NCERT Solutions for Class 6 Maths Chapter 10: The Other Side of Zero — Free PDF

Class 6 Maths Worksheet — Free PDF with Answers

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Chapter 10 Overview: The Other Side of Zero

Looking for a class 6 math worksheet or maths worksheet for class 6 with answers? This comprehensive resource covers everything you need. Chapter 10 of the NCERT Class 6 Maths textbook (2024-25) introduces students to the world of negative numbers. Until now, students have worked only with whole numbers ($0, 1, 2, 3, \ldots$). This chapter extends the number line to the left of zero, opening up an entirely new set of numbers.

The key topics covered are:
- Need for negative numbers — temperature, debt, altitude below sea level
- Integers — the set $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$
- Representing integers on the number line
- Ordering and comparing integers
- Addition of integers — same sign and different sign
- Subtraction of integers — using the additive inverse
- Real-world applications — temperature changes, bank balances, altitude

Integers are the gateway to algebra and are used extensively in science, economics, and computing.

Beyond this worksheet, SparkEd offers interactive online practice for Integers with step-by-step solutions and progress tracking — perfect for daily revision.

Exercise 10.1 — Introduction to Negative Numbers

This exercise builds understanding of why we need negative numbers and how to represent them.

Problem: Real-life situations with negative numbers

Question: Express the following using integers: (a) A temperature of $5°$ below zero. (b) A deposit of Rs. $200$. (c) A withdrawal of Rs. $150$. (d) $3$ floors below ground level.

Solution:

(a) $5°$ below zero $= -5°$C

(b) Deposit (money in) $= +200$ or simply $200$

(c) Withdrawal (money out) $= -150$

(d) $3$ floors below ground $= -3$

Answer: Negative numbers represent values below a reference point (zero). Positive numbers represent values above it.

Convention: Positive values often represent: above sea level, profit, deposit, temperature above zero. Negative values represent the opposite.

Problem: Integers on the number line

Question: Represent $-4, -1, 0, 2, 5$ on a number line. Which is the smallest? Which is the largest?

Solution:

On the number line:

Numbers increase as we move to the right and decrease as we move to the left.

From left to right: $-4, -1, 0, 2, 5$

Answer: Smallest $= -4$ (farthest left), Largest $= 5$ (farthest right).

Key rule: On the number line, every number to the right is greater than every number to the left. So $-1 > -4$ and $0 > -1$.

Problem: Comparing integers

Question: Fill in with $<$, $>$, or $=$: (a) $-3 \enspace\square\enspace 2$ (b) $-5 \enspace\square\enspace -2$ (c) $0 \enspace\square\enspace -7$ (d) $-3 \enspace\square\enspace -3$

Solution:

(a) $-3 < 2$ — any negative number is less than any positive number.

(b) $-5 < -2$ — on the number line, $-5$ is to the left of $-2$. Among negative numbers, the one with the larger absolute value is smaller.

(c) $0 > -7$ — zero is greater than any negative number.

(d) $-3 = -3$ — a number equals itself.

Answer: (a) $<$, (b) $<$, (c) $>$, (d) $=$.

Important insight: $-5 < -2$ even though $5 > 2$. For negative numbers, "more negative" means smaller.

Exercise 10.2 — Addition of Integers

This exercise teaches how to add integers, including cases with different signs.

Problem: Adding integers with the same sign

Question: Compute: (a) $(+5) + (+3)$ (b) $(-4) + (-6)$

Solution:

(a) Both positive:

Add the values and keep the positive sign.

(b) Both negative:

Add the absolute values and keep the negative sign.

Rule for same signs: Add the absolute values and give the result the common sign.

Answer: (a) $8$, (b) $-10$.

Problem: Adding integers with different signs

Question: Compute: (a) $(+7) + (-3)$ (b) $(-8) + (+5)$ (c) $(-6) + (+6)$

Solution:

(a) $7 + (-3)$: Subtract the smaller absolute value from the larger. $7 - 3 = 4$. The number with the larger absolute value ($7$) is positive, so the answer is $+4$.

(b) $(-8) + 5$: $8 - 5 = 3$. The number with the larger absolute value ($8$) is negative, so the answer is $-3$.

(c) $(-6) + 6 = 0$. A number and its opposite (additive inverse) always sum to $0$.

Rule for different signs: Subtract the smaller absolute value from the larger. Give the result the sign of the number with the larger absolute value.

Answer: (a) $4$, (b) $-3$, (c) $0$.

Problem: Addition on the number line

Question: Use a number line to compute $(-3) + 5$.

Solution:

Step 1: Start at $-3$ on the number line.

Step 2: Adding $5$ means moving $5$ steps to the right.

We land on $2$.

Answer: $(-3) + 5 = 2$.

Rules for the number line:
- Adding a positive number: move right
- Adding a negative number: move left

Exercise 10.3 — Subtraction of Integers

This exercise covers subtracting integers by converting to addition.

Problem: Subtracting a positive integer

Question: Compute: $3 - 7$.

Solution:

Now apply the rule for different signs: $|7| - |3| = 4$, and the larger absolute value ($7$) is negative.

Answer: $3 - 7 = -4$.

On the number line: start at $3$, move $7$ steps left, land on $-4$.

Problem: Subtracting a negative integer

Question: Compute: (a) $5 - (-3)$ (b) $(-4) - (-6)$

Solution:

The key rule: Subtracting a negative number is the same as adding its positive.

(a) $5 - (-3) = 5 + 3 = 8$

(b) $(-4) - (-6) = -4 + 6 = 2$

Answer: (a) $8$, (b) $2$.

Why does this work? Subtraction means "finding the difference." Moving $-3$ in the negative direction is the same as adding $3$ in the positive direction. Think of it as "two negatives make a positive."

Problem: Word problem with integer subtraction

Question: The temperature at noon was $8°$C. By midnight, it dropped by $13°$C. What is the temperature at midnight?

Solution:

The temperature dropped below zero.

Answer: The temperature at midnight is $-5°$C.

Another example: The temperature at the top of a mountain is $-12°$C and at the base is $6°$C. The difference is:

Key Concepts and Formulas

Tips for Solving Integer Problems

1. Use the number line. When in doubt, draw a number line and count steps. Right is positive, left is negative.

2. Convert all subtraction to addition. Change $a - b$ to $a + (-b)$. This reduces all problems to addition rules.

3. Remember: subtracting a negative is adding. $5 - (-3) = 5 + 3 = 8$.

4. For comparing negative numbers, think of the number line. $-2 > -5$ because $-2$ is to the right of $-5$.

5. Check your sign. After computing, ask: "Does the sign make sense?" If you start at $3$ and subtract $7$, you should go below zero.

6. Think of real-life contexts. Temperature, altitude, and bank balances make integers intuitive.

Practice on SparkEd

Integers are the foundation for algebra, coordinate geometry, and much more. SparkEd has 60 practice questions on Integers for Class 6 CBSE, with detailed step-by-step solutions for every question.

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