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Solutions

Overview

Step-by-step NCERT solutions for The Other Side of Zero (Chapter 10, NCERT Class 6 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the The Other Side of Zero textbook chapter.

Solved

What these solutions cover

All 44 questions in The Other Side of Zero are solved in the PDF. Here's what's inside, exercise by exercise:

Addition to Keep Track of Movement

  1. You start from Floor +2 and press –3 in the lift. Where will you reach? Write an expression for this movement.
  2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). a. (+1) + (+4) = _______ b. (+4) + (+1) = _______ c. (+4) + (–3) = _______ d. (–1) + (+2) = _______ e. (–1) + (+1) = _______ f. 0 + (+2) = _______ g. 0 + (–2) = _______
  3. Starting from different floors, find the movements required to reach Floor –5. For example, if I start at Floor +2, I must press –7 to reach Floor –5. The expression is (+2) + (–7) = –5. Find more such starting positions and the movements needed to reach Floor –5 and write the expressions.

Combining Button Presses is Also Addition

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

Comparing Numbers Using Floors

  1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. a. –2 [ ] +5 b. –5 [ ] +4 c. –5 [ ] –3 d. +6 [ ] –6 e. 0 [ ] –4 f. 0 [ ] +4
  2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: a. –10 [ ] –12 b. +17 [ ] –10 c. 0 [ ] –20 d. +9 [ ] –9 e. –25 [ ] –7 f. +15 [ ] –17
  3. 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.
  4. Mark the following floors of the building shown on the right. a. –7 b. –4 c. +3 d. –10

Subtraction to Find Which Button to Press

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

Adding and Subtracting Larger Numbers (Mineshaft)

  1. Complete these expressions. a. (+40) + ______ = +200 b. (+40) + ______ = –200 c. (–50) + ______ = +200 d. (–50) + ______ = –200 e. (–200) – (–40) = _______ f. (+200) – (+40) = _______ g. (–200) – (+40) = _______

Adding, Subtracting, and Comparing Any Numbers (Infinite Lift)

  1. 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)

Back to the Number Line

  1. Mark 3 positive numbers and 3 negative numbers on the number line above (from –10 to 10).
  2. Write down the above 3 marked negative numbers in the following boxes (in increasing order).
  3. Is 2 > –3? Why? Is –2 < 3? Why?
  4. What are: a. –5 + 0 b. 7 + (–7) c. –10 + 20 d. 10 – 20 e. 7 – (–7) f. –8 – (–10)?

Using the Unmarked Number Line to Add and Subtract

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

The Token Model — Using Tokens for Addition

  1. Complete the additions using tokens. a. (+6) + (+4) b. (–3) + (–2) c. (+5) + (–7) d. (–2) + (+6)
  2. 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? (a. shows 3 positive and 5 negative tokens; b. shows 6 positive and 3 negative tokens)

Using Tokens for Subtraction — Same-Sign Cases

  1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: a. (+10) – (+7) b. (–8) – (–4) c. (–9) – (–4) d. (+9) – (+12) e. (–5) – (–7) f. (–2) – (–6)
  2. Complete the subtractions: a. (–5) – (–7) b. (+10) – (+13) c. (–7) – (–9) d. (+3) – (+8) e. (–2) – (–7) f. (+3) – (+15)

Using Tokens for Subtraction — Mixed-Sign Cases

  1. Try to subtract: –3 – (+5). How many zero pairs will you have to put in? What is the result?
  2. Evaluate the following using tokens. a. (–3) – (+10) b. (+8) – (–7) c. (–5) – (+9) d. (–9) – (+10) e. (+6) – (–4) f. (–2) – (+7)

Integers in Other Places — Credits and Debits

  1. 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?
  2. 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?

Integers in Other Places — Geographical Cross Sections

  1. Looking at the geographical cross section, fill in the respective heights: a. b. c. d. e. f. g.
  2. Which is the highest point in this geographical cross section? Which is the lowest point?
  3. Can you write the points A, B, …, G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?
  4. What is the highest point above sea level on Earth? What is its height?
  5. What is the lowest point with respect to sea level on land or on the ocean floor? What is its height? (This height should be negative).

Integers in Other Places — Temperature

  1. 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. Temperatures: 14°C, 8°C, –2°C, –4°C Times: 02:00 a.m., 11:00 p.m., 02:00 p.m., 11:00 a.m.

Explorations with Integers — A Hollow Integer Grid

  1. Do the calculations for the second grid and find the border sum. The second grid is: 5 –3 –5 0 [ ] –5 –8 –2 7
  2. Complete the grids to make the required border sum: Grid 1 (given –10, –5, 9; border sum +4) Grid 2 (given 6, 8, –5, –2; border sum –2) Grid 3 (given 7, –5; border sum –4)

Explorations with Integers — An Amazing Grid of Numbers

  1. Play the same game with the grids below. What answer did you get? Grid A: 7 10 13 16 –2 1 4 7 –11 –8 –5 –2 –20 –17 –14 –11 Grid B: –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4

Explorations with Integers — Mixed Problems

  1. Write all the integers between the given pairs, in increasing order. a. 0 and –7 b. –4 and 4 c. –8 and –15 d. –30 and –23
  2. Give three numbers such that their sum is –8.
  3. There are two dice whose faces have these numbers: –1, 2, –3, 4, –5, 6. The smallest possible sum upon rolling these dice is –10 = (–5) + (–5) and the largest possible sum is 12 = (6) + (6). Some numbers between (–10) and (+12) are not possible to get by adding numbers on these two dice. Find those numbers.
  4. Solve these: 8 – 13 | (–8) – (13) | (–13) – (–8) | (–13) + (–8) 8 + (–13) | (–8) – (–13) | (13) – 8 | 13 – (–8)
  5. Find the years below. a. From the present year, which year was it 150 years ago? b. From the present year, which year was it 2200 years ago? (Recall that there was no year 0.) c. What will be the year 320 years after 680 BCE?
  6. Complete the following sequences: a. (–40), (–34), (–28), (–22), _____, _____, _____ b. 3, 4, 2, 5, 1, 6, 0, 7, _____, _____, _____ c. _____, _____, 12, 6, 1, (–3), (–6), _____, _____, _____
  7. Here are six integer cards: (+1), (+7), (+18), (–5), (–2), (–9). You can pick any of these and make an expression using addition(s) and subtraction(s). Here is an expression: (+18) + (+1) – (+7) – (–2) which gives a value (+14). Now, pick cards and make an expression such that its value is closer to (–30).
  8. The sum of two positive integers is always positive but a (positive integer) – (positive integer) can be positive or negative. What about: a. (positive) – (negative) b. (positive) + (negative) c. (negative) + (negative) d. (negative) – (negative) e. (negative) – (positive) f. (negative) + (positive)
  9. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string? (The pattern shown is: +,+,+,–,–,+,+,+,–,–, … repeating)

A Pinch of History — Brahmagupta's Rules

  1. Can you explain each of Brahmagupta's rules in terms of Bela's Building of Fun, or in terms of a number line?
  2. Give your own examples of each rule.
Sample

Worked examples

3 questions from The Other Side of Zero, solved step by step.

Addition to Keep Track of Movement

Starting from different floors, find the movements required to reach Floor –5. For example, if I start at Floor +2, I must press –7 to reach Floor –5. The expression is (+2) + (–7) = –5. Find more such starting positions and the movements needed to reach Floor –5 and write the expressions.

Use: Starting Floor + Movement = –5, so Movement = –5 – Starting Floor.

Starting at +1: (+1) + (–6) = –5

Starting at 0: (0) + (–5) = –5

Starting at –2: (–2) + (–3) = –5

Starting at –3: (–3) + (–2) = –5

(Many other starting positions are possible.)

Answer: Sample expressions: (+1)+(–6)=–5; (0)+(–5)=–5; (–2)+(–3)=–5; (–3)+(–2)=–5

Adding, Subtracting, and Comparing Any Numbers (Infinite Lift)

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)

Recall: subtracting a negative = adding the corresponding positive.

a. –125 + (–30) = –155

b. +105 – (–55) = +105 + 55 = +160

c. +105 + (+55) = +160

d. +80 – (–150) = +80 + 150 = +230

e. +80 + (+150) = +230

f. –99 – (–200) = –99 + 200 = +101

g. –99 + (+200) = +101

h. +1500 – (–1500) = +1500 + 1500 = +3000

Answer: a. –155 b. +160 c. +160 d. +230 e. +230 f. +101 g. +101 h. +3000

Back to the Number Line

Mark 3 positive numbers and 3 negative numbers on the number line above (from –10 to 10).

Any three positive integers and three negative integers can be marked.

Example: positive numbers 2, 5, 8; negative numbers –1, –3, –7.

Mark them as points on the number line at their respective positions.

Answer: Sample: positive: 2, 5, 8; negative: –1, –3, –7 (other choices are valid)

All 44 questions in The Other Side of Zero are solved step by step in the Solutions PDF above.

Keep solving

More solutions in Ganita Prakash

01Patterns in Mathematics02Lines and Angles03Number Play04Data Handling and Presentation05Prime Time06Perimeter and Area07Fractions08Playing with Constructions09Symmetry
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