 5) +
(– 7) = – 2.

Page 28 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Why zero pairs may be thrown away: one press of ‘+’ followed by one press of ‘–’

as e
com the value of the pocket. l
brings the lift attendant back to where he was. The pair is worth 0, so removing it
does not .change a g
a s em
agl

co m
ag
Cancel the zero pairs in the following two sets of tokens. On what floor is the lift
.
Q2

em
as
attendant in each case? What is the corresponding addition statement in each
case?
a g l

co m
Set (a): 3 green tokens and 5 red tokens.
e m.
m l as
.co a g
em pairs cancel → 2 red tokens are left
3szero
a
a gl (+ 3) + (– 5) = – 2 → the attendant is on Floor – 2 (Video Games)

om a s
Set (b): 6 green tokens and 3 red tokens..c
em agl
a s
agl
3 zero pairs cancel → 3 green tokens are left

co m
.
(+ 6) + (– 3) = + 3 → the attendant is on Floor + 3 (Book Store)
e m
m l as
m .co a g
l a se
ag (b)
+ + + + + + se m
com g l a
m . a
ase
left: 3 green
l
– ag –
=+3
–

co m
m .
m as e
.co a g l
m
Set (b): three zero pairs are struck out and three green tokens remain, so the attendant is on Floor +

l a se 3.

ag
.c
s e m
m a
co agl
Check it yourself: in set (a) he pressed ‘+’ 3 times and ‘–’ 5 times — two more downs

m .
e
than ups, so he must be 2 floors below the ground. ✔

g l as
a

co m
m .
m ase
.co


a g l Page 29 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

In-text Question — Page 257
Using tokens for subtraction

Q1 Is (– 7) – (– 5) the same as (– 7) + (+ 5)?

Yes — both give – 2.

With tokens: put down 7 red. Take away 5 red → 2 red are left → – 2

With tokens: put down 7 red and add 5 green → 5 zero pairs cancel → 2 red are left → – 2

Why the two are the same: removing 5 negative tokens and adding 5 positive
tokens have exactly the same effect on the pocket — each of them pushes the value
up by 5. This is the token-model proof of the rule a – (– b) = a + b: subtracting a
number is the same as adding its inverse.

Figure it Out — Page 258

Page 30 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Using tokens for subtraction

Q1 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)

EXPRESSION WHAT YOU DO WITH THE TOKENS ANSWER

a. (+ 10) – (+ 7) Put 10 green, take away 7 green → 3 green left +3

b. (– 8) – (– 4) Put 8 red, take away 4 red → 4 red left –4

c. (– 9) – (– 4) Put 9 red, take away 4 red → 5 red left –5

d. (+ 9) – (+ 12) Put 9 green; add 3 zero pairs to get 12 green, take them away → –3
3 red left

e. (– 5) – (– 7) Put 5 red; add 2 zero pairs to get 7 red, take them away → 2 +2
green left

f. (– 2) – (– 6) Put 2 red; add 4 zero pairs to get 6 red, take them away → 4 +4
green left

Check with the ‘add the inverse’ rule:

(– 8) – (– 4) = – 8 + 4 = – 4 ✔ (+ 9) – (+ 12) = 9 + (– 12) = – 3 ✔

(– 5) – (– 7) = – 5 + 7 = + 2 ✔ (– 2) – (– 6) = – 2 + 6 = + 4 ✔

Why zero pairs are needed in d, e and f: you are asked to take away more tokens
than you have. Adding zero pairs does not change the value (each pair is worth 0)
but it gives you the tokens you need to remove.

Q2 Complete the subtractions: a. (– 5) – (– 7) b. (+ 10) – (+ 13) c. (– 7) – (– 9) d. (+ 3) – (+ 8) e.
(– 2) – (– 7) f. (+ 3) – (+ 15)

Change every subtraction into ‘add the inverse’ and finish it in one line.

Page 31 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

a. (– 5) – (– 7) = – 5 + 7 = + 2

b. (+ 10) – (+ 13) = 10 + (– 13) = – 3

c. (– 7) – (– 9) = – 7 + 9 = + 2
d. (+ 3) – (+ 8) = 3 + (– 8) = – 5

e. (– 2) – (– 7) = – 2 + 7 = + 5

f. (+ 3) – (+ 15) = 3 + (– 15) = – 12

Tip: Notice that in (a), (c) and (e) a bigger negative is being subtracted from a smaller
one, and the answer comes out positive every time. In (b), (d) and (f) a bigger positive
is subtracted from a smaller one, so the answer is negative.

Figure it Out — Page 259
Using tokens for subtraction

Q1 Try to subtract: – 3 – (+ 5). How many zero pairs will you have to put in? What is the
result?

Start with 3 red tokens. You have to take away 5 green tokens — but there is not a single green
token there!

Put in 5 zero pairs (5 green with 5 red). The value is still – 3.

Now the set is 5 green + 8 red.

Take away the 5 green → 8 red are left.

Answer: 5 zero pairs are needed, and – 3 – (+ 5) = – 8.

Page 32 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

5 zero pairs added, then the green ones are taken away

+ + + + +

– – – – – – – – =–8

The three original red tokens plus the five red tokens from the zero pairs make 8 red tokens — the
answer – 8.

Shorter way: – 3 – (+ 5) = – 3 + (– 5) = – 8. Subtracting a positive number is the same
as adding a negative one, so you go 5 steps further down.

Q2 Evaluate the following using tokens. a. (– 3) – (+ 10) b. (+ 8) – (– 7) c. (– 5) – (+ 9) d. (– 9)
– (+ 10) e. (+ 6) – (– 4) f. (– 2) – (+ 7)

EXPRESSION ZERO PAIRS NEEDED SAME AS ANSWER

a. (– 3) – (+ 10) 10 – 3 + (– 10) – 13

b. (+ 8) – (– 7) 7 8+7 + 15

c. (– 5) – (+ 9) 9 – 5 + (– 9) – 14

d. (– 9) – (+ 10) 10 – 9 + (– 10) – 19

e. (+ 6) – (– 4) 4 6+4 + 10

f. (– 2) – (+ 7) 7 – 2 + (– 7) –9

The pattern: whenever the tokens you must remove are not there, put in exactly
that many zero pairs. Every time, the work ends up being the same as ‘add the
inverse’ — which is why that one rule is enough for all subtractions of integers.

Page 33 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
In-text Questions — Pages 259 & 260
m as e
.co
Section 10.3 Integers in Other Places — credits and debits
a g l
a s em
aQ1gl You open a bank account with the ₹100 you had saved. Then you make ₹60 at your
job the next day and you deposit it in your account. This is shown in your bank

co
passbook as a ‘credit’. Your new bank balance is _______.
m
e m . ag
g l as
a

A credit is money coming in, so it is a positive number.

co m
em.
as
₹100 + ₹60 = ₹160
m l
m .co a g
l a se
g
Answer: your new balance is ₹160.
a
m a s
m .co agl
se
Q2 The next day you pay your electric bill of ₹30 using your bank account. This is shown

g l a
in your bank passbook as a ‘debit’. Your bank balance is now ______.
a
m

. co
A debit is money going out, so it is a negative number.
e m
m l as
m .co a g
l a se ₹160 + (– ₹30) = ₹160 – ₹30 = ₹130

ag
Answer: your balance is now ₹130.
se m
com g l a
m . a
ase
Q3 agl
The next day you make a major purchase for your business of ₹150. Again this is
shown as a debit. What is your bank balance now? ______ Is this possible?

co m
m .
m as e
.co l

a g
se m
g l a ₹130 + (– ₹150) = – ₹20
a c
m .
s e
. c o m
Your balance is – ₹20, that is you are ₹20 short.
a g la
s e
Yes, this is possible. Some banks mdo allow the balance to go below zero for a short time — it is
ausually charges an extra amount as ‘interest’ or a ‘fee’ while the
agl
called an overdraft. The bank
balance stays negative.

co m
m .
m ase
.co


a g l Page 34 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Why a negative balance makes sense: the balance is simply the total of all credits
(+) and debits (–). If the debits are larger, the total lands on the other side of zero —
exactly like a lift going below the ground floor.

Q4 Your strategic large purchase the previous day allows you to make ₹200 at your
business the next day. What is your balance now? ______

– ₹20 + ₹200 = ₹180

Answer: ₹180. The first ₹20 of the earning clears the negative balance and brings you to 0; the
remaining ₹180 is real money in the account.

Tip: Write the whole month as one addition — 100 + 60 – 30 – 150 + 200 = 180.
Credits with a +, debits with a –, and add.

Figure it Out — Page 260
Credits and debits

Q1 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?

Credits are positive, debits are negative. Add them all.

Credits: 30 + 40 + 50 = + 120

Debits: 40 + 50 + 60 = – 150

Balance = (+ 120) + (– 150) = – ₹30

Answer: the balance is – ₹30 — the account is ₹30 in debt.

Page 35 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Why negative: more money went out than came in. 150 – 120 = 30, and since the
debits are larger, the answer carries the debit's minus sign.

Q2 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?

Total of the debits: 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128

= 3 + 4 + 8 + 16 + 32 + 64 + 128

= 7 + 8 + 16 + 32 + 64 + 128 = 15 + 16 + 32 + 64 + 128

= 31 + 32 + 64 + 128 = 63 + 64 + 128 = 127 + 128 = 255

Balance = (– 255) + (+ 256) = + ₹1

Answer: the balance is ₹1.

Did you know? The debits are the powers of 2, and their running total is always one
less than the next power of 2: 1 + 2 + 4 + … + 128 = 256 – 1 = 255. That is why a single
credit of ₹256 leaves exactly ₹1.

Q3 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?

Why a positive balance is better:

A negative balance means you owe money to the bank, and the bank charges interest or a
fee on it — so you end up paying extra.
Money kept in the account may even earn interest instead of costing it.
A positive balance is a safety net for an emergency — a hospital bill, a broken cycle, school
fees.
Banks trust customers who keep a positive balance, so getting a loan later becomes easier.

Page 36 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

When a temporary negative balance can still be worthwhile:

To buy stock or a machine for your business that will earn back more than it cost — exactly
what happened on page 259, where the ₹150 purchase brought ₹200 the next day.
In an emergency — medicines, an operation, or urgent travel — where waiting is not an
option.
For something that will save money later, such as paying school fees on time to avoid a late-
fee penalty.

The rule of thumb: going below zero is sensible only when the money borrowed
brings back more than the interest and fees you pay for it, and only for a short time.

Figure it Out — Page 261
Geographical cross sections

Q1 Looking at the geographical cross section, fill in the respective heights: a. b. c. d. e.
f. g.

Read each marked point against the height scale on the left. Sea level is 0 m; above it the
heights are positive, below it they are negative.

Height (m)
A + 1500
1500
E + 1200

1000

500 C + 300
G + 100
sea level
0

F – 200
– 500
B – 500

– 1000

D – 1200
– 1500

Page 37 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

The geographical cross section with the seven points read off the height scale. Peaks are positive,
valleys and ocean floors are negative.

POINT A. A B. B C. C D. D E. E F. F G. G

Height + 1500 m – 500 m + 300 m – 1200 m + 1200 m – 200 m + 100 m

Tip: These are readings from a drawing, so answers a little different from these (say
+ 1450 m for A) are also acceptable — what matters is the sign and the rough size.

Q2 Which is the highest point in this geographical cross section? Which is the lowest
point?

Highest: A, at + 1500 m above sea level

Lowest: D, at – 1200 m below sea level

A is the tallest peak in the slice, and D is the deepest trench.

Why D and not B: both are below sea level, but – 1200 < – 500. With negative
numbers the one further from 0 is the smaller — and here it is also the deeper one.

Q3 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?

List the seven heights: + 1500, – 500, + 300, – 1200, + 1200, – 200, + 100.

Decreasing order: A (+ 1500) > E (+ 1200) > C (+ 300) > G (+ 100) > F (– 200) > B (– 500)

> D (– 1200)

Increasing order: D (– 1200) < B (– 500) < F (– 200) < G (+ 100) < C (+ 300) < E (+ 1200)

< A (+ 1500)

Page 38 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Tip: Sort the positive heights normally, sort the negative heights in the reverse way,

m as e
l
and remember that every positive height beats every negative one.

m .co a g
l a se
a g
Q4 What is the highest point above sea level on Earth? What is its height?

com
m . ag
l a se
agEverest (Sagarmatha / Chomolungma), in the Himalayas
The highest point on Earth is Mount
on the Nepal–China border.

co m
em.
m l as
.co
Height of Mount Everest = + 8848 m above sea level
a g
se m
(the 2020 joint survey by India's neighbours put it at 8848.86 m)
g l a
a
m a s
agl
Did you know? India's highest peak is Kangchenjunga in Sikkim, at + 8586 m — the
third highest mountain in the world.
m.co
l a se
ag
What is the lowest point with respect to sea level on land or on the ocean floor?
co m
.
Q5

e m
as
What is its height? (This height should be negative).
m l
.co a g
a s em
gl The lowest point anywhere on Earth is the Challenger Deep, at the southern end of the

a
Mariana Trench in the Pacific Ocean.
se m
com g l a
m . a
ase
agl
Depth of the Challenger Deep ≈ – 11 000 m (about – 10 935 m)

m
On land, the lowest exposed point is the shore of the Dead Sea, at about – 430 m.

. co
se m
o m l a
Why the height is written as negative: heights are measured from sea level, which
aglike the basement floors of
.cis 0 m. Anything below that mark gets a minus sign, just
m Bela's building.
l a se
ag
.c
s e m
. c om
Compare: Everest is + 8848 m and the Challenger Deep is about – 11 000 m. The
a g la
difference between them is m
a s e 8848 + 11 000 = 19 848 m — nearly 20 km from the top

agl
of the world to the bottom!

co m
m .
m ase
.co


a g l Page 39 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

In-text Questions — Page 262
Temperature

Q1 During summer time you would have heard in the news that there is a ‘heat wave’.
What do you think will be the temperature during the summer when you feel very
hot?

On a heat-wave day the temperature usually crosses 45 °C.

A normal hot summer day in most of India: about 38 °C to 42 °C.
A heat wave is declared when the maximum goes far above the normal — often 45 °C or
more.
Places such as Churu (Rajasthan), Nagpur and Delhi have recorded around 48 °C to 50 °C.

Tip: All of these are positive temperatures — far above 0 °C, the freezing point of
water. Negative temperatures come only in very cold places.

Q2 What has been the maximum temperature during the summer and the minimum
temperature during the winter last year in your area? Find out.

This one you must find out for your own town — from the newspaper, the TV weather bulletin,
the IMD (India Meteorological Department) website, or by asking at school.

PLACE (EXAMPLE) SUMMER MAXIMUM WINTER MINIMUM DIFFERENCE

Delhi about 45 °C about 4 °C 41 °C

Chennai about 40 °C about 20 °C 20 °C

Shimla about 28 °C about – 2 °C 30 °C

Leh about 25 °C about – 15 °C 40 °C

How to find the difference: use subtraction as ‘movement’. For Leh, 25 – (– 15) = 25
+ 15 = 40 °C. The thermometer has to climb 15 degrees just to reach 0, and 25 more
after that.

Page 40 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Figure it Out — Page 262
Temperature

MATH TALK

Q1 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?

Places in India where the temperature falls below 0 °C:

Ladakh — Leh, Kargil, Drass (Drass is one of the coldest inhabited places on Earth, once
recording about – 45 °C).
Jammu & Kashmir — Srinagar, Gulmarg, Pahalgam.
Himachal Pradesh — Keylong, Manali, Spiti, Kalpa.
Uttarakhand — Auli, Munsiyari, Badrinath.
Sikkim and Arunachal Pradesh — Lachung, Tawang, Sela Pass.

What is common among them: every one of these places is in the Himalayas — high up in the
mountains, and in the far north of the country.

Why it gets colder there:

Altitude. The air gets thinner and cooler as you go up — roughly 6 °C colder for
every 1000 m of height. Leh is about 3500 m up, so it is about 20 °C colder than a
plain at the same latitude.
Snow. White snow reflects most of the sunlight back instead of absorbing it, so
the ground never warms up.
Dry, thin air. Whatever little heat the ground gains in the day escapes quickly at
night.
Latitude. These places are the furthest north in India, so the winter sun stays low
and weak.

Math Talk: Discuss in class — is 0 °C ‘no temperature’? No! It is only the mark where
water freezes. Temperatures continue below it, just as floors continue below the
ground.

Page 41 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q2 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. (14 °C, 8 °C, – 2 °C, – 4 °C with 02:00 a.m., 11:00 p.m., 02:00 p.m., 11:00 a.m.)
The two lists as printed on page 262:

TEMPERATURE TIME

14 °C 02:00 a.m.

8 °C 11:00 p.m.

– 2 °C 02:00 p.m.

– 4 °C 11:00 a.m.

Think about how a day warms up and cools down: coldest just before dawn, warmest in the
early afternoon.

TEMPERATURE TIME WHY

14 °C 02:00 p.m. Warmest reading — early afternoon, the sun is strongest

8 °C 11:00 a.m. Still warming up in the late morning

– 2 °C 11:00 p.m. Below freezing at night, but the ground is still losing its heat

– 4 °C 02:00 a.m. Coldest reading — the middle of the night, hours after sunset

Order of the four readings: – 4 °C < – 2 °C < 8 °C < 14 °C

Fall from 2 p.m. to 2 a.m.: – 4 – 14 = – 18 °C — the thermometer drops 18 degrees

Why the two negative readings belong to the night: after sunset there is no
sunlight to heat the ground, so in a dry, high place like Leh the temperature keeps
falling until sunrise. – 4 °C means 4 degrees below the freezing point of water.

Figure it Out — Pages 263 & 264

Page 42 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Section 10.4 Explorations with Integers — a hollow integer grid

Q1 Do the calculations for the second grid above and find the border sum.

4 –1 –3 5 –3 –5

–3 1 0 –5

–1 –1 2 –8 –2 7

First grid Second grid

The two hollow integer grids on page 263.

The second grid is

5 –3 –5

0 –5

–8 –2 7

Top row: 5 + (– 3) + (– 5) = – 3

Bottom row: (– 8) + (– 2) + 7 = – 3

Left column: 5 + 0 + (– 8) = – 3

Right column: (– 5) + (– 5) + 7 = – 3

Answer: the border sum of the second grid is – 3.

Why it is called a ‘hollow’ grid: only the eight numbers on the border are used —
the middle cell plays no part in any of the four sums.

Page 43 of 61

ase
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Complete the grids to make the required border sum: (i) – 10, ___, ___ / ___, ___, – 5 / 9,
e
Q2

com___ / ___, ___, – 5 / ___, ___, ___ with border sum – 4 l as
___, ___ with border sum + 4 (ii) 6, 8, ___ / ___, ___, – 5 / ___, – 2, ___ with border sum – 2
(iii) 7,.___, g
em a
a s
a gl

m
Fill in whichever line has only one number missing, then move to the next.
. co ag
e m
(i) Border sum + 4 — start with the left column: – 10 + ? + 9 = 4, so the middle-left cell is 5.

g l as
– 10 a10 4

co m
m.
5 –5

m as e
9
.co
– 10

a g
5
l
se m
g l a
a Top: – 10 + 10 + 4 = 4 ✔ Right: 4 + (– 5) + 5 = 4 ✔ Bottom: 9 + (– 10) + 5 = 4 ✔ Left: –

m a s
.co agl
10 + 5 + 9 = 4 ✔

se m
g l a
a
(ii) Border sum – 2 — the top row has only one gap: 6 + 8 + ? = – 2, so it is – 16. Everything else
then follows, and this grid has only one answer.

co m
6 8 – 16
m .
m as e
.co a g l
e–m
11 –5

a s
agl 19 –2 19

se m
o m g l a
m .c
Right: – 16 + (– 5) + 19 = – 2 ✔ Bottom: – 19 + (– 2) + 19 = – 2 ✔ Left: 6 + 11 + (– 19) =
a
l a se
ag
–2✔

(iii) Border sum – 4 — here you may choose freely. One filling is:
co m
m .
–s9e
m l a
.co ag
7 –2

a s em – 3
gl
–5

a c
m .
e
–8 –6 10

m a s
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 44 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Top: 7 + (– 2) + (– 9) = – 4 ✔ Right: – 9 + (– 5) + 10 = – 4 ✔ Bottom: – 8 + (– 6) + 10 = –

4 ✔ Left: 7 + (– 3) + (– 8) = – 4 ✔

Q3 For the last grid above, find more than one way of filling the numbers to get border
sum – 4.

7

–5

The last of the three grids in Question 2, page 264 — only two entries are given.

In the last grid only two numbers are fixed — 7 in the top-left corner and – 5 in the middle of the
right column. That leaves plenty of freedom. Here are two more fillings:
Way 2

Page 45 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

7 – 11 0

– 11 –5

0 –5 1

Top: 7 – 11 + 0 = – 4 ✔ Right: 0 – 5 + 1 = – 4 ✔ Bottom: 0 – 5 + 1 = – 4 ✔ Left: 7 – 11

+0=–4✔

Way 3

7 –6 –5

–9 –5

–2 –8 6

Top: 7 – 6 – 5 = – 4 ✔ Right: – 5 – 5 + 6 = – 4 ✔ Bottom: – 2 – 8 + 6 = – 4 ✔ Left: 7 – 9

–2=–4✔

How to make your own: pick any two numbers for the top-right and bottom-left
corners. The rest of the grid is then forced — each remaining cell sits in a line with
only one gap left.

Q4 Which other grids can be filled in multiple ways? What could be the reason?

The first grid (border sum + 4) and the third grid (border sum – 4) can be filled in many ways.
The second grid (border sum – 2) has only one answer.

GRID NUMBERS ALREADY GIVEN FREE CHOICES LEFT HOW MANY FILLINGS

First (+ 4) 3 numbers 1 many

Second (– 2) 4 numbers 0 exactly one

Third (– 4) 2 numbers 2 many

Page 46 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

The reason: there are four conditions (top row, bottom row, left column, right
column). Each condition can fill in one unknown, so four conditions can pin down
four unknowns. In the second grid exactly four numbers are missing on the border,
so everything is forced. In the other two grids more numbers are missing than there
are conditions, and the extra ones can be chosen as you like — after that the rest is
decided.

Another way to say it: in the first grid you may choose the top-right corner freely;
in the third grid you may choose two of the corners freely.

Q5 Make a border integer square puzzle and challenge your classmates.

Here is a simple recipe for making a puzzle that is sure to work.

1. Choose a border sum, say – 6.
2. Choose the four corner numbers first, for example 2, – 4 (top) and – 3, 1 (bottom).
3. Now fill the middle cell of each line so that the line adds up to – 6:

Top: 2 + ? + (– 4) = – 6 → ? = – 4

Bottom: – 3 + ? + 1 = – 6 → ? = – 4

Left: 2 + ? + (– 3) = – 6 → ? = – 5

Right: – 4 + ? + 1 = – 6 → ? = – 3

4. The complete grid is:

2 –4 –4

–5 –3

–3 –4 1

Now rub out three or four of the numbers, write ‘Border sum is – 6’ underneath, and hand it to a
friend.

Try This: Rub out exactly four numbers and your puzzle will have one answer. Rub
out five or more and your friend will find several answers — both kinds are fun.

Page 47 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Figure it Out — Page 265
An amazing grid of numbers!

TRY THIS

Q1 Try afresh, choose different numbers this time. What sum did you get? Was it
different from the first time? Try a few more times!

The sum is always – 1, no matter which numbers you circle. Here are three fresh games played
on the grid of page 264 (3, 4, 0, 9 / – 2, – 1, – 5, 4 / 1, 2, – 2, 7 / – 7, – 6, – 10, – 1):

GAME CIRCLED NUMBERS (ONE FROM EACH ROW AND EACH SUM
COLUMN)

The book's (– 1) + 9 + (– 7) + (– 2) –1
game

Game 2 3 + 4 + 2 + (– 10) –1

Game 3 0 + (– 2) + 7 + (– 6) –1

Game 4 9 + (– 5) + 1 + (– 6) –1

In Game 2, for example, 3 comes from row 1 column 1, 4 from row 2 column 4, 2 from row 3
column 2 and – 10 from row 4 column 3 — the striking-out rule makes sure you use every row
once and every column once.

3 + 4 + 2 + (– 10) = – 1 0 + (– 2) + 7 + (– 6) = – 1 9 + (– 5) + 1 + (– 6) = – 1

Answer: the sum is – 1 every single time. It is never different, however you play.

Q2 Play the same game with the grids below. What answer did you get?

First grid (7, 10, 13, 16 / – 2, 1, 4, 7 / – 11, – 8, – 5, – 2 / – 20, – 7, – 14, – 11): the answer is – 8
every time.

Page 48 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
e m.
16 + 4 + (– 8) + (– 20) = – 8
m l as
.co
10 + 7 + (– 11) + (– 14) = – 8
m a g
l a se
g
16 + 1 + (– 11) + (– 14) = – 8
a

co m
Second grid (– 11, – 10, – 9, – 8 / – 7, – 6, – 5, – 4 / – 3, – 2, – 1, 0 / 1, 2, 3, 4):

e m . ag
g l as
a
(– 11) + (– 6) + (– 1) + 4 = – 14 (– 8) + (– 5) + (– 2) + 1 = – 14 (– 9) + (– 4) + (– 3) + 2 =

– 14

co m
e m.
m l as
Answer: the first grid always gives – 8 and the second grid always gives – 14.
.co a g
a s emon a printing slip: in the first grid the last row is printed as – 20, – 7, – 14, – 11.
gl
Note
a To fit the pattern of the grid (each row going up in steps of 3) that second entry

m
should be – 17. If you happen to circle the printed – 7, your total will come out + 2
a s
m .co
instead of – 8. With – 17 in that cell, every game gives – 8.
agl
l a se
ag
Q3 What could be so special about these grids? Is the magic in the numbers or the way
co m
m .
se
they are arranged or both? Can you make more such grids?

o m g l a
m .c a
se magic is in the arrangement — the numbers are built from a row list and a column list.

g l a
a The
Look at the second grid. Write 1, 2, 3, 4 above the columns and – 11, – 7, – 3, 1 beside the rows.
se m
com g l a
.
Every entry is simply row number + column number — for instance the entry in row 2, column
m a
ase
3 is – 7 + 2 = – 5. ✔

+ 0 agl 1 2 3

co m –8
– 11
.
– 11 – 10 –9

em
m l as
.co a g
–7 –7 –6 –5 –4

a s em – 3
agl
–3 –2 –1 0

.c
1 1 2 3 4
s e m
m a
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 49 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Why the sum can never change: the striking-out rule forces you to take exactly one
number from each row and one from each column. So the four circled numbers use
every row number once and every column number once:

Sum = (– 11 – 7 – 3 + 1) + (0 + 1 + 2 + 3) = (– 20) + 6 = – 14

It does not matter which cells you pick — the same eight numbers always get added.

Make your own: choose any four row numbers and any four column numbers, add them in a
grid, and the answer of your game will always be (sum of the rows) + (sum of the columns). For
example rows 5, 0, – 2, 10 and columns 1, – 1, 4, 0 give a grid whose game always totals 13 + 4 =
17.

Try This: Make a 4 × 4 grid whose game always gives 0. Just pick row numbers and
column numbers that together add up to 0 — say rows 3, – 1, 2, – 4 and columns 5, –
5, 1, – 1.

Figure it Out — Pages 265 & 266
Practice with integers

Q1 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

‘Between’ means the two given numbers themselves are not written. Increasing order = left to
right on the number line.

a. Between 0 and – 7: – 6, – 5, – 4, – 3, – 2, – 1

b. Between – 4 and 4: – 3, – 2, – 1, 0, 1, 2, 3

c. Between – 8 and – 15: – 14, – 13, – 12, – 11, – 10, – 9

d. Between – 30 and – 23: – 29, – 28, – 27, – 26, – 25, – 24

Page 50 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Careful with (c) and (d): the smaller number is written second in the question. – 15
is smaller than – 8, so the list must start at – 14 and climb up to – 9. Do not forget
that 0 belongs in the list in part (b).

Q2 Give three numbers such that their sum is – 8.

Many answers are possible. Here are a few, each checked:

(– 3) + (– 4) + (– 1) = – 8

5 + (– 6) + (– 7) = – 8

10 + (– 12) + (– 6) = – 8

0 + (– 8) + 0 = – 8

100 + (– 50) + (– 58) = – 8

How to build your own: write down any two numbers, add them, and then choose
the third number so that the total moves to – 8. For 5 and – 6 the running total is – 1,
and – 1 needs a further – 7 to reach – 8.

Q3 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.

Make the addition table of every possible roll. The faces are – 5, – 3, – 1, 2, 4, 6.

Page 51 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

+ –5 –3 –1 2 4 6

–5 – 10 –8 –6 –3 –1 1

–3 –8 –6 –4 –1 1 3

–1 –6 –4 –2 1 3 5

2 –3 –1 1 4 6 8

4 –1 1 3 6 8 10

6 1 3 5 8 10 12

The sums that can appear are – 10, – 8, – 6, – 4, – 3, – 2, – 1, 1, 3, 4, 5, 6, 8, 10, 12.

So the numbers between – 10 and + 12 that are not possible are

– 9, – 7, – 5, 0, 2, 7, 9, 11

Why 0 cannot appear: to get 0 the two faces would have to be inverses of each
other — like + 5 and – 5, or + 1 and – 1. On these dice the positive faces are 2, 4, 6
and the negative faces are – 1, – 3, – 5, so no pair cancels out.

Check it yourself: that is 8 impossible values out of the 21 whole numbers from – 9
to + 11.

Q4 Solve these: 8 – 13, (– 8) – (13), (– 13) – (– 8), (– 13) + (– 8), 8 + (– 13), (– 8) – (– 13), (13) – 8,
13 – (– 8)

Change every subtraction to ‘add the inverse’ first.

Page 52 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

EXPRESSION WRITTEN AS AN ADDITION ANSWER

8 – 13 8 + (– 13) –5

(– 8) – (13) – 8 + (– 13) – 21

(– 13) – (– 8) – 13 + 8 –5

(– 13) + (– 8) already an addition – 21

8 + (– 13) already an addition –5

(– 8) – (– 13) – 8 + 13 +5

(13) – 8 13 + (– 8) +5

13 – (– 8) 13 + 8 + 21

Look at the pattern: only four different answers appear — – 5, – 21, + 5 and + 21.
The digits are always 5 or 21; only the signs change. 13 and 8 differ by 5 and add to
21, and the sign depends on which of them ‘wins’.

Q5 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?

Take the present year as 2026 CE (do the same working with whatever year it is when you solve
this).
a. 150 years back is still in the CE years, so plain subtraction works.

2026 – 150 = 1876 CE

b. 2200 years back takes us past the year 1 CE and into the BCE years. Do it in two steps.

Page 53 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
e m.
From 2026 CE back to 1 CE: 2026 – 1 = 2025 years
m l as
.co
Years still to go back: 2200 – 2025 = 175
m a g
l a se
g
Because there is no year 0, the year just before 1 CE is 1 BCE, the one before that is 2 BCE
a…

com
. ag
So we reach 175 BCE

se m
l a
ag us towards the year 1, so the BCE number gets smaller.
c. 680 BCE with 320 years added moves

co m
m.
680 – 320 = 360 BCE

m as e
.co a g l
a s emBCE years count backwards: think of BCE years as negative and CE years as
gl
Why
a positive on a time line, with no 0 in between. Adding years always means moving to

s
the right — 680 BCE + 320 years lands on 360 BCE, which is later in history.
m a
m.co agl
l a se
g
Note: The book's answer key writes ‘– 360 BCE’ for part (c). The minus sign is not
a
needed — ‘BCE’ already says that the year is before the Common Era. The correct

m
answer is 360 BCE.
. co
e m
m l as
.co a g
a s em Complete the following sequences: a. (– 40), (– 34), (– 28), (– 22), ___, ___, ___ b. 3, 4, 2, 5,
gl
Q6

a 1, 6, 0, 7, ___, ___, ___ c. ___, ___, 12, 6, 1, (– 3), (– 6), ___, ___, ___

se m
com g l a
. a

m
ase
agl
a. Find the difference between one term and the next.

m
– 40 + 6→ – 34 + 6→ – 28 + 6→ – 22 → – 16, – 10, – 4

. co
em
m l as
.co g
b. This sequence has two threads running through it, one going down and one going up.

m a
l a se
ag Odd places: 3, 2, 1, 0 … each one less → next is – 1, then – 2
.c
s e m
a
Even places: 4, 5, 6, 7 … each one more → next is 8
m
So the sequence continues – 1, 8, – 2
e m . co agl
g l as
a
c. Look at the differences: they grow by 1 each time.

co m
m .
m ase
.co


a g l Page 54 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

12 – 6→ 6 – 5→ 1 – 4→ – 3 – 3→ – 6

Going backwards the steps were – 7 and – 8, so the two missing numbers in front are 27, 19

Going forward the steps are – 2, – 1, 0, so the sequence continues – 8, – 9, – 9 (and then – 8

again)

Full sequence: 27, 19, 12, 6, 1, – 3, – 6, – 8, – 9, – 9, – 8, – 6 …

Why (c) turns around: the amount being subtracted gets smaller and smaller (6, 5,
4, 3, 2, 1), then becomes 0, and after that the steps turn positive. So the sequence
falls, flattens out at – 9, and starts climbing again.

Q7 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).

We can hit exactly – 30. Two neat ways:

Way 1: (– 5) – (+ 7) – (+ 18)

= – 5 – 7 – 18 = – 30 ✔

Way 2: (– 2) + (– 9) – (+ 18) – (+ 1)

= – 2 – 9 – 18 – 1 = – 30 ✔

How to search: to make a very negative value, add the negative cards and subtract
the positive ones — subtracting a positive number pushes the value down. The big
card (+ 18) is the most useful of all, provided it is subtracted.

Try This: Using the same six cards, can you make + 30? Try (+ 18) + (+ 7) + (+ 1) – (– 5)
+ … and see how close you get.

Page 55 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

Q8 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)

KIND OF RESULT EXAMPLES
EXPRESSION

a. (positive) – (negative) always positive 7 – (– 2) = 9, 1 – (– 100) = 101

b. (positive) + (negative) positive, negative or 7 + (– 2) = 5, 2 + (– 7) = – 5, 7 + (– 7) = 0
zero

c. (negative) + (negative) always negative – 3 + (– 4) = – 7

d. (negative) – (negative) positive, negative or – 3 – (– 8) = 5, – 8 – (– 3) = – 5, – 3 – (– 3) =
zero 0

e. (negative) – (positive) always negative –3–4=–7

f. (negative) + (positive) positive, negative or – 3 + 8 = 5, – 8 + 3 = – 5, – 3 + 3 = 0
zero

Why some are certain and others are not: in (a), (c) and (e) both moves push the
same way — in (a) you start above zero and go further up; in (c) and (e) you start
below zero and go further down. In (b), (d) and (f) the two numbers pull in opposite
directions, so the winner depends on which one has the bigger digits.

Note: (b), (d) and (f) are really the same situation, because d and f can be turned into
b by ‘adding the inverse’.

Q9 This string has a total of 100 tokens arranged in a particular pattern. What is the
value of the string?

Look at the pattern on the string: 3 green tokens then 2 red tokens, repeating again and
again.

Page 56 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

one block of 5 tokens = + 1

+ + + – – + + + – – …

The string repeats a block of 5 tokens — three green and two red. Each block is worth (+ 3) + (– 2) = +
1.

Value of one block = (+ 3) + (– 2) = + 1

Number of blocks in the string = 100 ÷ 5 = 20

Value of the string = 20 × (+ 1) = + 20

Answer: the value of the string is + 20.

Another way to see it: in 20 blocks there are 20 × 3 = 60 green tokens and 20 × 2 =
40 red tokens. Cancel 40 zero pairs and 20 green tokens are left, so the value is + 20.
✔

Figure it Out — Page 268
Section 10.5 A Pinch of History — Brahmagupta's rules

Q1 Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or in
terms of a number line?

Yes — every rule is just a journey in the lift, or a walk on the number line.
Rules for addition

Page 57 of 61

Class 6 Maths Chapter 10 The Other Side of Zero AglaSem · NCERT Solutions

BRAHMAGUPTA'S RULE IN THE BUILDING OF FUN / ON THE NUMBER
LINE

1. Positive + positive is positive (2 + 3 = 5) Go 2 floors up, then 3 more up — you are 5 floors above
the ground.

2. Negative + negative is negative (– 2 + (– 3) = Go 2 floors down, then 3 more down — you are 5 floors
– 5) below the ground.

3. Positive + negative: subtract the digits, keep Go 5 floors down, then 3 back up. The 3 ups cancel 3 of
the sign of the bigger (– 5 + 3 = – 2) the downs and 2 downs are left over, so you finish on
Floor – 2.

4. A number + its inverse = 0 (2 + (– 2) = 0) Press + 2 then – 2 and you are back at the Welcome Hall,
Floor 0.

5. A number + 0 = the same number (– 2 + 0 = – Press the button zero times — you stay exactly where you
2) were.

Rules for subtraction (remember: Target – Starting = Movement needed)

BRAHMAGUPTA'S RULE IN THE BUILDING OF FUN / ON THE NUMBER LINE

1. Smaller positive from larger positive is From Floor 2 to Floor 3 you must go up: press + 1.
positive (3 – 2 = 1)

2. Larger positive from smaller positive is From Floor 3 to Floor 2 you must come down: press – 1.
negative (2 – 3 = – 1)

3. Subtracting a negative is adding the From Floor – 3 to Floor 2 you must first climb 3 floors to reach
positive (2 – (– 3) = 2 + 3) the ground and 2 more after that — 5 floors up in all.

4. A number minus itself is 0 (– 2 – (– 2) = 0) You are already on Floor – 2 and want to be on Floor – 2 — no
button press at all.

5. Number – 0 = the number; 0 – number = From Floor – 2 to the ground floor you must press + 2, the
its inverse (0 – (– 2) = 2) inverse of – 2.

Why this matters: Brahmagupta wrote these rules in the Brāhma-sphuṭa-siddhānta
in 628 CE — the first time in the world that positive numbers, negative numbers and
zero were treated as equally real numbers. The lift model of this chapter is just a
picture of the same rules, more than 1300 years later.

Page 58 of 61

as e
Class 6 Maths Chapter 10 The Other Side of Zero
a g l AglaSem · NCERT Solutions

co m
m.
Give your own examples of each rule.
e
Q2

m l as
m .co a g
a se

ag l
Addition

co m
. ag
1. Two positives: 14 + 25 = 39 (a credit of ₹14 and another of ₹25)
e m
g l as
2. Two negatives: (– 12) + (– 8) = – 20 (a debit of ₹12 and another of ₹8)
a
3. One of each: (– 20) + 7 = – 13 and 20 + (– 7) = + 13

co m
m.
4. A number and its inverse: 45 + (– 45) = 0

m as e
.co
5. With zero: (– 31) + 0 = – 31
a g l
a s em
a l
gSubtraction
m a s
.co agl
1. 19 – 6 = 13 (both positive, the smaller taken away)

se m
l a
2. 6 – 19 = – 13 (the larger taken away from the smaller)
g
a
3. 15 – (– 9) = 15 + 9 = 24 (temperature rising from – 9 °C to 15 °C)

co m
.
4. (– 17) – (– 17) = 0

e m
m l as
.co
5. (– 6) – 0 = – 6 and 0 – (– 6) = + 6
a g
a s em
agl Try This: Make up a story for each of your examples — a bank passbook, a
thermometer in Leh, a lift in a shopping mall, or a mine shaft. A rule you can tell a
se m
com g l a
. a
story about is a rule you will not forget.
m
ase
agl

co m
m .
m
Chapter at a glance
o l a se
m .cThere are numbers less than zero. They are written with aa‘–’g sign in front, are called
l a se
g
negative numbers, and lie to the left of 0 on the number line. In Bela's Building of Fun
a c
.
they are the floors below the ground.

s e m
m a
The numbers … –3, –2, –1, 0, 1, 2, 3 … are the integers. Positive integers are 1, 2, 3, …;

m . co
negative integers are –1, –2, –3, …; and 0 is neither positive nor negative — it is the
agl
reference point.
l a se
agadditive inverse — the number that gives 0 when added to it. The
Every number has an
inverse of +4 is –4, the inverse of –543 is 543, and the inverse of 0 is 0.

co m
m .
m ase
.co


a g l Page 59 of 61