Q3: Connect the inverses by drawing lines.

To Do: Match each number with its additive inverse from the given set. Solution: Based on the previous question, we can pair the numbers with their inverses: +4 is the inverse of -4. +3 is the inverse of -3. +2 is the inverse of -2. +1 is the inverse of -1. 0 is the inverse of 0. Final Answer: The…

The Other Side Of Zero Class 6 Mathematics NCERT Solutions

Q1Chapter 10 Questions

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

Solution

Given: Movement of four floors up. Movement of three floors down.

To Find: The button presses required for each movement.

Solution: Based on the text, the '+' button is used to go up and the '-' button is used to go down.

To go four floors up, you must press the '+' button four times. This can be written as ++++ or +4.
To go three floors down, you must press the '-' button three times. This can be written as --- or -3.

Final Answer: To go four floors up, you press +4. To go three floors down, you press -3.

Q2Chapter 10 Questions

Number all the floors in the Building of Fun.

Solution

Given: The 'Building of Fun' layout with various attractions on different floors.

To Find: The floor number for each location.

Solution: Based on the descriptions and diagrams in the text:

The ground floor is the 'Welcome Hall' and is designated as Floor 0.
Floors above the ground are positive, and floors below are negative.
Book Store: The text mentions reaching the Book Store by the expression (+1)+(+2)=+3. So, the Book Store is on Floor +3.
Art Centre: Reached by pressing +2 from the ground floor. So, the Art Centre is on Floor +2.
Food Court: Reached by pressing +1 from the ground floor. So, the Food Court is on Floor +1.
Welcome Hall: This is the ground floor. So, the Welcome Hall is on Floor 0.
Toy Store: Reached by pressing -1 from the ground floor. So, the Toy Store is on Floor -1.
Video Games shop: Reached by pressing -2 from the ground floor. So, the Video Games shop is on Floor -2.

Final Answer:

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

Q1Chapter 10 Questions (Page 248)

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

Solution

To Do: Evaluate the subtractions using the 'finding the missing number to be added' perspective.

Concept: The subtraction a - b = ? is equivalent to the addition b + ? = a.

Solution:

15 - 5: This can be expressed as 5 + ? = 15. We need to find what number to add to 5 to get 15. The answer is 10. So, 15 - 5 = 10.
100 - 10: This can be expressed as 10 + ? = 100. We need to find what number to add to 10 to get 100. The answer is 90. So, 100 - 10 = 90.
74 - 34: This can be expressed as 34 + ? = 74. We need to find what number to add to 34 to get 74. The answer is 40. So, 74 - 34 = 40.

Final Answer:

15 - 5 = 10 because 5 + 10 = 15.
100 - 10 = 90 because 10 + 90 = 100.
74 - 34 = 40 because 34 + 40 = 74.

Q1Chapter 10 Questions (Page 252)

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?

Solution

Answer: Yes. For example, in the exercises on page 249:

(0) - (-2) was calculated as +2, which is the same as 0 + (+2).
(+4) - (-3) was calculated as +7, which is the same as (+4) + (+3). This shows that subtracting a negative number is equivalent to adding its positive counterpart.

Q2Chapter 10 Questions (Page 252)

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

Solution

Given: Starting Number = 5 Target Number = 9

To Find: The movement needed.

Formula: Movement = Target Number - Starting Number

Solution: Movement = 9 - 5 = 4. This means you must travel 4 steps in the positive direction (to the right).

Final Answer: You must travel 4 steps.

Q3Chapter 10 Questions (Page 252)

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

Solution

Given: Starting Number = 9 Target Number = 3

To Find: The movement needed.

Formula: Movement = Target Number - Starting Number

Solution: Movement = 3 - 9 = -6. This means you must travel 6 steps in the negative direction (backward).

Final Answer: You must move 6 steps backward, or move -6.

Q4Chapter 10 Questions (Page 252)

Now, from 3, if you wish to go to -2, how far must you travel?

Solution

Given: Starting Number = 3 Target Number = -2

To Find: The movement needed.

Formula: Movement = Target Number - Starting Number

Solution: Movement = -2 - 3 = -5. This means you must travel 5 steps in the negative direction (backward).

Final Answer: You must travel 5 steps backward, or move -5.

Q1Chapter 10 Questions (Page 255)

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

Solution

To Find: The value of each expression, visualized using an unmarked number line (UNL).

Solution:

a. -125 + (-30)

On a UNL, start at -125.
+(-30) means moving 30 units to the left (negative direction).
Moving 30 units left from -125 lands you on -155.
So, -125 + (-30) = -155.

b. +105 - (-55)

This is a subtraction problem, which can be thought of as finding the movement from -55 to +105.
To go from -55 to 0 is a movement of +55.
To go from 0 to +105 is a movement of +105.
Total movement = 55 + 105 = 160.
Alternatively, +105 - (-55) = 105 + 55 = 160.
So, +105 - (-55) = 160.

c. +80 - (-150)

This is the movement from -150 to +80.
To go from -150 to 0 is a movement of +150.
To go from 0 to +80 is a movement of +80.
Total movement = 150 + 80 = 230.
Alternatively, +80 - (-150) = 80 + 150 = 230.
So, +80 - (-150) = 230.

d. -99 - (-200)

This is the movement from -200 to -99.
To go from -200 to -99 is a movement in the positive direction.
The distance is 200 - 99 = 101.
Alternatively, -99 - (-200) = -99 + 200 = 101.
So, -99 - (-200) = 101.

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

Q1Chapter 10 Questions (Pages 259-260)

Your new bank balance is _____.

Solution

Given: Starting balance = ₹100 Deposit (credit) = ₹60

Solution: New Balance = Starting Balance + Deposit New Balance = 100 + 60 = 160

Final Answer: Your new bank balance is ₹160.

Q2Chapter 10 Questions (Pages 259-260)

Your bank balance is now _____.

Solution

Given: Previous balance = ₹160 Payment (debit) = ₹30

Solution: New Balance = Previous Balance - Payment New Balance = 160 - 30 = 130

Final Answer: Your bank balance is now ₹130.

Q3Chapter 10 Questions (Pages 259-260)

What is your bank balance now? _____ Is this possible?

Solution

Given: Previous balance = ₹130 Purchase (debit) = ₹150

Solution: New Balance = Previous Balance - Purchase New Balance = 130 - 150 = -20

Yes, this is possible. Some bank accounts allow for overdrafts, where the balance can become negative. This usually incurs fees or interest.

Final Answer: Your bank balance is now -₹20. Yes, this is possible.

Q4Chapter 10 Questions (Pages 259-260)

What is your balance now？ _____

Solution

Given: Previous balance = -₹20 Earning (credit) = ₹200

Solution: New Balance = Previous Balance + Earning New Balance = -20 + 200 = 180

Final Answer: Your balance now is ₹180.

Q1Figure it Out (Page 245)

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

Solution

Given: Starting Floor = +2 Movement = -3 (pressing the '-' button three times)

To Find: The target floor and the expression for the movement.

Formula: Starting Floor + Movement = Target Floor

Solution: We start at Floor +2 and move down 3 floors. The expression is (+2) + (-3). Calculating the result: 2 - 3 = -1. Floor -1 is the Toy Store.

Final Answer: You will reach Floor -1, the Toy Store. The expression is (+2) + (-3) = -1.

Q2Figure it Out (Page 245)

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

To Find: The value of the given expressions.

Solution: We can solve these by performing integer addition.

a. (+1) + (+4) = 1 + 4 = 5 or +5 b. (+4) + (+1) = 4 + 1 = 5 or +5 c. (+4) + (-3) = 4 - 3 = 1 or +1 d. (-1) + (+2) = -1 + 2 = 1 or +1 e. (-1) + (+1) = -1 + 1 = 0 f. 0 + (+2) = 0 + 2 = 2 or +2 g. 0 + (-2) = 0 - 2 = -2

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

Q3Figure it Out (Page 245)

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

Given: Target Floor = -5

To Find: Different combinations of Starting Floor and Movement that result in the Target Floor -5.

Formula: Starting Floor + Movement = Target Floor

Solution: We need to find pairs of numbers that add up to -5. Here are a few examples:

If the starting floor is 0 (Welcome Hall): 0 + ? = -5 The movement required is -5. Expression: (0) + (-5) = -5
If the starting floor is +1 (Food Court): +1 + ? = -5 The movement required is -6. Expression: (+1) + (-6) = -5
If the starting floor is -2 (Video Games shop): -2 + ? = -5 The movement required is -3. Expression: (-2) + (-3) = -5
If the starting floor is +4: +4 + ? = -5 The movement required is -9. Expression: (+4) + (-9) = -5

Final Answer: Here are some possible expressions:

(0) + (-5) = -5
(+1) + (-6) = -5
(-2) + (-3) = -5

Q1Figure it out (Page 246)

Evaluate these expressions by thinking of them as the resulting movement of combining button presses: a. $(+1)+(+4)=$ b. $(+4)+(+1)=$ c. $(+4)+(-3)+(-2)=$ d. $(-1)+(+2)+(-3)=$

Solution

To Find: The value of the given expressions, representing combined movements.

Solution: We perform the integer addition for each expression.

a. (+1) + (+4) = 1 + 4 = 5 or +5 (move up 1, then up 4, total up 5) b. (+4) + (+1) = 4 + 1 = 5 or +5 (move up 4, then up 1, total up 5) c. (+4) + (-3) + (-2) = 4 - 3 - 2 = 1 - 2 = -1 (move up 4, down 3, down 2, total down 1) d. (-1) + (+2) + (-3) = -1 + 2 - 3 = 1 - 3 = -2 (move down 1, up 2, down 3, total down 2)

Final Answer: a. +5 b. +5 c. -1 d. -2

Q2Figure it out (Page 246)

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

Solution

Given: A set of integers: +4, -4, -3, 0, +2, -1.

To Find: The additive inverse of each number.

Concept: The additive inverse of a number is the number that, when added to it, results in zero. For any non-zero number a, its inverse is -a. The inverse of 0 is 0.

Solution:

The inverse of +4 is -4, because (+4) + (-4) = 0.
The inverse of -4 is +4, because (-4) + (+4) = 0.
The inverse of -3 is +3, because (-3) + (+3) = 0.
The inverse of 0 is 0, because 0 + 0 = 0.
The inverse of +2 is -2, because (+2) + (-2) = 0.
The inverse of -1 is +1, because (-1) + (+1) = 0.

Final Answer:

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
Inverse of -1 is +1

Q3Figure it out (Page 246)

Connect the inverses by drawing lines.

Solution

To Do: Match each number with its additive inverse from the given set.

Solution: Based on the previous question, we can pair the numbers with their inverses:

+4 is the inverse of -4.
+3 is the inverse of -3.
+2 is the inverse of -2.
+1 is the inverse of -1.
0 is the inverse of 0.

Final Answer: The pairs of inverses are:

(+4, -4)
(+3, -3)
(+2, -2)
(+1, -1)
(0, 0)

Q4Figure it out (Page 246)

Who is on the lowest floor? Jay is in the Art Centre. So, he is on Floor +2. Asin is in the Sports Centre. So, she is on Floor ___. Binnu is in the Cinema Centre. So, she is on Floor _. Aman is in the Toys Store. So, he is on Floor ___.

Solution

To Find: The floor number for Asin, Binnu, and Aman, and then determine who is on the lowest floor.

Solution: First, we identify the floor number for each person based on the building diagram provided in the chapter.

Jay is in the Art Centre, which is on Floor +2.
Asin is in the Sports Centre, which is on Floor +5.
Binnu is in the Cinema Centre, which is on Floor -3.
Aman is in the Toy Store, which is on Floor -1.

Now, we compare the floor numbers: +2, +5, -3, -1. On a number line, the number furthest to the left is the smallest (lowest). Comparing the numbers: -3 < -1 < +2 < +5. The lowest floor number is -3. Binnu is on Floor -3.

Final Answer: Asin is on Floor +5. Binnu is on Floor -3. Aman is on Floor -1.

Binnu is on the lowest floor.

Q1Figure it Out (Page 247)

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

Solution

To Do: Compare the pairs of integers and use the correct inequality symbol (< for less than, > for greater than).

Concept: A number is greater than another if it is on a higher floor (or further to the right on a number line).

Solution: a. Floor -2 is below Floor +5. So, -2 < +5. b. Floor -5 is below Floor +4. So, -5 < +4. c. Floor -5 is below Floor -3. So, -5 < -3. d. Floor +6 is above Floor -6. So, +6 > -6. e. Floor 0 (ground floor) is above Floor -4. So, 0 > -4. f. Floor 0 (ground floor) is below Floor +4. So, 0 < +4.

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

Q2Figure it Out (Page 247)

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

Solution

To Do: Compare the pairs of integers and use the correct inequality symbol.

Solution: a. -10 is greater than -12 (Floor -10 is above Floor -12). So, -10 > -12. b. Any positive number is greater than any negative number. So, +17 > -10. c. 0 is greater than any negative number. So, 0 > -20. d. Any positive number is greater than any negative number. So, +9 > -9. e. -25 is less than -7 (Floor -25 is below Floor -7). So, -25 < -7. f. Any positive number is greater than any negative number. So, +15 > -17.

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

Q3Figure it Out (Page 247)

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

Given: A vertical number line representing floors. A = -12, D = -1, E = +1. The markings are evenly spaced.

To Find: The floor numbers for B, C, F, G, and H.

Solution: First, let's establish the scale. The value changes from -1 (Floor D) to +1 (Floor E). The diagram shows two markings between D and E. So the change in value is +1 - (-1) = 2. This means each marking represents 1 unit.

Now we can find the other values by counting from a known point.

Floor B: Floor A is at -12. Floor B is 3 markings above A. So, B = -12 + 3 = -9.
Floor C: Floor B is at -9. Floor C is 3 markings above B. So, C = -9 + 3 = -6.
Floor F: Floor E is at +1. Floor F is 1 marking above E. So, F = +1 + 1 = +2.
Floor G: Floor F is at +2. Floor G is 4 markings above F. So, G = +2 + 4 = +6.
Floor H: Floor G is at +6. Floor H is 5 markings above G. So, H = +6 + 5 = +11.

Final Answer:

B = -9
C = -6
F = +2
G = +6
H = +11

Q4Figure it Out (Page 247)

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

Solution

To Do: Describe the position of the given floor numbers on the vertical number line relative to the marked points (A, B, C, etc.).

Solution: Using the values found in the previous question (A=-12, B=-9, C=-6, D=-1, E=+1, F=+2, G=+6, H=+11):

a. -7: This is between Floor C (-6) and Floor B (-9). It is one marking below C. b. -4: This is between Floor C (-6) and Floor D (-1). It is two markings above C. c. +3: This is between Floor F (+2) and Floor G (+6). It is one marking above F. d. -10: This is between Floor B (-9) and Floor A (-12). It is one marking below B.

Final Answer: The locations are: a. -7 is one floor below Floor C. b. -4 is two floors above Floor C. c. +3 is one floor above Floor F. d. -10 is one floor below Floor B.

Q1Figure it Out (Page 249)

Complete these expressions. You may think of them as finding the movement needed to reach the Target Floor from the Starting Floor. a. $(+1)-(+4)=$ b. $(0)-(+2)=$ c. $(+4)-(+1)=$ d. $(0)-(-2)=$ e. $(+4)-(-3)=$ f. $(-4)-(-3)=$ g. $(-1)-(+2)=$ h. $(-2)-(-2)=$ i. $(-1)-(+1)=$ j. $(+3)-(-3)=

Solution

To Find: The value of the given subtraction expressions.

Formula: Target Floor - Starting Floor = Movement needed

Solution: a. (+1) - (+4) = 1 - 4 = -3 (To go from floor 4 to floor 1, move down 3 floors) b. (0) - (+2) = 0 - 2 = -2 (To go from floor 2 to floor 0, move down 2 floors) c. (+4) - (+1) = 4 - 1 = +3 (To go from floor 1 to floor 4, move up 3 floors) d. (0) - (-2) = 0 + 2 = +2 (To go from floor -2 to floor 0, move up 2 floors) e. (+4) - (-3) = 4 + 3 = +7 (To go from floor -3 to floor 4, move up 7 floors) f. (-4) - (-3) = -4 + 3 = -1 (To go from floor -3 to floor -4, move down 1 floor) g. (-1) - (+2) = -1 - 2 = -3 (To go from floor 2 to floor -1, move down 3 floors) h. (-2) - (-2) = -2 + 2 = 0 (To go from floor -2 to floor -2, move 0 floors) i. (-1) - (+1) = -1 - 1 = -2 (To go from floor 1 to floor -1, move down 2 floors) j. (+3) - (-3) = 3 + 3 = +6 (To go from floor -3 to floor 3, move up 6 floors)

Final Answer: a. -3 b. -2 c. +3 d. +2 e. +7 f. -1 g. -3 h. 0 i. -2 j. +6

Q1Figure it out (Page 251)

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

Solution

To Find: The missing numbers and the values of the expressions.

Solution: a. (+40) + ? = +200. We need to find the movement from +40 to +200. 200 - 40 = 160. So, ? = +160. b. (+40) + ? = -200. We need to find the movement from +40 to -200. -200 - 40 = -240. So, ? = -240. c. (-50) + ? = +200. We need to find the movement from -50 to +200. 200 - (-50) = 200 + 50 = 250. So, ? = +250. d. (-50) + ? = -200. We need to find the movement from -50 to -200. -200 - (-50) = -200 + 50 = -150. So, ? = -150. e. (-200) - (-40) = -200 + 40 = -160. (Movement from -40 to -200 is down 160). f. (+200) - (+40) = 200 - 40 = +160. (Movement from +40 to +200 is up 160). g. (-200) - (+40) = -200 - 40 = -240. (Movement from +40 to -200 is down 240).

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

Q1Figure it Out (Page 253)

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

Solution

To Do: Provide examples of 3 positive and 3 negative numbers.

Solution: Here are some examples:

Three positive numbers: 2, 5, 8
Three negative numbers: -1, -3, -7

On a number line, 2, 5, 8 would be to the right of 0, and -1, -3, -7 would be to the left of 0.

Final Answer: Positive numbers: 2, 5, 8 Negative numbers: -1, -3, -7 (Other answers are possible).

Q2Figure it Out (Page 253)

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

Solution

Solution: Using the example numbers from the previous question, the three negative numbers are: -7, -3, -1

Final Answer: -7, -3, -1

Q3Figure it Out (Page 253)

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

Solution

To Answer: Determine if the inequalities are true and provide the reason.

Solution:

Is $2 > -3$ ? Yes, this is true. On a number line, numbers to the right are greater. Since 2 is to the right of -3, 2 is greater than -3. Also, any positive number is greater than any negative number.
Is $-2 < 3$ ? Yes, this is true. On a number line, numbers to the left are smaller. Since -2 is to the left of 3, -2 is less than 3. Also, any negative number is less than any positive number.

Final Answer:

Yes, 2 > -3 because 2 lies to the right of -3 on the number line.
Yes, -2 < 3 because -2 lies to the left of 3 on the number line.

Q4Figure it Out (Page 253)

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

Solution

To Find: The value of each expression.

Solution: a. -5 + 0 = -5 (Additive property of zero) b. 7 + (-7) = 0 (A number plus its additive inverse is zero) c. -10 + 20 = 10 d. 10 - 20 = -10 e. 7 - (-7) = 7 + 7 = 14 (Subtracting a negative is the same as adding a positive) f. -8 - (-10) = -8 + 10 = 2

Final Answer: a. -5 b. 0 c. 10 d. -10 e. 14 f. 2

Q1Figure it Out (Page 257)

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

Solution

To Find: The sum of the expressions using the token model.

Solution: a. (+6) + (+4)

Combine 6 positive tokens and 4 positive tokens.
The result is 10 positive tokens.
(+6) + (+4) = +10.

b. (-3) + (-2)

Combine 3 negative tokens and 2 negative tokens.
The result is 5 negative tokens.
(-3) + (-2) = -5.

c. (+5) + (-7)

Combine 5 positive tokens and 7 negative tokens.
Form 5 'zero pairs' (one positive + one negative).
After removing the 5 zero pairs, 2 negative tokens are left.
(+5) + (-7) = -2.

d. (-2) + (+6)

Combine 2 negative tokens and 6 positive tokens.
Form 2 'zero pairs'.
After removing the 2 zero pairs, 4 positive tokens are left.
(-2) + (+6) = +4.

Final Answer: a. +10 b. -5 c. -2 d. +4

Q2Figure it Out (Page 257)

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?

Solution

To Do: Analyze two sets of tokens, find the net value, determine the floor, and write the addition statement.

Solution:

Set a:

Tokens: The image shows 3 positive (red) tokens and 5 negative (green) tokens.
Zero Pairs: We can form 3 zero pairs (3 positive + 3 negative).
Remaining Tokens: After cancelling the zero pairs, 2 negative tokens remain.
Floor: The value is -2, so the lift attendant is on Floor -2.
Addition Statement: The initial set of tokens represents the sum of 3 positives and 5 negatives. The statement is (+3) + (-5) = -2.

Set b:

Tokens: The image shows 6 positive (red) tokens and 3 negative (green) tokens.
Zero Pairs: We can form 3 zero pairs (3 positive + 3 negative).
Remaining Tokens: After cancelling the zero pairs, 3 positive tokens remain.
Floor: The value is +3, so the lift attendant is on Floor +3.
Addition Statement: The initial set of tokens represents the sum of 6 positives and 3 negatives. The statement is (+6) + (-3) = +3.

Final Answer:

Case a: The attendant is on Floor -2. The addition statement is (+3) + (-5) = -2.
Case b: The attendant is on Floor +3. The addition statement is (+6) + (-3) = +3.

Q1Figure it Out (Page 258)

Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: a. $(+10)-(+7)$ b. $(-8)-(-4)$ c. $(-9)-(-4)$ d. $(+9)-(+12)$ e. $(-5)-(-7)$ f. $(-2)-(-6)$

Solution

To Find: The value of each subtraction expression using the token model.

Solution: a. (+10) - (+7): Start with 10 positive tokens. Take away 7 positive tokens. 3 positive tokens remain. Result: +3. b. (-8) - (-4): Start with 8 negative tokens. Take away 4 negative tokens. 4 negative tokens remain. Result: -4. c. (-9) - (-4): Start with 9 negative tokens. Take away 4 negative tokens. 5 negative tokens remain. Result: -5. d. (+9) - (+12): Start with 9 positive tokens. We need to take away 12. Add 3 zero pairs (3 positive, 3 negative). Now we have 12 positive and 3 negative tokens. Take away 12 positive tokens. 3 negative tokens remain. Result: -3. e. (-5) - (-7): Start with 5 negative tokens. We need to take away 7. Add 2 zero pairs (2 positive, 2 negative). Now we have 7 negative and 2 positive tokens. Take away 7 negative tokens. 2 positive tokens remain. Result: +2. f. (-2) - (-6): Start with 2 negative tokens. We need to take away 6. Add 4 zero pairs (4 positive, 4 negative). Now we have 6 negative and 4 positive tokens. Take away 6 negative tokens. 4 positive tokens remain. Result: +4.

Checking with other methods (converting to addition): a. (+10) + (-7) = +3 ✓ b. (-8) + (+4) = -4 ✓ c. (-9) + (+4) = -5 ✓ d. (+9) + (-12) = -3 ✓ e. (-5) + (+7) = +2 ✓ f. (-2) + (+6) = +4 ✓

Final Answer: a. +3 b. -4 c. -5 d. -3 e. +2 f. +4

Q2Figure it Out (Page 258)

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

Solution

To Find: The value of each subtraction expression.

Method: Convert subtraction to addition of the inverse: a - b = a + (-b).

Solution: 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

Final Answer: a. +2 b. -3 c. +2 d. -5 e. +5 f. -12

Q1Figure it out (Page 259)

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

Solution

To Find: The value of -3 - (+5) using the token model.

Solution:

Start: We begin with 3 negative tokens to represent -3.
Goal: We need to take away 5 positive tokens.
Problem: We have no positive tokens to take away.
Action: To create positive tokens without changing the value, we add 'zero pairs'. Since we need to remove 5 positive tokens, we must add 5 zero pairs (5 positive and 5 negative tokens).
New State: Our collection now has the original 3 negative tokens plus the 5 positive and 5 negative tokens from the zero pairs. Total: 8 negative tokens and 5 positive tokens. The value is still -3.
Subtract: Now we can take away the 5 positive tokens.
Result: We are left with 8 negative tokens.

Therefore, -3 - (+5) = -8.

Final Answer: You will have to put in 5 zero pairs. The result is -8.

Q2Figure it out (Page 259)

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

Solution

To Find: The value of each subtraction expression.

Method: For each problem, we convert the subtraction into an addition problem by adding the inverse of the second number. a - b = a + (-b) and a - (-b) = a + b.

Solution: a. (-3) - (+10) = -3 - 10 = -13 b. (+8) - (-7) = 8 + 7 = 15 c. (-5) - (+9) = -5 - 9 = -14 d. (-9) - (+10) = -9 - 10 = -19 e. (+6) - (-4) = 6 + 4 = 10 f. (-2) - (+7) = -2 - 7 = -9

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

Q1Figure it Out (Page 260)

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

Given: Starting Balance = ₹0 Credits: +₹30, +₹40, +₹50 Debits: -₹40, -₹50, -₹60

To Find: The final bank account balance.

Solution: We add all the credits and debits to the starting balance. Balance = 0 + (+30) + (+40) + (+50) + (-40) + (-50) + (-60) Combine the positive and negative numbers: Total Credits = 30 + 40 + 50 = 120 Total Debits = -40 - 50 - 60 = -150 Final Balance = 120 - 150 = -30

Final Answer: Your bank account balance is now -₹30.

Q2Figure it Out (Page 260)

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

Given: Starting Balance = ₹0 Debits: -₹1, -₹2, -₹4, -₹8, -₹16, -₹32, -₹64, -₹128 Credit: +₹256

To Find: The final bank account balance.

Solution: First, let's find the sum of all the debits. Total Debits = -(1 + 2 + 4 + 8 + 16 + 32 + 64 + 128) This is a geometric series. The sum is 2^8 - 1 = 256 - 1 = 255. So, Total Debits = -255.

Now, calculate the final balance: Final Balance = Starting Balance + Total Debits + Total Credits Final Balance = 0 + (-255) + (+256) Final Balance = 256 - 255 = 1

Final Answer: Your bank account balance is now ₹1.

Q3Figure it Out (Page 260)

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

Answer:

Why a positive balance is better:

Avoiding Fees: Banks often charge overdraft fees or penalties when an account balance goes negative.
Avoiding Interest: A negative balance is essentially a loan from the bank, which accrues interest, making you pay back more than you used.
Financial Health: A consistent positive balance indicates good financial management and stability.
Access to Funds: You always have your own money available for expenses without needing to borrow.

When a negative balance might be worthwhile:

Strategic Investment: A person might temporarily overdraw their account to make a large purchase or investment that is expected to generate a profit greater than the fees and interest incurred.
Emergency Situations: In an unexpected emergency, it might be necessary to spend more money than is in the account, making a temporary negative balance a necessary choice.
Bridging a Gap: If a large payment is expected to arrive very soon, one might go into a negative balance for a short period to cover an urgent expense, knowing the funds to cover it are on the way.

Q1Figure it Out (Page 261)

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

Solution

To Do: Estimate the heights of points A through G from the provided diagram relative to sea level (0 m).

Solution: By observing the vertical scale in the diagram:

A: is at the level marked +1500 m.
B: is at the level marked -500 m.
C: is slightly above the 0 m line, approximately at +300 m.
D: is below -1000 m, approximately at -1200 m.
E: is above +1000 m, approximately at +1200 m.
F: is slightly below the 0 m line, approximately at -200 m.
G: is slightly above the 0 m line, approximately at +100 m.

Final Answer: a. A: +1500 m b. B: -500 m c. C: +300 m d. D: -1200 m e. E: +1200 m f. F: -200 m g. G: +100 m

Q2Figure it Out (Page 261)

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

Solution

Given: The heights of points A through G.

To Find: The highest and lowest points.

Solution: The heights are: A(+1500), B(-500), C(+300), D(-1200), E(+1200), F(-200), G(+100).

The highest point is the one with the largest positive value, which is +1500 m (Point A).
The lowest point is the one with the most negative value (largest magnitude negative number), which is -1200 m (Point D).

Final Answer: The highest point is A. The lowest point is D.

Q3Figure it Out (Page 261)

Can you write the points A, B, ..., G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?

Solution

Given: The heights of points A through G: A(+1500), E(+1200), C(+300), G(+100), F(-200), B(-500), D(-1200).

To Do: Arrange the points in decreasing and increasing order of height.

Solution:

Decreasing Order (highest to lowest): We arrange the heights from the largest positive to the most negative. +1500 > +1200 > +300 > +100 > -200 > -500 > -1200 The sequence of points is: A, E, C, G, F, B, D.
Increasing Order (lowest to highest): We arrange the heights from the most negative to the largest positive. -1200 < -500 < -200 < +100 < +300 < +1200 < +1500 The sequence of points is: D, B, F, G, C, E, A.

Final Answer:

Decreasing order: A, E, C, G, F, B, D
Increasing order: D, B, F, G, C, E, A

Q4Figure it Out (Page 261)

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

Solution

Answer: The highest point above sea level on Earth is the summit of Mount Everest. Its height is approximately 8,848 meters above sea level.

Q5Figure it Out (Page 261)

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

Answer: The lowest known point on Earth is the Challenger Deep, located in the Mariana Trench in the western Pacific Ocean. Its depth is approximately -10,994 meters (or about 11 kilometers below sea level).

Q1Figure it Out (Page 262)

Do you know that there are some places in India where temperatures can go below $0^\circ C$ ? Find out the places in India where temperatures sometimes go below $0^\circ C$ . What is common among these places? Why does it become colder there and not in other places?

Solution

Answer:

Yes, there are several places in India where temperatures can go below

0^\circ C

, especially during the winter.

Examples of such places:

Dras, Jammu and Kashmir (often cited as one of the coldest inhabited places in India)
Leh and other parts of Ladakh
Srinagar, Gulmarg, and Pahalgam in Jammu and Kashmir
Shimla, Manali, and Spiti Valley in Himachal Pradesh
Auli and Munsiyari in Uttarakhand
Tawang in Arunachal Pradesh

What is common among these places?

High Altitude: All these places are located at high altitudes in the Himalayan mountain range.

Why does it become colder there?

Altitude Effect: As altitude increases, the air becomes thinner and less dense. Thinner air cannot hold heat as effectively as denser air at sea level, causing temperatures to drop significantly, especially at night.
Proximity to Snow: These regions receive heavy snowfall in winter. The snow and ice cover reflects sunlight and keeps the ground and surrounding air very cold.
Geographical Location: Their location in the northern latitudes of India also contributes to colder winters.

Q2Figure it Out (Page 262)

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. Temperature: $14^\circ C, 8^\circ C, -2^\circ C, -4^\circ C$ Time: 02:00 a.m., 11:00 p.m., 02:00 p.m., 11:00 a.m.

Solution

To Do: Match each temperature to the most likely time of day based on a typical daily temperature cycle.

Reasoning:

The temperature is usually highest in the early afternoon.
The temperature is usually lowest in the early morning, just before sunrise.
The temperature drops through the evening and night.
The temperature rises through the morning.

Matching:

The highest temperature is $14^\circ C$ . This would most likely occur in the early afternoon, at 02:00 p.m..
The next warmest temperature is $8^\circ C$ . This would likely be in the late morning, at 11:00 a.m..
The colder of the two negative temperatures is $-4^\circ C$ . This would be the coldest time, in the early morning at 02:00 a.m..
The remaining temperature is $-2^\circ C$ , which would fit the late evening time of 11:00 p.m., as it has cooled down but is not yet at its coldest point.

Final Answer:

$14^\circ C$ → 02:00 p.m.
$8^\circ C$ → 11:00 a.m.
$-2^\circ C$ → 11:00 p.m.
$-4^\circ C$ → 02:00 a.m.

Q1Figure it Out (Page 263)

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

Solution

Given: The grid: | 5 | -3 | -5 | | 0 | | -5 | |-8 | -2 | 7 |

To Find: The sum of the top row, bottom row, left column, and right column, known as the 'border sum'.

Solution:

Top row sum: 5 + (-3) + (-5) = 5 - 3 - 5 = 2 - 5 = -3
Bottom row sum: (-8) + (-2) + 7 = -10 + 7 = -3
Left column sum: 5 + 0 + (-8) = 5 - 8 = -3
Right column sum: (-5) + (-5) + 7 = -10 + 7 = -3

All the sums are equal to -3.

Final Answer: The border sum is -3.

Q2Figure it Out (Page 263)

Complete the grids to make the required border sum.

Solution

To Do: Fill in the blank cells of each grid so that the top row, bottom row, left column, and right column each add up to the specified 'border sum'. Note that multiple solutions may exist.

Solution:

Grid 1: Border sum is +4 | -10 | A | B | | C | | -5 | | 9 | D | E |

Top row: -10 + A + B = 4
Bottom row: 9 + D + E = 4
Left column: -10 + C + 9 = 4 => C - 1 = 4 => C = 5
Right column: B - 5 + E = 4 Let's pick a value. Let A = 10. Then -10 + 10 + B = 4 => B = 4. Let D = -10. Then 9 - 10 + E = 4 => -1 + E = 4 => E = 5. Check right column: B - 5 + E = 4 - 5 + 5 = 4. It works.

Grid 2: Border sum is -2 | 6 | 8 | A | | B | | -5| | C | -2| D |

Top row: 6 + 8 + A = -2 => 14 + A = -2 => A = -16
Left column: 6 + B + C = -2
Bottom row: C - 2 + D = -2
Right column: A - 5 + D = -2 => -16 - 5 + D = -2 => -21 + D = -2 => D = 19 Now use D in bottom row: C - 2 + 19 = -2 => C + 17 = -2 => C = -19 Now use C in left column: 6 + B - 19 = -2 => B - 13 = -2 => B = 11

Grid 3: Border sum is -4 | 7 | A | B | | C | | -5| | D | E | F |

Top row: 7 + A + B = -4
Left column: 7 + C + D = -4
Bottom row: D + E + F = -4
Right column: B - 5 + F = -4 Let A = -2. Then 7 - 2 + B = -4 => 5 + B = -4 => B = -9. Let C = -3. Then 7 - 3 + D = -4 => 4 + D = -4 => D = -8. Use D in bottom row: -8 + E + F = -4. Use B in right column: -9 - 5 + F = -4 => -14 + F = -4 => F = 10. Use F in bottom row: -8 + E + 10 = -4 => E + 2 = -4 => E = -6.

Final Answer: One possible solution for each grid:

Border sum +4 | -10 | 10 | 4 | | 5 | | -5| | 9 | -10| 5 |

Border sum -2 | 6 | 8 | -16 | | 11| | -5 | | -19|-2 | 19 |

Border sum -4 | 7 | -2 | -9 | | -3| | -5 | | -8| -6 | 10 |

Q3Figure it Out (Page 263)

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

Solution

To Find: Another valid solution for the third grid with a border sum of -4.

Solution: Let's try different initial choices for the blank cells. Original grid: | 7 | A | B | | C | | -5| | D | E | F |

Let's choose A = 0. Then 7 + 0 + B = -4 => B = -11. Let's choose C = 0. Then 7 + 0 + D = -4 => D = -11. Now use the right column: B - 5 + F = -4 => -11 - 5 + F = -4 => -16 + F = -4 => F = 12. Now use the bottom row: D + E + F = -4 => -11 + E + 12 = -4 => E + 1 = -4 => E = -5.

This gives a new valid solution.

Final Answer: Here are two possible ways to fill the grid:

Solution 1: | 7 | -2 | -9 | | -3| | -5 | | -8| -6 | 10 |

Solution 2: | 7 | 0 | -11 | | 0 | | -5 | |-11| -5 | 12 |

Q4Figure it Out (Page 263)

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

Solution

Answer:

Any grid where there is more than one empty cell that needs to be chosen to start the solving process can be filled in multiple ways.

In our case:

Grid 1 (Border sum +4): Could be filled in multiple ways.
Grid 2 (Border sum -2): This grid had only one possible solution because the values in the corners and sides created a system of equations that could be solved sequentially, leaving no free choices.
Grid 3 (Border sum -4): Could be filled in multiple ways.

Reason: The reason is the number of 'degrees of freedom'. If the number of unknown values is greater than the number of independent equations that constrain them, there will be multiple solutions. In these puzzles, we can often make an arbitrary choice for one blank cell, which then determines the values of the others. A different initial choice will lead to a different, but still valid, final grid.

Q5Figure it Out (Page 263)

Make a border integer square puzzle and challenge your classmates.

Solution

Answer: This is an activity for the student to create their own puzzle. Here is an example of a puzzle that could be created:

Puzzle: Complete the grid for a border sum of +10.

| 5 | | | | | | -3| |-2 | | |

Solution to the created puzzle: | 5 | 8 | -3 | | 9 | | -3 | |-2 | 5 | 7 |

Q1Figure it Out (Page 265 - Amazing Grid)

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

Given: The grid: | 3 | 4 | 0 | 9 | |-2 |-1 |-5 | 4 | | 1 | 2 |-2 | 7 | |-7 |-6 |-10|-1 |

To Do: Play the game with a different starting choice and find the sum.

Solution: Let's try a new path.

Circle 3 in the first row, first column. Strike out its row and column. Grid becomes: | | | | | | |-1 |-5 | 4 | | | 2 |-2 | 7 | | |-6 |-10|-1 |
Circle -1 from the unstruck numbers. Strike out its row and column. Grid becomes: | | | | | | | | | | | | |-2 | 7 | | | |-10|-1 |
Circle 7 from the unstruck numbers. Strike out its row and column. The only number left is -10.
Circle -10.
Add the circled numbers: 3 + (-1) + 7 + (-10) = 2 + 7 - 10 = 9 - 10 = -1.

The sum is -1, which is the same as the example in the book.

Final Answer: The sum I got is -1. It was not different from the first time.

Q2Figure it Out (Page 265 - Amazing Grid)

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

Solution

To Do: Find the constant sum for each of the two new grids.

Solution:

Grid 1: | 7 | 10 | 13 | 16 | |-2 | 1 | 4 | 7 | |-11|-8 |-5 |-2 | |-20|-7 |-14|-11|

Let's pick one number from each row and column. For example, the diagonal: 7, 1, -5, -11. Sum = 7 + 1 + (-5) + (-11) = 8 - 5 - 11 = 3 - 11 = -8. Let's try another combination: 10, -2, -5, -11. Oops, -5 and -11 are in the same column. Invalid choice. Let's try 10, -2, -14, -2. No, -2 is in the same column. Let's try 10 (row 1, col 2), -2 (row 2, col 1), -5 (row 3, col 3), -11 (row 4, col 4). This is not a valid selection as -11 and -5 are not in the last column. Let's try 10 (R1C2), -2 (R2C1), -5 (R3C3), -11 (R4C4)... this is not right. Let's re-read the rules. Circle one, strike out row/col. Ok. Let's pick: 7 (R1C1), 1 (R2C2), -5 (R3C3), -11 (R4C4) is not a valid play. Let's play correctly. Circle 7 (R1C1). Circle 1 (R2C2). Circle -5 (R3C3). Circle -11 (R4C4) is not possible. The last choice must be (R4C4). Let's re-play.

Circle 7 (R1C1). Unstruck numbers are in a 3x3 grid.
Circle 1 (R2C2). Unstruck numbers are in a 2x2 grid: (-5, -2) and (-14, -11).
Circle -5 (R3C3). Unstruck number is -11 (R4C4).
Circle -11. Sum = 7 + 1 + (-5) + (-11) = 8 - 16 = -8.

Grid 2: | -11 | -10 | -9 | -8 | | -7 | -6 | -5 | -4 | | -3 | -2 | -1 | 0 | | 1 | 2 | 3 | 4 |

Let's play:

Circle -10 (R1C2).
Circle -7 (R2C1).
Circle -1 (R3C3).
Circle 4 (R4C4). Sum = (-10) + (-7) + (-1) + 4 = -17 - 1 + 4 = -18 + 4 = -14.

Final Answer: For the first grid, the sum is -8. For the second grid, the sum is -14.

Q3Figure it Out (Page 265 - Amazing Grid)

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

Answer:

The magic is in both the numbers and their arrangement. These grids are constructed in a specific way.

How they are made: Each number in the grid is the sum of a 'row number' and a 'column number' that are hidden. For example, for a 4x4 grid, you can choose four row numbers (let's say r1, r2, r3, r4) and four column numbers (c1, c2, c3, c4). The number in row i and column j is r_i + c_j.

When you play the game, you pick exactly one number from each row and each column. So your final selection will be (r1 + c_a), (r2 + c_b), (r3 + c_c), (r4 + c_d), where a, b, c, d is some permutation of 1, 2, 3, 4.

The sum of your chosen numbers will be: (r1 + c_a) + (r2 + c_b) + (r3 + c_c) + (r4 + c_d) = (r1 + r2 + r3 + r4) + (c_a + c_b + c_c + c_d) Since a,b,c,d is just a reordering of 1,2,3,4, the sum of the column numbers is always the same: c1+c2+c3+c4.

So, the total sum is always (sum of all row numbers) + (sum of all column numbers), which is a constant, no matter which path you take.

Yes, you can make more such grids. Just pick a set of row numbers and a set of column numbers and add them to fill the grid.

Q1Figure it Out (Page 265 - Final Section)

Write all the integers between the given pairs, in increasing order. a. 0 and -7 b. -4 and 4 c. -8 and -15 d. -30 and -23

Solution

To Do: List the integers that lie strictly between the two given numbers, from smallest to largest.

Solution: a. 0 and -7: The integers are -6, -5, -4, -3, -2, -1. b. -4 and 4: The integers are -3, -2, -1, 0, 1, 2, 3. c. -8 and -15: The integers are -14, -13, -12, -11, -10, -9. d. -30 and -23: The integers are -29, -28, -27, -26, -25, -24.

Final Answer: 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

Q2Figure it Out (Page 265 - Final Section)

Give three numbers such that their sum is -8.

Solution

To Find: Any set of three integers that add up to -8.

Solution: There are infinite possibilities. Here are a few examples:

Example 1: -1 + (-2) + (-5) = -8
Example 2: 0 + (-10) + 2 = -8
Example 3: 5 + (-13) + 0 = -8

Final Answer: One possible set of numbers is -5, 7, -10.

Q3Figure it Out (Page 265 - Final Section)

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

Given: Two dice with faces: -1, 2, -3, 4, -5, 6.

To Find: The integer sums between -10 and 12 that cannot be obtained by rolling the two dice.

Solution: Let's list all possible sums. We can create an addition table:

+	-1	2	-3	4	-5	6
-1	-2	1	-4	3	-6	5
2	1	4	-1	6	-3	8
-3	-4	-1	-6	1	-8	3
4	3	6	1	8	-1	10
-5	-6	-3	-8	-1	-10	1
6	5	8	3	10	1	12

The possible sums are: -10, -8, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 10, 12.

Now let's list all integers from -10 to 12 and identify the missing ones:

-10 (Possible)
-9 (Not possible)
-8 (Possible)
-7 (Not possible)
-6 (Possible)
-5 (Not possible)
-4 (Possible)
-3 (Possible)
-2 (Possible)
-1 (Possible)
0 (Not possible)
1 (Possible)
2 (Not possible)
3 (Possible)
4 (Possible)
5 (Possible)
6 (Possible)
7 (Not possible)
8 (Possible)
9 (Not possible)
10 (Possible)
11 (Not possible)
12 (Possible)

Final Answer: The numbers that are not possible to get are: -9, -7, -5, 0, 2, 7, 9, 11.

Q4Figure it Out (Page 265 - Final Section)

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

Solution

To Do: Calculate the value of each expression in the grid.

Solution:

Top Row:

8 - 13 = -5
(-8) - (13) = -8 - 13 = -21
(-13) - (-8) = -13 + 8 = -5
(-13) + (-8) = -13 - 8 = -21

Bottom Row:

8 + (-13) = 8 - 13 = -5
(-8) - (-13) = -8 + 13 = 5
(13) - 8 = 5
13 - (-8) = 13 + 8 = 21

Final Answer: The completed grid is:

| -5 | -21 | -5 | -21 | | -5 | 5 | 5 | 21 |

Q5Figure it Out (Page 265 - Final Section)

Find the years below. a. From the present year, which year was it 150 years ago? b. From the present year, which year was it 2200 years ago? Hint: Recall that there was no year 0. c. What will be the year 320 years after 680 BCE?

Solution

Solution: (Assuming the present year is 2024. The answer will vary depending on the current year.)

a. 150 years ago: 2024 - 150 = 1874. It was the year 1874 CE.

b. 2200 years ago: 2024 - 2200 = -176. Since there is no year 0, the year before 1 CE was 1 BCE. So, year -1 corresponds to 2 BCE, -2 to 3 BCE, and so on. Year -176 corresponds to 177 BCE. It was the year 177 BCE.

c. 320 years after 680 BCE: We can represent 680 BCE as the year -679 (since 1 BCE is year 0 in this continuous model, but the hint says no year 0, so let's use integers where -1 is 1 BCE). Let's treat BCE as negative years. Year = -680. 320 years after means we add 320. -680 + 320 = -360. A result of -360 corresponds to the year 360 BCE.

Final Answer: a. 1874 (assuming the present is 2024) b. 177 BCE (assuming the present is 2024) c. 360 BCE

Q6Figure it Out (Page 265 - Final Section)

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

Solution

To Do: Identify the pattern in each sequence and find the next three terms.

Solution: a. The pattern is adding 6 to the previous term. (-40) + 6 = -34; (-34) + 6 = -28.

(-22) + 6 = -16
(-16) + 6 = -10
(-10) + 6 = -4

b. This sequence consists of two interleaved sequences.

Sequence 1 (odd positions): 3, 2, 1, 0, ... (subtracting 1). The next term is -1.
Sequence 2 (even positions): 4, 5, 6, 7, ... (adding 1). The next terms are 8 and 9. The completed sequence is: 3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2. Wait, the solution key says -1, 8, -2. Let's re-check the pattern. Yes, odd positions are 3, 2, 1, 0, -1, -2. Even positions are 4, 5, 6, 7, 8. So the next three terms are -1, 8, -2.

c. Let's find the difference between consecutive terms:

6 - 12 = -6
1 - 6 = -5
(-3) - 1 = -4
(-6) - (-3) = -3 The difference is increasing by 1 each time. The pattern of differences is ..., -8, -7, -6, -5, -4, -3, ...
To find the terms after -6: The next differences will be -2, -1, 0.
- (-6) + (-2) = -8
- (-8) + (-1) = -9
- (-9) + (0) = -9
To find the terms before 12: The differences were -7 and -8.
- 12 - (-7) = 12 + 7 = 19
- 19 - (-8) = 19 + 8 = 27

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

Q7Figure it Out (Page 265 - Final Section)

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

Given: Integer cards: +1, +7, +18, -5, -2, -9. Target value: Close to -30.

Solution: To get a large negative number, we should subtract the large positive numbers and add the negative numbers.

Let's try combining the most negative numbers and subtracting the most positive number: (-9) + (-5) + (-2) - (+18) = -16 - 18 = -34 (This is close)

Let's try another combination: (-9) + (-2) - (+18) = -11 - 18 = -29 (This is very close, only 1 away from -30)

Let's see if we can get exactly -30. We need to find a combination of numbers that sum to -30. Let's try (-2) + (-9) - (+18) - (+1) = -11 - 18 - 1 = -30. This expression uses four of the cards and results in exactly -30.

Final Answer: An expression that gives a value of exactly -30 is (-2) + (-9) - (+18) - (+1) = -30.

Q8Figure it Out (Page 265 - Final Section)

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

To Do: Determine the sign of the result for each operation.

Solution: a. (positive) - (negative): This is equivalent to (positive) + (positive). The result is always positive. Example: 5 - (-3) = 5 + 3 = 8.

b. (positive) + (negative): The result depends on which number has a larger absolute value. It can be positive or negative (or zero). Example: 5 + (-3) = 2 (positive), 3 + (-5) = -2 (negative).

c. (negative) + (negative): The sum of two negative numbers is always negative. Example: (-5) + (-3) = -8.

d. (negative) - (negative): This is equivalent to (negative) + (positive). The result can be positive or negative (or zero). Example: (-5) - (-3) = -5 + 3 = -2 (negative), (-3) - (-5) = -3 + 5 = 2 (positive).

e. (negative) - (positive): This is equivalent to (negative) + (negative). The result is always negative. Example: (-5) - (3) = -5 + (-3) = -8.

f. (negative) + (positive): Same as case (b) and (d). The result can be positive or negative (or zero). Example: (-5) + 3 = -2 (negative), (-3) + 5 = 2 (positive).

Final 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

Q9Figure it Out (Page 265 - Final Section)

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

Solution

Given: A string of 100 tokens. The pattern shown is: Green, Green, Red, Red, Red. This pattern repeats. From the chapter, Green tokens are negative (-1) and Red tokens are positive (+1).

To Find: The total value of the string of 100 tokens.

Solution:

Identify the repeating block: The pattern is (-1), (-1), (+1), (+1), (+1).
Calculate the value of one block: The value of one block of 5 tokens is (-1) + (-1) + (+1) + (+1) + (+1) = -2 + 3 = +1.
Find the number of blocks: The total number of tokens is 100. Each block has 5 tokens. Number of blocks = 100 \div 5 = 20 blocks.
Calculate the total value: The total value is the value of one block multiplied by the number of blocks. Total Value = (+1) \times 20 = 20.

Final Answer: The value of the string is 20.

Q1Figure it Out (Page 267)

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

Solution

To Do: Explain Brahmagupta's rules using the Building of Fun or number line analogy.

Solution: Here are explanations for some of Brahmagupta's rules:

Addition Rules:

The sum of two positives is positive.
- Building: Starting on a positive floor (e.g., +2) and going up more floors (e.g., +3) will always land you on a higher positive floor (+5).
- Number Line: Starting at a positive number and moving to the right (positive direction) will always result in a number further to the right, which is a larger positive number.
The sum of two negatives is negative.
- Building: Starting on a negative floor (e.g., -1) and going down more floors (e.g., -2) will always land you on a lower negative floor (-3).
- Number Line: Starting at a negative number and moving to the left (negative direction) will always result in a number further to the left, which is a more negative number.
To add a positive and a negative number...
- Building: Starting on floor +5 and going down 3 floors (-3) lands you on +2. The movement down was smaller than the starting height, so you stay positive. Starting on +3 and going down 5 floors (-5) lands you on -2. The movement down was larger than the starting height, so you end up on a negative floor.
- Number Line: Adding a positive and a negative is like moving right and then left. The final position depends on which move was longer.

Subtraction Rules:

Subtracting a negative number is the same as adding the corresponding positive number.
- Building: The expression (+2) - (-3) asks for the movement needed to go from Floor -3 to Floor +2. To do this, you must go up 5 floors. So, (+2) - (-3) = +5. This is the same result as adding the positive: (+2) + (+3) = +5.
- Number Line: 2 - (-3) asks for the distance and direction from -3 to 2. This is a move of 5 units to the right, so +5.

Q2Figure it Out (Page 267)

Give your own examples of each rule.

Solution

To Do: Provide a numerical example for each of Brahmagupta's rules.

Solution:

Brahmagupta's Rules for Addition:

Sum of two positives is positive: 10 + 5 = 15
Sum of two negatives is negative: (-10) + (-5) = -15
Sum of a positive and a negative:
- (-10) + 5 = -5 (Sign of the greater number, -10, is negative)
- 10 + (-5) = 5 (Sign of the greater number, 10, is positive)
Sum of a number and its inverse is zero: 10 + (-10) = 0
Sum of any number and zero is the same number: (-10) + 0 = -10

Brahmagupta's Rules for Subtraction:

Smaller positive from larger positive is positive: 10 - 5 = 5
Larger positive from smaller positive is negative: 5 - 10 = -5
Subtracting a negative is adding the positive: 10 - (-5) = 10 + 5 = 15
Subtracting a number from itself gives zero: (-10) - (-10) = 0
Subtracting zero from a number gives the same number: (-10) - 0 = -10
Subtracting a number from zero gives its inverse: 0 - (-10) = 10

+	-1	2	-3	4	-5	6
-1	-2	1	-4	3	-6	5
2	1	4	-1	6	-3	8
-3	-4	-1	-6	1	-8	3
4	3	6	1	8	-1	10
-5	-6	-3	-8	-1	-10	1
6	5	8	3	10	1	12

+	-1	2	-3	4	-5	6
-1	-2	1	-4	3	-6	5
2	1	4	-1	6	-3	8
-3	-4	-1	-6	1	-8	3
4	3	6	1	8	-1	10
-5	-6	-3	-8	-1	-10	1
6	5	8	3	10	1	12

The Other Side Of ZeroClass 6 Mathematics NCERT Solutions

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

Solution

Number all the floors in the Building of Fun.

Solution

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

Solution

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?

Solution

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

Solution

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

Solution

Now, from 3, if you wish to go to -2, how far must you travel?

Solution

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

Solution

Your new bank balance is _____.

Solution

Your bank balance is now _____.

Solution

What is your bank balance now? _____ Is this possible?

Solution

What is your balance now？ _____

Solution

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

Solution

Solution

Solution

Evaluate these expressions by thinking of them as the resulting movement of combining button presses: a. (+1)+(+4)=(+1)+(+4)=(+1)+(+4)= b. (+4)+(+1)=(+4)+(+1)=(+4)+(+1)= c. (+4)+(−3)+(−2)=(+4)+(-3)+(-2)=(+4)+(−3)+(−2)= d. (−1)+(+2)+(−3)=(-1)+(+2)+(-3)=(−1)+(+2)+(−3)=

Solution

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

Solution

Connect the inverses by drawing lines.

Solution

Who is on the lowest floor? Jay is in the Art Centre. So, he is on Floor +2. Asin is in the Sports Centre. So, she is on Floor _____. Binnu is in the Cinema Centre. So, she is on Floor _____. Aman is in the Toys Store. So, he is on Floor _____.

Solution

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

Solution

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

Solution

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

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

Solution

Solution

Solution

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

Solution

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

Solution

Is 2>−32 > -32>−3? Why? Is −2<3-2 < 3−2<3? Why?

Solution

What are a. −5+0-5+0−5+0 b. 7+(−7)7+ (-7)7+(−7) c. −10+20-10+20−10+20 d. 10−2010-2010−20 e. 7−(−7)7-(-7)7−(−7) f. −8−(−10)-8-(-10)−8−(−10)?

Solution

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

Solution

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?

Solution

Solution

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

Solution

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

Solution

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

Solution

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

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

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

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

Solution

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

Solution

Can you write the points A, B, ..., G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?

Solution

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

Solution

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

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. $(-1)+(+2)+(-3)=$

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

Who is on the lowest floor? Jay is in the Art Centre. So, he is on Floor +2. Asin is in the Sports Centre. So, she is on Floor ___. Binnu is in the Cinema Centre. So, she is on Floor _. Aman is in the Toys Store. So, he is on Floor ___.

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

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

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

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

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

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

Do you know that there are some places in India where temperatures can go below $0^\circ C$ ? Find out the places in India where temperatures sometimes go below $0^\circ C$ . What is common among these places? Why does it become colder there and not in other places?

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

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

+	-1	2	-3	4	-5	6
-1	-2	1	-4	3	-6	5
2	1	4	-1	6	-3	8
-3	-4	-1	-6	1	-8	3
4	3	6	1	8	-1	10
-5	-6	-3	-8	-1	-10	1
6	5	8	3	10	1	12