---
title: "NCERT Solutions Class 6 Maths Chapter 10 The Other Side of Zero"
url: https://www.swavid.com/maths/class/6/chapter/the-other-side-of-zero/ncert-solutions
dateModified: 2026-10-07T14:55:35+00:00
---

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

This chapter's questions cover the foundational concepts of integers, including the introduction of negative numbers, representation on number lines, and operations of addition and subtraction.

Free PDF (36 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-6/swavid-ncert-solutions-class-6-maths-chapter-10-the-other-side-of-zero-43e9315323.pdf

## 10.1 Bela's Building of Fun

### Question 1

*2 marks · Very short answer*

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

**Solution**

1. To go four floors up from the ground floor, we must press the '+' button four times, which is written as +4.
2. To go three floors down from the ground floor, we must press the '-' button three times, which is written as -3.

**Answer:** Press + + + + or +4 to go four floors up, and - - - or -3 to go three floors down.

> Common mistake: Confusing the number of button presses with the sign or mixing up the plus and minus directions.

## Numbering the floors in the building of fun

### Question 1

*2 marks · Very short answer*

Number all the floors in the Building of Fun.

**Solution**

1. The ground floor is called Floor 0.
2. The floors above the ground floor are numbered with positive numbers (+1, +2, +3, ...) and the floors below the ground are numbered with negative numbers (-1, -2, ...).

**Answer:** Book Store: +3, Art Centre: +2, Food Court: +1, Welcome Hall: 0, Toy Store: -1, Video Games shop: -2

> Common mistake: Confusing floors above and below ground by assigning negative numbers to floors above ground.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Starting floor = $+2$
2. Movement = $-3$
3. Target floor = $\text{Starting Floor} + \text{Movement} = (+2) + (-3) = -1$
4. You will reach Floor $-1$, which is the Toy Store.

**Answer:** $(+2) + (-3) = -1$; Toy Store

> Common mistake: Confusing the direction of movement and adding instead of subtracting.

### Question 2

*3 marks · Short answer*

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**

1. For a, $(+1) + (+4) = +5$
2. For b, $(+4) + (+1) = +5$
3. For c, $(+4) + (-3) = +1$
4. For d, $(-1) + (+2) = +1$
5. For e, $(-1) + (+1) = 0$
6. For f, $0 + (+2) = +2$
7. For g, $0 + (-2) = -2$

**Answer:** a. $+5$, b. $+5$, c. $+1$, d. $+1$, e. $0$, f. $+2$, g. $-2$

> Common mistake: Mixing up signs when combining positive and negative values.

### Question 3

*3 marks · Short answer*

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**

1. We use the relation: $\text{Starting Floor} + \text{Movement} = \text{Target Floor}$ (-5).
2. If we start at Floor $+1$, the expression is $(+1) + (-6) = -5$.
3. If we start at Floor $-2$, the expression is $(-2) + (-3) = -5$.
4. If we start at Floor $0$, the expression is $0 + (-5) = -5$.

**Answer:** $(+1) + (-6) = -5$, $(-2) + (-3) = -5$, $0 + (-5) = -5$

> Common mistake: Forgetting that moving down requires a negative number.

## Figure it out

### Question 1

*3 marks · Short answer*

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) = $

**Part a (0.75 marks)**

1. Combine the button presses: $(+1) + (+4) = +5$.

Answer a: (+5)

**Part b (0.75 marks)**

1. Combine the button presses: $(+4) + (+1) = +5$.

Answer b: (+5)

**Part c (0.75 marks)**

1. Combine the button presses: $(+4) + (-3) + (-2) = (+1) + (-2) = -1$.

Answer c: (-1)

**Part d (0.75 marks)**

1. Combine the button presses: $(-1) + (+2) + (-3) = (+1) + (-3) = -2$.

Answer d: (-2)

**Answer:** a. (+5), b. (+5), c. (-1), d. (-2)

> Common mistake: Adding or subtracting numbers without considering the signs properly.

### Question 2

*2 marks · Very short answer*

Write the inverses of these numbers: $+4, -4, -3, 0, +2, -1$.

**Solution**

1. The additive inverse of a number is the number that, when added to the given number, gives zero.
2. Inverse of +4 is -4, inverse of -4 is +4, inverse of -3 is +3, inverse of 0 is 0, inverse of +2 is -2, and inverse of -1 is +1.

**Answer:** Inverse of +4 is -4, of -4 is +4, of -3 is +3, of 0 is 0, of +2 is -2, of -1 is +1.

> Common mistake: Writing the inverse without the correct sign.

### Question 3

*Activity*

Connect the inverses by drawing lines.

**Solution**

1. Identify the additive inverse for each given number.
2. Draw lines connecting each number to its corresponding inverse.

**Answer:** Connect each number with its inverse: +9 with -9, +7 with -7, -8 with +8, and -5 with +5.

## Comparing numbers using floors

### Question 1

*3 marks · Short answer*

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 Toys Store. So, he is on Floor ____.

**Part 1**

1. Jay is in the Art Centre, which is on Floor $+2$.

Answer 1: Floor $+2$

**Part 2 (1 mark)**

1. Asin is in the Sports Centre, which is on Floor $+5$.

Answer 2: Floor $+5$

**Part 3 (1 mark)**

1. Binnu is in the Cinema Centre, which is on Floor $-3$.

Answer 3: Floor $-3$

**Part 4 (1 mark)**

1. Aman is in the Toys Store, which is on Floor $-1$.

Answer 4: Floor $-1$

**Answer:** Binnu is on the lowest floor on Floor $-3$.

> Common mistake: Confusing floors below ground as positive numbers or mixing up the order of negative numbers.

## Figure it Out

### Question 1

*3 marks · Short answer*

Compare the following numbers using the Building of Fun and fill in the boxes with $<$ or $>$.
a. $-2 \square +5$
b. $-5 \square +4$
c. $-5 \square -3$
d. $+6 \square -6$
e. $0 \square -4$
f. $0 \square +4$

**Part a (0.5 marks)**

1. Negative number $-2$ is below ground and positive number $+5$ is above ground.
2. Therefore, $-2 < +5$.

Answer a: $-2 < +5$

**Part b (0.5 marks)**

1. Negative number $-5$ is lower than positive number $+4$.
2. Therefore, $-5 < +4$.

Answer b: $-5 < +4$

**Part c (0.5 marks)**

1. Floor $-5$ is lower than Floor $-3$ in the Building of Fun.
2. Therefore, $-5 < -3$.

Answer c: $-5 < -3$

**Part d (0.5 marks)**

1. Floor $+6$ is above ground and Floor $-6$ is below ground.
2. Therefore, $+6 > -6$.

Answer d: $+6 > -6$

**Part e (0.5 marks)**

1. Ground floor $0$ is above any negative floor like $-4$.
2. Therefore, $0 > -4$.

Answer e: $0 > -4$

**Part f (0.5 marks)**

1. Ground floor $0$ is below positive floor $+4$.
2. Therefore, $0 < +4$.

Answer f: $0 < +4$

**Answer:** a. $-2 < +5$, b. $-5 < +4$, c. $-5 < -3$, d. $+6 > -6$, e. $0 > -4$, f. $0 < +4$

> Common mistake: Confusing the direction of inequality symbols for negative numbers.

### Question 2

*3 marks · Short answer*

Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with $<$ or $>$:
a. $-10 \square -12$
b. $+17 \square -10$
c. $0 \square -20$
d. $+9 \square -9$
e. $-25 \square -7$
f. $+15 \square -17$

**Part a (0.5 marks)**

1. Floor $-10$ is higher than Floor $-12$ on the building line.
2. Therefore, $-10 > -12$.

Answer a: $-10 > -12$

**Part b (0.5 marks)**

1. Any positive number is greater than any negative number.
2. Therefore, $+17 > -10$.

Answer b: $+17 > -10$

**Part c (0.5 marks)**

1. Zero is greater than any negative number.
2. Therefore, $0 > -20$.

Answer c: $0 > -20$

**Part d (0.5 marks)**

1. Positive number $+9$ is greater than negative number $-9$.
2. Therefore, $+9 > -9$.

Answer d: $+9 > -9$

**Part e (0.5 marks)**

1. Floor $-25$ is lower than Floor $-7$ in the building.
2. Therefore, $-25 < -7$.

Answer e: $-25 < -7$

**Part f (0.5 marks)**

1. Positive number $+15$ is greater than negative number $-17$.
2. Therefore, $+15 > -17$.

Answer f: $+15 > -17$

**Answer:** a. $-10 > -12$, b. $+17 > -10$, c. $0 > -20$, d. $+9 > -9$, e. $-25 < -7$, f. $+15 > -17$

> Common mistake: Thinking $-12$ is greater than $-10$ because $12 > 10$.

### Question 3

*3 marks · Short answer*

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

**Solution**

1. Given Floor $A = -12$, Floor $D = -1$, and Floor $E = +1$.
2. Observing the vertical spacing from the figure in the textbook (Fig. 10.6), each tick mark represents $1$ unit.
3. Counting down from $D = -1$ by $8$ units to $B$, we get $B = -9$.
4. Counting down from $D = -1$ by $5$ units to $C$, we get $C = -6$.
5. Counting up from $E = +1$ by $1$ unit to $F$, we get $F = +2$.
6. Counting up from $E = +1$ by $5$ units to $G$, we get $G = +6$.
7. Counting up from $E = +1$ by $10$ units to $H$, we get $H = +11$.

**Answer:** $B = -9$, $C = -6$, $F = +2$, $G = +6$, $H = +11$

> Common mistake: Miscounting the scale units between the marked floors.

### Question 4

*Activity*

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

**Solution**

1. Locate $0$ on the vertical building line shown in the textbook (Fig. 10.6).
2. Mark $-7$ by counting $7$ units down from $0$.
3. Mark $-4$ by counting $4$ units down from $0$.
4. Mark $+3$ by counting $3$ units up from $0$.
5. Mark $-10$ by counting $10$ units down from $0$.

**Answer:** Marked floors $-7, -4, +3,$ and $-10$ correctly on the vertical number line.

> Common mistake: Counting upwards instead of downwards for negative floor numbers.

## Subtraction to find which button to press

### Question 1

*3 marks · Short answer*

Evaluate $15-5$, $100-10$ and $74-34$ from this perspective.

**Part (i) (1 mark)**

1. Express $15 - 5$ as finding the missing addend: $5 + ? = 15$.
2. Find the missing number to get $15 - 5 = 10$.

Answer (i): 10

**Part (ii) (1 mark)**

1. Express $100 - 10$ as finding the missing addend: $10 + ? = 100$.
2. Find the missing number to get $100 - 10 = 90$.

Answer (ii): 90

**Part (iii) (1 mark)**

1. Express $74 - 34$ as finding the missing addend: $34 + ? = 74$.
2. Find the missing number to get $74 - 34 = 40$.

Answer (iii): 40

**Answer:** $15 - 5 = 10$, $100 - 10 = 90$, and $74 - 34 = 40$

> Common mistake: Treating subtraction only as take away instead of recognizing it as finding the missing number to be added.

## Figure it Out

### Question 1

*3 marks · Short answer*

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) = $

**Part a (0.3 marks)**

1. Starting floor is $+1$ and target floor is $+4$, requiring a movement of $-3$ floors.

Answer a: $(-3)$

**Part b (0.3 marks)**

1. Starting floor is $0$ and target floor is $+2$, requiring a movement of $-2$ floors.

Answer b: $(-2)$

**Part c (0.3 marks)**

1. Starting floor is $+4$ and target floor is $+1$, requiring a movement of $+3$ floors.

Answer c: $(+3)$

**Part d (0.3 marks)**

1. Starting floor is $0$ and target floor is $-2$, requiring a movement of $+2$ floors.

Answer d: $(+2)$

**Part e (0.3 marks)**

1. Starting floor is $-3$ and target floor is $+4$, requiring a movement of $+7$ floors.

Answer e: $(+7)$

**Part f (0.3 marks)**

1. Starting floor is $-3$ and target floor is $-4$, requiring a movement of $-1$ floor.

Answer f: $(-1)$

**Part g (0.3 marks)**

1. Starting floor is $+2$ and target floor is $-1$, requiring a movement of $-3$ floors.

Answer g: $(-3)$

**Part h (0.3 marks)**

1. Starting floor is $-2$ and target floor is $-2$, requiring no movement.

Answer h: $0$

**Part i (0.3 marks)**

1. Starting floor is $+1$ and target floor is $-1$, requiring a movement of $-2$ floors.

Answer i: $(-2)$

**Part j (0.3 marks)**

1. Starting floor is $-3$ and target floor is $+3$, requiring a movement of $+6$ floors.

Answer j: $(+6)$

**Answer:** a. $(-3)$, b. $(-2)$, c. $(+3)$, d. $(+2)$, e. $(+7)$, f. $(-1)$, g. $(-3)$, h. $0$, i. $(-2)$, j. $(+6)$

> Common mistake: Confusing the sign of the movement by subtracting in the wrong direction.

## Figure it Out

### Question 1

*3 marks · Short answer*

Complete these expressions.
a. $(+40) + \_\_\_\_\_\_ = +200$
b. $(+40) + \_\_\_\_\_\_\_ = -200$
c. $(-50) + \_\_\_\_\_\_ = +200$
d. $(-50) + \_\_\_\_\_\_\_ = -200$
e. $(-200) - (-40) = $
f. $(+200) - (+40) = $
g. $(-200) - (+40) = $

**Solution**

1. For part a, $40 + ? = 200$, so the missing number is $200 - 40 = +160$.
2. For part b, $40 + ? = -200$, so the missing number is $-200 - 40 = -240$.
3. For part c, $-50 + ? = 200$, so the missing number is $200 - (-50) = 200 + 50 = +250$.
4. For part d, $-50 + ? = -200$, so the missing number is $-200 - (-50) = -200 + 50 = -150$.
5. For part e, $(-200) - (-40) = -200 + 40 = -160$.
6. For part f, $(+200) - (+40) = 200 - 40 = +160$.
7. For part g, $(-200) - (+40) = -200 - 40 = -240$.
8. Thus, the completed values are a. $+160$, b. $-240$, c. $+250$, d. $-150$, e. $-160$, f. $+160$, g. $-240$.

**Answer:** a. $+160$, b. $-240$, c. $+250$, d. $-150$, e. $-160$, f. $+160$, g. $-240$

> Common mistake: Confusing signs when subtracting a negative number or finding the missing addend.

## Figure it Out

### Question 1

*6 marks · Long answer*

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. $+1500 - (-1500)$

**Part a (0.75 marks)**

1. Start at level -125 and move 30 steps down in the infinite lift.
2. The result is -125 + (-30) = -155.

Answer a: -155

**Part b (0.75 marks)**

1. Subtracting a negative number is the same as adding the corresponding positive number.
2. Rewrite +105 - (-55) as +105 + (+55) = 160.

Answer b: 160

**Part c (0.75 marks)**

1. Start at level +105 and move 55 steps up in the infinite lift.
2. The result is +105 + (+55) = 160.

Answer c: 160

**Part d (0.75 marks)**

1. Rewrite subtraction of a negative number as addition of the corresponding positive number.
2. Evaluate +80 + (+150) = 230.

Answer d: 230

**Part e (0.75 marks)**

1. Start at level +80 and move 150 steps up.
2. The result is +80 + (+150) = 230.

Answer e: 230

**Part f (0.75 marks)**

1. Replace the subtraction of -200 with the addition of its inverse +200.
2. Evaluate -99 + 200 = 101.

Answer f: 101

**Part g (0.75 marks)**

1. Start at level -99 and move 200 steps up in the infinite lift.
2. The result is -99 + (+200) = 101.

Answer g: 101

**Part h (0.75 marks)**

1. Rewrite +1500 - (-1500) as addition of the positive number +1500.
2. Evaluate +1500 + (+1500) = 3000.

Answer h: 3000

**Answer:** a. -155, b. 160, c. 160, d. 230, e. 230, f. 101, g. 101, h. 3000

> Common mistake: Forgetting that subtracting a negative number is equivalent to adding the corresponding positive number.

## Figure it Out

### Question 1

*Activity*

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

**Solution**

1. Observe the number line ranging from $-10$ to $10$ in the figure.
2. Mark any three positive numbers (such as $2, 5, 8$) to the right of zero and any three negative numbers (such as $-1, -3, -7$) to the left of zero.

**Answer:** Three positive numbers and three negative numbers are marked on the number line.

### Question 2

*2 marks · Very short answer*

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

**Solution**

1. Look at the three negative numbers marked on the number line in the previous activity.
2. Write down those negative numbers in the given boxes, for example $-7$, $-3$, and $-1$.

**Answer:** $$-7, -3, -1$$

> Common mistake: Writing positive numbers instead of negative numbers.

### Question 3

*3 marks · Short answer*

Is $2 > -3$? Why? Is $-2 < 3$? Why?

**Solution**

1. Yes, $2 > -3$ because on the number line, $2$ lies to the right of $-3$.
2. Yes, $-2 < 3$ because on the number line, $-2$ lies to the left of $3$.
3. Recall that smaller numbers are to the left of larger numbers on the number line.

**Answer:** $2 > -3$ and $-2 < 3$ because smaller numbers lie to the left of larger numbers on the number line.

> Common mistake: Confusing the direction of inequality signs.

### Question 4

*3 marks · Short answer*

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

**Solution**

1. For $-5 + 0$, adding zero to any number gives back the same number, so the result is $-5$.
2. For $7 + (-7)$, a number plus its additive inverse is zero, so the result is $0$.
3. For $-10 + 20$, subtracting the smaller number from the greater without signs and keeping the sign of the greater gives $+10$.
4. For $10 - 20$, using addition rules or number line movement gives $-10$.
5. For $7 - (-7)$, subtracting a negative number is the same as adding the corresponding positive number, $7 + 7 = 14$.
6. For $-8 - (-10)$, converting to addition gives $-8 + 10 = 2$.

**Answer:** (i) $-5$, (ii) $0$, (iii) $10$, (iv) $-10$, (v) $14$, (vi) $2$

> Common mistake: Incorrectly handling signs when subtracting a negative integer.

## Use unmarked number lines to evaluate these expressions:

### Question 1

*3 marks · Short answer*

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

**Part a (0.75 marks)**

1. Start at $-125$ on the unmarked number line.
2. Move 30 units further in the negative (left) direction.
3. The result is $-125 + (-30) = -155$.

Answer a: -155

**Part b (0.75 marks)**

1. Start at $+105$ on the unmarked number line.
2. Subtracting a negative number is the same as adding the corresponding positive number.
3. The expression becomes $+105 + (+55) = 160$.

Answer b: 160

**Part c (0.75 marks)**

1. Start at $+80$ on the unmarked number line.
2. Replace subtraction of $-150$ with addition of its inverse, $+150$.
3. The expression becomes $+80 + (+150) = 230$.

Answer c: 230

**Part d (0.75 marks)**

1. Start at $-99$ on the unmarked number line.
2. Replace subtraction of $-200$ with addition of $+200$.
3. The expression becomes $-99 + (+200) = 101$.

Answer d: 101

**Answer:** a. -155, b. 160, c. 230, d. 101

> Common mistake: Confusing signs when subtracting a negative number, treating it as subtraction instead of addition.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Part a (0.75 marks)**

1. We have $6$ positive tokens and $4$ positive tokens.
2. Combining them gives $6 + 4 = 10$ positive tokens.
3. The result is $+10$.

Answer a: $+10$

**Part b (0.75 marks)**

1. We have $3$ negative tokens and $2$ negative tokens.
2. Combining them gives $3 + 2 = 5$ negative tokens.
3. The result is $-5$.

Answer b: $-5$

**Part c (0.75 marks)**

1. We have $5$ positive tokens and $7$ negative tokens.
2. Five zero pairs are formed and removed, leaving $2$ negative tokens.
3. The result is $-2$.

Answer c: $-2$

**Part d (0.75 marks)**

1. We have $2$ negative tokens and $6$ positive tokens.
2. Two zero pairs are formed and removed, leaving $4$ positive tokens.
3. The result is $+4$.

Answer d: $+4$

**Answer:** a. $+10$, b. $-5$, c. $-2$, d. $+4$

> Common mistake: Confusing signs when combining positive and negative tokens.

### Question 2

*3 marks · Short answer*

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

**Part a (1.5 marks)**

1. The set has $3$ positive tokens and $5$ negative tokens as shown in the textbook (Fig. 10.2).
2. After canceling $3$ zero pairs, we are left with $2$ negative tokens.
3. The lift attendant is on Floor $-2$ and the addition statement is $(+3) + (-5) = -2$.

Answer a: Floor $-2$, expression $(+3) + (-5) = -2$

**Part b (1.5 marks)**

1. The set has $6$ positive tokens and $3$ negative tokens as shown in the textbook (Fig. 10.2).
2. After canceling $3$ zero pairs, we are left with $3$ positive tokens.
3. The lift attendant is on Floor $+3$ and the addition statement is $(+6) + (-3) = +3$.

Answer b: Floor $+3$, expression $(+6) + (-3) = +3$

**Answer:** a. $(+3) + (-5) = -2$ (Floor $-2$); b. $(+6) + (-3) = +3$ (Floor $+3$)

> Common mistake: Counting the wrong sign after removing zero pairs.

## Figure it Out

### Question 1

*3 marks · Short answer*

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. $(-2)-(-6)$

**Part a (0.5 marks)**

1. Start with 10 positive tokens and take away 7 positive tokens.
2. 3 positive tokens are left.

Answer a: (+3)

**Part b (0.5 marks)**

1. Start with 8 negative tokens and take away 4 negative tokens.
2. 4 negative tokens are left.

Answer b: (-4)

**Part c (0.5 marks)**

1. Start with 9 negative tokens and take away 4 negative tokens.
2. 5 negative tokens are left.

Answer c: (-5)

**Part d (0.5 marks)**

1. Start with 9 positive tokens, but we cannot take away 12 positives without adding zero pairs.
2. After adding 3 zero pairs and taking away 12 positives, 3 negative tokens are left.

Answer d: (-3)

**Part e (0.5 marks)**

1. Start with 5 negative tokens, but we need to take away 7 negatives.
2. By adding 2 zero pairs, we can take away 7 negatives and leave 2 positive tokens.

Answer e: (+2)

**Part f (0.5 marks)**

1. Start with 2 negative tokens, but we need to take away 6 negatives.
2. By adding 4 zero pairs, we can take away 6 negatives and leave 4 positive tokens.

Answer f: (+4)

**Answer:** a. (+3), b. (-4), c. (-5), d. (-3), e. (+2), f. (+4)

> Common mistake: Forgetting that subtracting more negative tokens than present leaves positive tokens.

### Question 2

*3 marks · Short answer*

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

**Part a (0.5 marks)**

1. Convert the subtraction to addition of its inverse: $(-5) + (+7)$.
2. Subtract the smaller number from the greater and give the sign of the greater number: $+2$.

Answer a: $+2$

**Part b (0.5 marks)**

1. Convert to addition: $(+10) + (-13)$.
2. Subtract the smaller from the greater and keep the sign of the greater: $-3$.

Answer b: $-3$

**Part c (0.5 marks)**

1. Convert to addition: $(-7) + (+9)$.
2. Subtract $7$ from $9$ to get $+2$.

Answer c: $+2$

**Part d (0.5 marks)**

1. Convert to addition: $(+3) + (-8)$.
2. Subtract $3$ from $8$ and place the sign of the greater number: $-5$.

Answer d: $-5$

**Part e (0.5 marks)**

1. Convert to addition: $(-2) + (+7)$.
2. Subtract $2$ from $7$ to get $+5$.

Answer e: $+5$

**Part f (0.5 marks)**

1. Convert to addition: $(+3) + (-15)$.
2. Subtract $3$ from $15$ and place the sign of the greater number: $-12$.

Answer f: $-12$

**Answer:** a. $+2$, b. $-3$, c. $+2$, d. $-5$, e. $+5$, f. $-12$

> Common mistake: Forgetting to change the sign of the number being subtracted when converting to addition.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Start with 3 negative tokens representing $-3$.
2. To subtract $+5$, we need to take away 5 positive tokens, but there are no positive tokens present.
3. We add 5 zero pairs (each consisting of one positive and one negative token) so that the value does not change, making a total of 8 negative tokens and 5 positive tokens.
4. We then take away the 5 positive tokens, leaving 8 negative tokens.
5. The result is $-8$ and we have to put in 5 zero pairs.

**Answer:** We have to put in 5 zero pairs, and the result is $-8$.

> Common mistake: Forgetting to add zero pairs when there are not enough tokens of the required type to take away.

### Question 2

*3 marks · Short answer*

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

**Part a (0.5 marks)**

1. Start with 3 negatives and add 10 zero pairs.
2. Take away 10 positives, leaving 13 negatives.

Answer a: -13

**Part b (0.5 marks)**

1. Start with 8 positives and add 7 zero pairs.
2. Take away 7 negatives, leaving 15 positives.

Answer b: +15

**Part c (0.5 marks)**

1. Start with 5 negatives and add 9 zero pairs.
2. Take away 9 positives, leaving 14 negatives.

Answer c: -14

**Part d (0.5 marks)**

1. Start with 9 negatives and add 10 zero pairs.
2. Take away 10 positives, leaving 19 negatives.

Answer d: -19

**Part e (0.5 marks)**

1. Start with 6 positives and add 4 zero pairs.
2. Take away 4 negatives, leaving 10 positives.

Answer e: +10

**Part f (0.5 marks)**

1. Start with 2 negatives and add 7 zero pairs.
2. Take away 7 positives, leaving 9 negatives.

Answer f: -9

**Answer:** a. -13, b. +15, c. -14, d. -19, e. +10, f. -9

> Common mistake: Forgetting to add sufficient zero pairs before taking away tokens of the opposite sign.

## Figure it Out

### Question 1

*3 marks · Short answer*

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?

**Solution**

1. Credits are represented as positive numbers and debits are represented as negative numbers.
2. The bank account balance is the total of all credits and debits starting from 0: $(+30) + (+40) + (+50) + (-40) + (-50) + (-60)$.
3. Calculating the sum gives $\text{\textcent}(-30)$.

**Answer:** ₹ (-30)

> Common mistake: Treating debits as positive additions instead of negative subtractions.

### Question 2

*3 marks · Short answer*

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?

**Solution**

1. Debits are represented as negative numbers and credits are represented as positive numbers.
2. The bank account balance is the total starting from 0: $(-1) + (-2) + (-4) + (-8) + (-16) + (-32) + (-64) + (-128) + (+256)$.
3. Summing the debits gives $-255$, and adding the credit of $+256$ results in $+1$.

**Answer:** ₹ 1

> Common mistake: Making arithmetic errors when adding multiple consecutive negative numbers.

### Question 3

*3 marks · Short answer*

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**

1. It is generally better to maintain a positive balance to avoid paying additional bank fees, interest, or penalties charged when an account becomes negative.
2. It may be worthwhile to temporarily have a negative balance (an overdraft) when making a crucial business or personal purchase that generates higher returns or income shortly after.

**Answer:** A positive balance avoids bank fees, while a temporary negative balance can be worthwhile for strategic investments that quickly generate income.

> Common mistake: Forgetting to mention the cost of fees or the purpose of strategic investments.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Part a**

1. Observe the height corresponding to point A from the given geographical cross-section.
2. The height of point A is above sea level at $+1500$ m.

Answer a: $+1500$ m

**Part b**

1. Observe the height corresponding to point B from the given geographical cross-section.
2. The height of point B is below sea level at $-500$ m.

Answer b: $-500$ m

**Part c**

1. Observe the height corresponding to point C from the given geographical cross-section.
2. The height of point C is above sea level at $+300$ m.

Answer c: $+300$ m

**Part d**

1. Observe the height corresponding to point D from the given geographical cross-section.
2. The height of point D is below sea level at $-1200$ m.

Answer d: $-1200$ m

**Part e**

1. Observe the height corresponding to point E from the given geographical cross-section.
2. The height of point E is above sea level at $+1200$ m.

Answer e: $+1200$ m

**Part f**

1. Observe the height corresponding to point F from the given geographical cross-section.
2. The height of point F is below sea level at $-200$ m.

Answer f: $-200$ m

**Part g**

1. Observe the height corresponding to point G from the given geographical cross-section.
2. The height of point G is above sea level at $+100$ m.

Answer g: $+100$ m

**Answer:** a. $+1500$ m, b. $-500$ m, c. $+300$ m, d. $-1200$ m, e. $+1200$ m, f. $-200$ m, g. $+100$ m

> Common mistake: Confusing positive heights above sea level with negative depths below sea level.

### Question 2

*2 marks · Very short answer*

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

**Solution**

1. Compare the heights of all points given in the cross-section.
2. The highest point is A ($+1500$ m) and the lowest point is D ($-1200$ m).

**Answer:** Highest point: A, Lowest point: D

> Common mistake: Confusing the lowest negative number with the smallest value magnitude-wise.

### Question 3

*3 marks · Short answer*

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?

**Part i (1.5 marks)**

1. Arrange the heights of points A, B, C, D, E, F, G from largest to smallest.
2. The decreasing order of heights is A, E, C, G, F, B, D.

Answer i: A, E, C, G, F, B, D

**Part ii (1.5 marks)**

1. Arrange the heights of points A, B, C, D, E, F, G from smallest to largest.
2. The increasing order of heights is D, B, F, G, C, E, A.

Answer ii: D, B, F, G, C, E, A

**Answer:** Decreasing: A, E, C, G, F, B, D; Increasing: D, B, F, G, C, E, A

> Common mistake: Listing negative integers in reverse order for increasing sequence.

### Question 4

*2 marks · Very short answer*

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

**Solution**

1. Identify the highest point above sea level on Earth.
2. The highest point is Mount Everest with a height of $8848$ m above sea level.

**Answer:** Mount Everest, $8848$ m

> Common mistake: Forgetting to include the unit meters.

### Question 5

*2 marks · Very short answer*

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**

1. The lowest point with respect to sea level on the ocean floor is the Challenger Deep in the Mariana Trench.
2. Its height is represented as a negative number, which is approximately $-11034\text{ m}$ (or $-10994\text{ m}$).

**Answer:** The lowest point is the Challenger Deep in the Mariana Trench, with a height of $-11034\text{ m}$.

> Common mistake: Forgetting to put a negative sign for depths below sea level.

## Figure it Out

### Question 1

*3 marks · Short answer*

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**

1. Places in India like Leh, Drass in Ladakh, and Gulmarg in Kashmir experience temperatures below $0^\circ\text{C}$ during winter.
2. All these places are located at very high altitudes in the Himalayan and mountain regions.
3. It becomes colder there because temperature decreases with an increase in altitude above sea level.

**Answer:** High-altitude places like Leh and Drass experience temperatures below $0^\circ\text{C}$ due to elevation.

> Common mistake: Confusing latitude with altitude as the reason for freezing temperatures.

### Question 2

*3 marks · Short answer*

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.

**Solution**

1. Observe the given temperature readings of Leh in November: 14°C, 8°C, -2°C, and -4°C.
2. During a winter day in Leh, the temperature is highest during the afternoon and lowest just before sunrise (early morning).
3. Matching the highest temperature (14°C) to 02:00 p.m., 8°C to 11:00 a.m., -2°C to 11:00 p.m., and the lowest temperature (-4°C) to 02:00 a.m.

**Answer:** 14°C matches 02:00 p.m., 8°C matches 11:00 a.m., -2°C matches 11:00 p.m., and -4°C matches 02:00 a.m.

> Common mistake: Matching higher positive temperatures to night-time or negative temperatures to afternoon.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Calculate the top row sum: $5 + (-3) + (-5) = -3$
2. Calculate the bottom row sum: $(-8) + (-2) + 7 = -3$
3. Calculate the left column sum: $5 + 0 + (-8) = -3$
4. Calculate the right column sum: $(-5) + (-5) + 7 = -3$
5. Since all rows and columns add up to the same number, the border sum is $-3$.

**Answer:** The border sum of the second grid is -3.

> Common mistake: Adding incorrectly with negative signs or missing one of the rows or columns.

## Figure it Out

### Question 2

*3 marks · Short answer*

Complete the grids to make the required border sum:

**Part a (1 mark)**

1. 1. Identify the target border sum as $+4$.
2. 2. Choose numbers for the empty cells such that each row and column adds up to $+4$.
3. 3. One possible filled grid is: Top row $-10, 10, 4$, Middle column $-5, -5$, Bottom row $9, -10, 5$.

Answer a: Grid completed with border sum $+4$.

**Part b (1 mark)**

1. 1. Identify the target border sum as $-2$.
2. 2. Choose numbers for the empty cells such that each row and column adds up to $-2$.
3. 3. One possible filled grid is: Top row $6, 8, -16$, Middle column $11, -5$, Bottom row $-19, -2, 19$.

Answer b: Grid completed with border sum $-2$.

**Part c (1 mark)**

1. 1. Identify the target border sum as $-4$.
2. 2. Choose numbers for the empty cells such that each row and column adds up to $-4$.
3. 3. One possible filled grid is: Top row $7, -2, -9$, Middle column $-3, -5$, Bottom row $-8, -6, 10$.

Answer c: Grid completed with border sum $-4$.

**Answer:** Filled grids with required border sums of $+4$, $-2$, and $-4$.

> Common mistake: Forgetting to check all four sides (top row, bottom row, left column, right column).

### Question 3

*3 marks · Short answer*

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

**Solution**

1. 1. Observe that an integer grid has multiple empty cells with only partial sum constraints.
2. 2. Changing one pair of numbers in opposite directions while keeping their sum constant allows alternative valid fillings.
3. 3. Thus, multiple combinations of numbers can achieve the same border sum of $-4$.

**Answer:** Multiple ways exist because changing interior or boundary values in complementary pairs maintains the same border sum.

> Common mistake: Not verifying if every row and column strictly equals the required border sum.

### Question 4

*3 marks · Short answer*

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

**Solution**

1. 1. All the grids with missing numbers can generally be filled in multiple ways.
2. 2. The reason is that fixing the border sum leaves freedom to adjust numbers as long as the row and column totals remain balanced.
3. 3. Opposite adjustments in paired cells cancel out and keep the total sum unchanged.

**Answer:** All grids can be filled in multiple ways because multiple integer combinations can yield the same row and column sums.

> Common mistake: Assuming there is only one correct set of numbers for a grid puzzle.

### Question 5

*Activity*

Make a border integer square puzzle and challenge your classmates.

**Solution**

1. 1. Draw a hollow square grid with empty cells on the border.
2. 2. Assign a target border sum using positive and negative integers.
3. 3. Challenge classmates to fill the empty cells such that the top, bottom, left, and right sums match the target.

**Answer:** An activity to create and solve border integer square puzzles.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Choose any number from the grid and cross out its row and column.
2. Repeat the process by choosing an unstruck number until no numbers are left.
3. Add all the circled numbers together to get the final sum.

**Answer:** The sum obtained depends on the numbers chosen in each step, but the game always results in a sum that can be explored by varying choices.

> Common mistake: Struck numbers being chosen again by mistake.

### Question 2

*3 marks · Short answer*

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

**Solution**

1. Play the same elimination game on the first given $4 \times 4$ grid by repeatedly circling numbers and striking out their rows and columns.
2. Add the final circled numbers for the first grid to get $-8$.
3. Repeat the game for the second $4 \times 4$ grid and add the circled numbers to get $-14$.

**Answer:** For the first grid the sum is $-8$, and for the second grid the sum is $-14$.

> Common mistake: Forgetting to strike out the complete row and column after circling a number.

### Question 3

*3 marks · Short answer*

What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

**Solution**

1. Observe that the magic in these grids comes from both the specific numbers chosen and the special way they are arranged in rows and columns.
2. Notice that every row and column sum properties remain balanced in these types of grids.
3. Yes, more such grids can be made by following arithmetic progressions and specific grid construction rules.

**Answer:** The magic lies in both the numbers and their careful arrangement, and more such grids can be created.

> Common mistake: Assuming the magic is only in the numbers without considering their arrangement.

## Figure it Out

### Question 1

*3 marks · Short answer*

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 $-23$

**Part a (1 mark)**

1. List the integers between $0$ and $-7$ in increasing order.
2. The integers are $-6, -5, -4, -3, -2, -1$.

Answer a: $-6, -5, -4, -3, -2, -1$

**Part b (1 mark)**

1. List the integers between $-4$ and $4$ in increasing order.
2. The integers are $-3, -2, -1, 0, 1, 2, 3$.

Answer b: $-3, -2, -1, 0, 1, 2, 3$

**Part c (0.5 marks)**

1. List the integers between $-8$ and $-15$ in increasing order.
2. The integers are $-14, -13, -12, -11, -10, -9$.

Answer c: $-14, -13, -12, -11, -10, -9$

**Part d (0.5 marks)**

1. List the integers between $-30$ and $-23$ in increasing order.
2. The integers are $-29, -28, -27, -26, -25, -24$.

Answer d: $-29, -28, -27, -26, -25, -24$

**Answer:** Integers in increasing order: (a) $-6, -5, -4, -3, -2, -1$ (b) $-3, -2, -1, 0, 1, 2, 3$ (c) $-14, -13, -12, -11, -10, -9$ (d) $-29, -28, -27, -26, -25, -24$

> Common mistake: Writing numbers in decreasing order instead of increasing order.

### Question 2

*3 marks · Short answer*

Give three numbers such that their sum is $-8$.

**Solution**

1. Choose any three integers such that their sum equals $-8$.
2. For example, consider $-5$, $7$, and $-10$.
3. Calculate the sum: $(-5) + 7 + (-10) = 2 + (-10) = -8$.

**Answer:** $-5, 7, -10$ (Other combinations are also possible)

> Common mistake: Incorrect sign addition while verifying the sum.

### Question 3

*3 marks · Short answer*

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**

1. The faces on each die are $-1, 2, -3, 4, -5, 6$.
2. We find all possible sums by adding any number from the first die and any number from the second die.
3. The obtainable sums are $-10, -8, -7, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 9, 10, 12$.
4. The numbers between $-10$ and $+12$ that cannot be formed by adding numbers from the two dice are $-9, -5, 0, 2, 7, 11$.

**Answer:** -9, -5, 0, 2, 7, 11

> Common mistake: Forgetting that zero does not exist on the number line or missing negative combinations.

### Question 4

*3 marks · Short answer*

Solve these:
$8 - 13$
$(-8) - (13)$
$(-13) - (-8)$
$(-13) + (-8)$
$8 + (-13)$
$(-8) - (-13)$
$(13) - 8$
$13 - (-8)$

**Part 1 (0.375 marks)**

1. Evaluate $8 - 13$ by converting subtraction to addition of inverse or using number line.
2. $8 + (-13) = -5$.

Answer 1: $-5$

**Part 2 (0.375 marks)**

1. Evaluate $(-8) - (13)$.
2. $(-8) + (-13) = -21$.

Answer 2: $-21$

**Part 3 (0.375 marks)**

1. Evaluate $(-13) - (-8)$ by replacing subtraction of negative with addition of positive.
2. $(-13) + (+8) = -5$.

Answer 3: $-5$

**Part 4 (0.375 marks)**

1. Evaluate $(-13) + (-8)$ by adding two negatives.
2. $(-21)$.

Answer 4: $-21$

**Part 5 (0.375 marks)**

1. Evaluate $8 + (-13)$.
2. $(-5)$.

Answer 5: $-5$

**Part 6 (0.375 marks)**

1. Evaluate $(-8) - (-13)$ by converting subtraction of negative to addition.
2. $(-8) + (+13) = 5$.

Answer 6: $5$

**Part 7 (0.375 marks)**

1. Evaluate $13 - 8$.
2. $5$.

Answer 7: $5$

**Part 8 (0.375 marks)**

1. Evaluate $13 - (-8)$ by converting subtraction to addition.
2. $13 + 8 = 21$.

Answer 8: $21$

**Answer:** Solutions: $8 - 13 = -5$, $(-8) - (13) = -21$, $(-13) - (-8) = -5$, $(-13) + (-8) = -21$, $8 + (-13) = -5$, $(-8) - (-13) = 5$, $13 - 8 = 5$, $13 - (-8) = 21$

> Common mistake: Confusion between signs when subtracting negative numbers.

### Question 5

*3 marks · Case-based*

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? ________

**Part a (1 mark)**

1. Subtract $150$ years from the present year $0$ (taking $1$, since there is no year $0$).
2. Counting back $150$ years gives $150$ BCE.

Answer a: 150 BCE

**Part b (1 mark)**

1. Subtract $2200$ years from the present year, crossing over year $1$ directly to $1$ BCE.
2. Counting back $2200$ years gives $2200$ BCE.

Answer b: 2200 BCE

**Part c (1 mark)**

1. Starting from $680$ BCE, going forward by $320$ years means moving towards zero.
2. Subtracting $320$ from $680$ gives $360$ BCE.

Answer c: 360 BCE

**Answer:** a. 150 CE (or years before present), b. 2200 BCE, c. 360 BCE

> Common mistake: Forgetting that there is no year 0 and incorrectly shifting the year count by 1.

### Question 6

*3 marks · Case-based*

Complete the following sequences:
a. $(-40), (-34), (-28), (-22), \_\_\_\_\_ , \_\_\_\_\_\_ , \_\_\_\_\_\_$
b. $3, 4, 2, 5, 1, 6, 0, 7, \_\_\_\_\_ , \_\_\_\_\_ , \_\_\_\_\_$
c. \_\_\_\_\_ , \_\_\_\_\_\_ , 12, 6, 1, (-3), (-6), \_\_\_\_\_ , \_\_\_\_\_ , \_\_\_\_\_\_

**Part a (1 mark)**

1. Observe the pattern in $(-40), (-34), (-28), (-22)$: each term increases by $+6$.
2. Add $6$ to the last given term $-22$ to get $-16$, then $-10$, and then $-4$.

Answer a: -16, -10, -4

**Part b (1 mark)**

1. Observe the alternating pattern or differences: $3+1=4$, $4-2=2$, $2+3=5$, $5-4=1$, $1+5=6$, $6-6=0$, $0+7=7$.
2. Continue the pattern by subtracting $8$, adding $9$, and subtracting $10$ to get $-1, 8, -2$.

Answer b: -1, 8, -2

**Part c (1 mark)**

1. Observe the differences between consecutive terms: $12-6=6$, $6-1=5$, $1-(-3)=4$, $(-3)-(-6)=3$.
2. The differences decrease by $1$ each time; working backwards and forwards gives the missing terms.

Answer c: 27, 19, -8, -9, -9

**Answer:** a. -16, -10, -4; b. -1, 8, -2; c. 27, 19, -8, -9, -9

> Common mistake: Miscalculating the difference between negative integers in arithmetic sequences.

### Question 7

*3 marks · Short answer*

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**

1. List the available integer cards: $(+1), (+7), (+18), (-5), (-2), (-9)$.
2. Choose cards and operations to get a value close to $(-30)$.
3. Consider the expression: $(-2) + (-9) - (+18) - (+1) = -30$.

**Answer:** $(-2) + (-9) - (+18) - (+1) = -30$

> Common mistake: Incorrectly combining signs when subtracting integers.

### Question 8

*3 marks · Short answer*

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**

1. Analyze each operation using the rules of integers.
2. a. (positive) $-$ (negative) becomes addition of a positive, which is always positive.
3. b. (positive) $+$ (negative) can be positive or negative depending on which magnitude is larger.
4. c. (negative) $+$ (negative) is always negative.
5. d. (negative) $-$ (negative) can be positive or negative depending on the magnitudes.
6. e. (negative) $-$ (positive) is always negative.
7. f. (negative) $+$ (positive) can be positive or negative depending on the magnitudes.

**Answer:** a. positive, b. can be positive or negative, c. negative, d. can be positive or negative, e. negative, f. can be positive or negative

> Common mistake: Confusing the rules for addition and subtraction of integers with unlike signs.

### Question 9

*3 marks · Short answer*

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

**Solution**

1. Observe the repeating pattern in the string of 100 tokens.
2. Determine the value of one repeating group of tokens.
3. Multiply the value of one group by the number of repetitions to find the total value of 20.

**Answer:** $20$

> Common mistake: Miscounting the number of tokens in the repeating pattern.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Rule 1: The sum of two positives means moving upwards twice in the Building of Fun, giving a positive floor.
2. Rule 2: The sum of two negatives means moving downwards twice, which results in a deeper negative floor.
3. Rule 3: A positive and a negative number cancel each other through zero pairs, leaving the sign of the greater floor.

**Answer:** Brahmagupta's rules can be understood using the lifts and floors of the Building of Fun.

> Common mistake: Confusing the sign of the result when adding a positive and a negative number.

### Question 2

*3 marks · Short answer*

Give your own examples of each rule.

**Solution**

1. Rule 1 example: $3 + 4 = 7$ (sum of two positive numbers).
2. Rule 2 example: $(-3) + (-4) = -7$ (sum of two negative numbers).
3. Rule 3 example: $-5 + 3 = -2$ (sum of a positive and a negative number).

**Answer:** Examples are $3+4=7$, $(-3)+(-4)=-7$, and $-5+3=-2$.

> Common mistake: Forgetting to place the minus sign when adding two negative numbers.

## Frequently asked questions

### How many total questions are there in NCERT Solutions for Class 6 Maths Chapter 10 The Other Side of Zero?

The chapter is structured across multiple practice sections known as Figure it Out and specific topic subsections in the new NCERT book for the 2026-27 session. It covers a comprehensive set of questions involving very short answer, short answer, long answer, activity, and case-based types. You can access SwaVid's free PDF and complete step-by-step solutions on this page only.

### Which major topics are covered in the Class 6 Maths Chapter 10 exercise questions?

The questions cover essential concepts like integers and floor movement, addition and subtraction using the Building of Fun, additive inverse, and comparing numbers on a vertical number line. Other topics include token models, banking and accounting applications, geographical heights and depths, negative temperatures, integer grids, and Brahmagupta's rules. SwaVid's free PDF and step-by-step solutions for all these topics are available on this page only.

### What are the hardest question types in this chapter and how should we approach them?

The more challenging questions involve case-based problems, complex integer grid border sums, cross-grid number elimination games, and calculating past and future years accounting for no year $0$. To approach them, break down the movement or mathematical rules carefully, such as using number lines or integer tokens step by step. SwaVid provides detailed explanations for these difficult questions in the free PDF and step-by-step solutions on this page only.

### How do I write answers to score full marks in Class 6 Maths Chapter 10 exams?

To score full marks, clearly show your working steps, especially when dealing with integer addition, subtraction, or finding missing movements to reach a target floor. Mention whether you are moving above or below zero and properly account for signs during calculations. You can study the ideal answer formats from SwaVid's free PDF and step-by-step solutions available on this page only.

### Is the free PDF for Chapter 10 The Other Side of Zero aligned with the new NCERT book?

Yes, all solutions are strictly based on the new NCERT book following the NCF 2023 curriculum for the 2026-27 academic session. It thoroughly covers every activity and exercise question, including Bela's Building of Fun and historical rules by Brahmagupta. You can download SwaVid's free PDF and view step-by-step solutions directly on this page only.

## Related pages

- [Class 6 Maths chapters](https://www.swavid.com/maths/class/6)

Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student